the kurosh subgroup theorem for proﬁnite groups€¦ · universidade federal de minas gerais...

UNIVERSIDADE FEDERAL DE MINAS GERAISINSTITUTO DE CIÊNCIAS EXATAS

DEPARTAMENTO DE PÓS-GRADUAÇÃO EM MATEMÁTICA

The Kurosh Subgroup Theorem forprofinite groups

Mattheus Pereira da Silva Aguiar

Belo Horizonte - MG

July 2019

UNIVERSIDADE FEDERAL DE MINAS GERAISINSTITUTO DE CIÊNCIAS EXATAS

DEPARTAMENTO DE PÓS-GRADUAÇÃO EM MATEMÁTICA

Mattheus Pereira da Silva AguiarSupervisor: PhD. John William MacQuarrie

The Kurosh Subgroup Theorem for profinite groups

Thesis submitted to the examination board assigned

by the Graduate Program in Mathematics of the

Institute of Exact Sciences - ICEX of the Federal

University of Minas Gerais, as a partial requirement

to obtain a Master Degree in Mathematics.

Belo Horizonte - MG

July 2019

“First, have a definite, clear practical ideal; a goal,

an objective. Second, have the necessary means to

achieve your ends; wisdom, money, materials, and

methods. Third, adjust all your means to that end.”.

Aristotle

ABSTRACT

A profinite graph Γ is a profinite space with a graph structure, i.e., a closed set named the

vertex set and two continuous incidence maps d0, d1 ∶ Γ → V (Γ). A profinite group G can act

on a profinite graph and G/Γ is called the quotient graph by the action of G. If G acts freely,

the map ζ ∶ Γ → G/Γ is called a Galois covering of the profinite graph Γ and the group G

the associated group of this Galois covering. We can also define a universal Galois covering,

where the associated group with this covering is the fundamental group of the profinite graph Γ,

denoted by πC1 (Γ). We give an original proof of the Nielsen-Schreier Theorem for free profinite

groups over a finite space, which states that every open profinite subgroup of a free profinite

group on a finite space is a free profinite subgroup on a finite space using these aforementioned

structures. We also define a sheaf of pro-C groups, the free pro-C product of a sheaf and use it

to define a graph of pro-C groups and the fundamental group of a graph of pro-C groups in a

similar manner we did previously. The analogue of the universal Galois covering is named the

standard graph, denoted by SC(G,Γ). Finally we show that the free product of pro-C groups

can be seen as the fundamental group of a graph of pro-C groups and use this fact to prove the

Kurosh Subgroup Theorem for profinite groups, the main result of this thesis.

Keywords: Profinite graphs, Profinite groups, Galois coverings, Fundamental group, Nielsen-

Schreier, Graph of pro-C groups, Kurosh.

RESUMO EXPANDIDO

A teoria de Bass-Serre para grafos abstratos foi inaugurada no livro ’Arbres, amalgames,

SL2’ (’Trees’ na versão em inglês), escrito por Jean-Pierre Serre em colaboração com Hyman

Bass (1977). A motivação original do Serre era entender a estrutura de certos grupos algébricos

cujas construções Bruhat-Tits são árvores. No entanto, a teoria rapidamente se tornou uma

ferramenta padrão para a teoria geométrica de grupos e a topologia geométrica, particularmente

no estudo de 3-variedades. Um trabalho posterior de Bass contribuiu substancialmente para

a formalização e o desenvolvimento das ferramentas básicas e atualmente o termo teoria de

Bass-Serre é amplamente empregado para descrever a disciplina.

Essa teoria busca explorar e generalizar as propriedades de duas construções já bem

conhecidas da teoria grupos: produto livre com amalgamação e extensões HNN. No entanto,

ao contrário dos estudos algébricos tradicionais dessas duas estruturas, a teoria de Bass-Serre

utiliza a linguagem geométrica de espaços de recobrimento e grupos fundamentais.

O análogo profinito iniciou em Gildenhuys e Ribes (1978) onde o nome booleano foi

utilizado. O objetivo era construir um paralelo entre a teoria de Bass-Serre de grupos abstratos

agindo sobre árvores abstratas para grupos profinitos e aplicações em grupos abstratos. O C-

recobrimento galoisiano universal de um grafo conexo também foi definido nesse artigo de

Gildenhuys e Ribes (1978).

Haran (1987) e Mel’nikov (1989) expandiram essas ideias de maneira independente e

desenvolveram abordagens gerais para produtos livres de grupos profinitos indexados por um

espaço profinito. O objetivo deles era ser capaz de descrever a estrutura de pelo menos

certos subgrupos fechados de produtos pro-p livres de grupos pro-p, demonstrando uma versão

profinita do Teorema do Subgrupo de Kurosh, que é o teorema principal dessa dissertação. Os

artigos de Haran e Mel’nikov obtém resultados similares. Nesse texto, adotamos a elegante

versão de Mel’nikov. As primeiras seções do capítulo 4 são baseadas em Zalesskii e Mel’nikov

(1989).

O primeiro capítulo dessa dissertação estabelece esse contexto histórico do desenvolvimento

da teoria de grafos profinitos. O segundo inicia com a definição de limite inverso com algumas

de suas propriedades, o que nos possibilita definir um espaço profinito, que surge como o limite

inverso de espaços topológicos finitos, cada um com a topologia discreta.

Fornecidas tais definições básicas, definimos um grafo profinito, que é um espaço profinito

com estrutura de grafo. O espaço de vértices é um subspaço fechado (logo também profinito) e

as aplicações de incidência são aplicações contínuas (no ponto de vista topológico), do espaço

todo para o espaço dos vértices. Então definimos um q-morfismo de grafos profinitos, que é uma

generalização do morfismo usual de grafos abstratos, pois permite que arestas sejam mapeadas

em vértices. Em seguida definimos ação de grupos profinitos em garfos profinitos e o grafo

quociente pela ação de um grupo profinito G, que é o espaço das órbitas pela ação de G e como

um exemplo construímos o grafo de Cayley de um grupo profinito.

No terceiro capítulo introduzimos o conceito de recobrimento galoisiano e do C-

recobrimento galoisiano, que é uma aplicação ζ ∶ Γ→∆ = G/Γ de um grafo profinito Γ no grafo

quociente pela ação do grupo profinito G, ∆. O grupo profinito G é dito o grupo associado ao

recobrimento galoisiano. Definimos então morfismos de recobrimentos galoisianos e o conceito

de C-recobrimento galoisiano universal, cujo grupo associado é o grupo fundamental πC1 (Γ)do grafo conexo Γ. Em seguida estabelecemos condições para que 0-seções e 0-transversais

existam e a construção dos recobrimentos galoisianos universais.

Terminamos o capítulo apresentando dois exemplos importantes: 3.4.5, 3.4.6 e uma

demonstração original da versão profinita do Teorema de Nielsen-Schreier (quando o espaço

profinito é finito), cujo enunciado estabelece que todo subgrupo aberto de um grupo profinito

livre é profinito livre em um espaço finito. Esse teorema é demonstrado em [2], do artigo de

Binz, Neukirch e Wenzel [1971] e utiliza a versão abstrata. Ribes e Steinberg (2010) deram uma

nova demonstração sem utilizar a versão abstrata, através de produtos entrtelaçados, mas não

é tão simples quanto a apresentada nessa dissertação. A abordagem também é completamente

diferente.

O último capítulo contém os principais tópicos dessa dissertação: a definição de um feixe

de pro-C grupos, o produto livre de um feixe de pro-C grupos, grafos de pro-C grupos e

especializações, o grupo fundamental de um grafo de pro-C grupos, o grafo padrão de um grafo

de pro-C grupos e o Teorema de Kurosh, que estabelece:

Theorem 1 (Kurosh). Seja C uma pseudo-variedade de grupos finitos fechada para extensão.

Seja G = ∐ni=1Gi o produto pro-C livre de um número finito de grupos pro-C Gi. Se H é um

subgrupo aberto de G, então

H =n

∐i=1

∐τ∈H/G/Gi

(H ∩ gi,τGig−1i,τ) ∐ F

é um produto pro-C livre de grupos H ∩ gi,τGig−1i,τ , onde, cada i = 1,⋯, n, gi,τ varia sobre um

sistema de representantes das classes laterais duplas H/G/Gi, e F é um grupo pro-C livre de

posto finito rF ,

rF = 1 − t +n

∑i=1

(t − ti),

onde t = [G ∶H] e ti = ∣H/G/Gi∣.

A ideia da demonstração é baseada em enxergar o produto livre de um número finito de

grupos pro-C como o grupo fundamental de um grafo de pro-C grupos em que todos os grupos

de aresta são triviais e H age sobre o grafo padrão de pro-C grupos.

ACKNOWLEDGEMENT

I would first like to thank God for the life, health, faith, determination and wisdom.

I would also like to thank my mother and my sister that always had been on my side

encouraging, supporting and giving me strength to surpass all the challenges I had to face in

this journey. I love you both.

To Professor Anderson Porto that was and is a friendly encouraging, since the mathematical

Olympiads, giving me the hand and the road map to achieve uncountable many realisations.

Also to Professors Leonardo Gomes and Ana Cristina Vieira for the guidance and to the

master and undergraduate colleagues, in special to Maria Cecíllia Alecrim, Moacir Aloísio,

José Alves and Mateus Figueira for the companionship and friendship.

Finally I would like to thank my thesis advisor Professor John MacQuarrie. The door to

Prof. MacQuarrie’s office was always open whenever I had a question or a new idea about my

research or writing.

The present work was realised with the support of the Coordenação de Aperfeiçoamento de

Pessoal de Nível Superior - Brasil (CAPES) - Code of financing 001.

Contents

1 Introduction 1

2 Profinite graphs 32.1 Inverse limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Profinite graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Groups acting on profinite graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 The fundamental group of a profinite graph 213.1 Galois coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Universal Galois coverings and fundamental groups . . . . . . . . . . . . . . . . . 26

3.3 0-transversals and 0-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Existence of universal Galois coverings and the Nielsen-Schreier Theorem for

pro-C groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Graphs of pro-C groups 394.1 Free pro-C product of a sheaf of pro-C groups . . . . . . . . . . . . . . . . . . . . 39

4.2 Graphs of pro-C groups and specialisations . . . . . . . . . . . . . . . . . . . . . . 45

4.3 The fundamental group of a graph of pro-C groups . . . . . . . . . . . . . . . . . 47

4.4 The standard graph of a graph of pro-C groups . . . . . . . . . . . . . . . . . . . . 51

4.5 The Kurosh Theorem for free pro-C products . . . . . . . . . . . . . . . . . . . . . 57

Bibliography 61

Chapter 1

Introduction

Bass–Serre theory for abstract graphs was presented in the book ’Arbres, amalgames, SL2’(Trees in the english version), written by Jean-Pierre Serre in collaboration with Hyman Bass(1977). Serre’s original motivation was to understand the structure of certain algebraic groupswhose Bruhat–Tits buildings are trees. However, the theory quickly became a standard toolof geometric group theory and geometric topology, particularly the study of 3-manifolds. Asubsequent work of Bass contributed substantially to the formalization and development of thebasic tools of the theory and currently the term ’Bass–Serre theory’ is widely used to describethe subject.

This theory builds on exploiting and generalizing the properties of two older group-theoreticconstructions: free products with amalgamation and HNN-extensions. However, unlike thetraditional algebraic study of these two constructions, Bass–Serre theory uses the geometriclanguage of covering theory and fundamental groups.

The notion of a profinite graph first appeared in Gildenhuys and Ribes (1978) where thename boolean was used. The aim was to develop a parallel to the Bass-Serre theory of abstractgroups acting on abstract trees for profinite groups and applications to abstract groups. Theirwork also introduces the universal Galois C-covering of a connected profinite graph, the conceptof a ’sheaf of pro-C groups’ and the corresponding free pro-C product. Later, Zalesskii andMel’nikov (1989) developed the concept of a graph of pro-C groups, the fundamental group ofa graph of pro-C groups and the standard graph. These resources will be fundamental to provethe main theorem of this thesis.

Chapter 2 provides a review of some important results of inverse limits, profinite spaces andpro-C groups. There is also an introduction to the concept of a profinite graph, that arises as aninverse limit of finite graphs, the action of a profinite group on a profinite graph and the Cayleygraph of a profinite group.

Chapter 3 starts with the definition of a Galois covering of a profinite graph Γ and theprofinite group associated with this Galois covering. Later this group is seen as a subgroup ofAut(Γ). A profinite graph Γ also has a universal Galois covering, constructed by a universalproperty. As the profinite group G has to act freely on Γ to be a Galois covering of a graph Γ,

1

it admits a section j and a G-transversal J , by Lemma 3.3.9. We finish the chapter giving anidea of the construction of universal Galois coverings, summarized in Construction 3.4.1 andTheorem 3.4.3. We also give two important examples: 3.4.5, 3.4.6 and an original proof of theNielsen-Schreier Theorem for free profinite groups on a finite space, which states that everyopen profinite subgroup of a free profinite group on a finite space is a free profinite subgroupon a finite space. This Theorem is proved in [2], from the article of Binz, Neukirch and Wenzel[1971] and uses the abstract version. Ribes and Steinberg (2010) gave a new proof withoutusing the abstract version, through wreath products, but it is not so simple as the one presentedin this thesis. The approach is entirely different as well.

The last chapter contains the main topics of this thesis: the definition of a sheaf of pro-Cgroups, the free product of a sheaf of pro-C groups, graphs of pro-C groups and specialisations,the fundamental group of a graph of pro-C groups, the standard graph of a graph of pro-C groupsand the Kurosh theorem for profinite groups that states:

Theorem 2 (Kurosh). Let C be an extension-closed pseudovariety of finite groups. Let G =∐ni=1Gi be a free pro-C product of a finite number of pro-C groups Gi. If H is an open subgroup

of G, then

H =n

∐i=1

∐τ∈H/G/Gi


is a free pro-C product of groups H ∩ gi,τGig−1i,τ , where, for each i = 1,⋯, n, gi,τ ranges over

a system of representatives of the double cosets H/G/Gi, and F is a free pro-C group of finiterank rF ,

rF = 1 − t +n

∑i=1

(t − ti),

where t = [G ∶H] and ti = ∣H/G/Gi∣.

The idea of the proof is based on seen the free product of finitely many pro-C groups as thefundamental group of a graph of pro-C groups where all the edge groups are trivial and H actson the standard graph of pro-C groups.

The Theorem was originally proved (the abstract case) by A. G. Kurosh in 1934. Afterthat, there were many subsequent proofs, including proofs of Harold W. Kuhn (1952), SaundersMac Lane (1958) and others. The theorem was also generalized for describing subgroups ofamalgamated free products and HNN extensions. One of this important generalizations is theprofinite case.

Haran (1987) and Mel’nikov (1989) independently expanded these ideas and developedrather general approaches to free products of profinite groups indexed by a profinite space; theiraim was to be able to describe the structure of at least certain closed subgroups of free pro-pproducts of pro-p groups, proving a profinite version of the Kurosh Subgroup Theorem, also themain theorem of this thesis. The pappers of Haran and Mel’nikov obtain similar group theoreticresults. Here we have adopted the rather elegant viewpoint of Mel’nikov. We will omit theproofs of the results that are not used directly in the main result.

2

Chapter 2

Profinite graphs

In this chapter we define the concepts of inverse limit, profinite space and profinite groupthat are fundamental for the definiton of a profinite graph and the quotient graph by an actionof a profinite group. We also present the definitions of a profinite graph, some properties andconcepts, the action of a profinite group on a profinite graph and the Cayley graph of a profinitegroup. Other properties can be found on chapters 1 and 2 of [1] and chapter 1 of [2].

2.1 Inverse limits

In this section we define the concept of inverse limit and establish some of its elementaryproperties that will be crucial to the definition of a profinite graph and a profinite group. Ratherthan developing the concept and establishing those properties under the most general conditions,we restrict ourselves to inverse limits of topological spaces or topological groups. The maintopics of this section can be found in [1] and [2].

Definition 2.1.1 (Directed poset). Let I = (I,⪯) be a set with binary relation ⪯. It is called adirected partially ordered set or a directed poset if the following holds:

(a) i ⪯ i for i ∈ I;

(b) i ⪯ j and j ⪯ i imply i = j for i, j ∈ I;

(c) i ⪯ j and j ⪯ k imply i ⪯ k for i, j, k ∈ I; and

(d) if i, j ∈ I , there exists some k ∈ I such that i, j ⪯ k.

Definition 2.1.2 (Inverse system). An inverse system of topological spaces over a directed posetI consists of a collection of topological spaces {Xi} indexed by I and collection of continuousmappings ϕij ∶Xi →Xj (defined when i ⪰ j) such that the diagrams of the form

Xiϕik //

ϕij

Xk

Xj

ϕjk

>>

3

commute whenever they are defined, i.e., whenever i, j, k ∈ I and i ⪰ j ⪰ k. It is denoted by{Xi, ϕij, I}.

Definition 2.1.3 (Compatible mappings). Let Y be a topological space, {Xi, ϕij, I} an inversesystem of topological spaces over a directed poset I and ψi ∶ Y → Xi be a continuous mappingfor each i ∈ I . These mappings are said to be compatible if the following diagram commutes:

Yψi //

ψj

��

Xi

ϕij

��

Xj

or equivalently, ϕijψi = ψj , whenever i ⪰ j.

Definition 2.1.4 (Inverse limit). A topological space X together with compatible continuousmappings

ϕi ∶X →Xi (i ∈ I)

is an inverse limit of the inverse system {Xi, ϕij, I} if the following universal property issatisfied:

Yψ //

ψi

��

X

ϕi

��

Xi

whenever Y is a topological space and ψi ∶ Y →Xi is a set of compatible continuous mappings,then there is a unique continuous mapping ψ ∶ Y → X such that ϕiψ = ψi for all i ∈ I . We saythat ψ is ’induced’ or ’determined’ by the compatible homomorphisms ψi.

The maps ϕi are called projections. The projection maps ϕi are not necessarily surjections.We denote the inverse limit by (X,ϕi), or often simply by X , by abuse of notation.

Proposition 2.1.5 ([2], Proposition 1.1.1). Let {Xi, ϕij, I} be an inverse system of topologicalspaces over a directed poset I . Then

(a) There exists an inverse limit of the inverse system {Xi, ϕij, I}.

(b) This limit is unique in the following sense: if (X,ϕi) and (Y,ψi) are two inverse limits ofthe inverse system {Xi, ϕij, I}, then there is a unique homeomorphism ϕ ∶X → Y such thatψiϕ = ϕi.

4

Proof. See [2], Proposition 1.1.1, page 2.

If {Xi, I} is a collection of topological spaces indexed by a set I , its direct product orcartesian product is the topological space Πi∈IXi, endowed with the product topology. In thecase of topological groups, the group operation is defined coordinatewise. The inverse limit ofsuch a system, denoted by Γ = lim←Ði∈I

Γi, is the subset of ∏i∈I Γi consisting of those tuples (mi)with ϕij(mi) =mj , whenever i ⪰ j.

For the next proposition, we can recall that

Definition 2.1.6 (Totally disconnected space). A topological space X is said totallydisconnected if every point in the space is it own connected component.

So, if we take the inverse system of a special type of topological spaces (Haussdorf, compactand totally disconnected), their inverse limit maintain these properties. It can be seen throughthe Tychonoff Theorem for the compact part and the fact that a closed subset of a compact spaceis compact. For the Hausdorff and totally disconnected parts, these are well known results. Westate this in a proposition:

Proposition 2.1.7 ([2], Proposition 1.1.3). Let {Xi, ϕij, I} be an inverse system of compact,Hausdorff and totally disconnected topological spaces over the directed set I . Then

lim←Ði∈I

Xi

is also a compact, Haussdorf and totally disconnected space.

The next proposition gives to us conditions where this inverse limit is not empty:

Proposition 2.1.8 ([2], Proposition 1.1.4). Let {Xi, ϕij, I} be an inverse system of compact,Haussdorf nonempty topological spaces Xi over the directed set I . Then

lim←Ði∈I

Xi

is nonempty. In particular, the inverse limit of an inverse system of nonempty finite sets with thediscrete topology is nonempty.


The next definition gives a necessary condition to an inverse system on a subset I ′ of thedirected poset I have the same inverse limit of the equivalent inverse system in I .

Definition 2.1.9 (Cofinal). Let (I,⪯) be a directed poset. Assume that I ′ is a subset of I in sucha way that (I ′,⪯) becomes a directed poset. We say that I ′ is cofinal in I if for every i ∈ I thereis some i′ ∈ I ′ such that i ⪯ i′.

5

Proposition 2.1.10 ([2], Proposition 1.1.9). Let {Xi, ϕij, I} be an inverse system of compacttopological spaces (respectively, compact topological groups) over a directed poset I andassume that I ′ is a cofinal subset of I . Then

lim←Ði∈I

Xi ≅ lim←Ði′∈I′

Xi′

.

Proof. See [2], Lemma 1.1.9, page 8.

Remark 2.1.11. On all definitions and propositions above, if X is also a topological group, thecontinuous mappings ϕij ∶Xi →Xj are continuous group homomorphisms.

For further properties of inverse limits, see [2], Section 1.1.

Definition 2.1.12 (Profinite space). A profinite spaceX is a topological space that is the inverselimit lim←Ði

Xi of finite spaces Xi endowed with the discrete topology.

By Proposition 2.1.8, if each finite space Xi is nonempty, then the profinite space X =lim←Ði

Xi is also nonempty. The next theorem provides a characterization of profinite spaces:

Theorem 2.1.13 ([2], Theorem 1.1.12). Let X be a topological space. Then the followingconditions are equivalent:

(a) X is a profinite space;

(b) X is compact, Hausdorff and totally disconnected;

(c) X is compact, Hausdorff and admits a base of clopen sets for its topology.

Proof. See [2], Theorem 1.1.12, page 10.

To define a pro-C group we need the notion of a pseudovariety. In [2] it is referred as avariety of profinite groups.

Definition 2.1.14 (Pseudovariety of finite groups). A nonempty class of finite groups C is apseudovariety if it is closed under taking subgroups, quotients and finite direct products. Apseudovariety of finite groups C is said to be extension-closed if whenever

1 K G H 1

is an exact sequence of finite groups with K,H ∈ C, then G ∈ C.

6

Now we can define a pro-C group:

Definition 2.1.15 (Pro-C group). Let C be a nonempty class of finite groups. Define a pro-Cgroup G as an inverse limit

G = lim←Ði∈I

Gi

of a surjective inverse system {Gi, ϕij, I} of groups Gi in C, where each group Gi is endowedwith the discrete topology.

We think of such a pro-C group G as a topological group, whose topology is inherited fromthe product topology on Πi∈IGi. If C is the pseudovariety of finite groups, we call a pro-C groupG a profinite group.

2.2 Profinite graphs

In this section we define profinite graphs and the Cayley graph of a profinite group.

Definition 2.2.1 (Profinite graph). A profinite graph is a profinite space Γ with a distinguishedclosed nonempty subset V (Γ) called the vertex set, E(Γ) = Γ − V (Γ) the edge set and twocontinuous maps d0, d1 ∶ Γ→ V (Γ) whose restrictions to V (Γ) are the identity map idV (Γ). Werefer to d0 and d1 as the incidence maps of the graph Γ.

A profinite graph is also an oriented abstract graph without the topology. As the set ofvertices is closed, the edges set is open, but it need not to be closed. When this happens, E(Γ)is also compact (and so profinite) and we only need to check the continuity of the incidencemaps on V (Γ) and E(Γ) separately, since then V (Γ) and E(Γ) are disjoint and clopen.

Associated with each edge e of Γ we introduce the symbols e1 = e and e−1. We define theincidence maps for these symbols as follows: d0(e−1) = d1(e) and d0(e) = d1(e−1). We candefine some structures immediately from the abstract Bass-Serre theory that behave exactly inthe same way.

Definition 2.2.2 (Path). Given vertices v and w of Γ, a path pvw from v to w is a finite sequenceeε11 ,⋯, eεmm , where m ≥ 0, ei ∈ E(Γ), εi = ±1, (i = 1,⋯,m) such that d0(eε11 ) = v, d1(eεmm ) = wand d1(eεii ) = d0(eεi+1i+1 ) for i = 1,⋯,m − 1. Such a path is said to have length m.

Note that a path is always meant to be finite. The underlying graph of the path pvw consistsof the edges eε11 ,⋯, eεmm and their vertices di(ej) (i = 0,1; j = 1,⋯,m).

Definition 2.2.3. The path pvw is called reduced if whenever ei = ei+1, then εi = εi+1 for alli = 1,⋯,m − 1.

In abstract Bass-Serre theory there is an algorithm to transform any path into a reduced one,called simple reduction of a path. It also holds for this definition, because it is the immediateanalogue, so we can consider all paths as a reduced path, named the reduced form. We can alsoprove that this reduced form is unique. For more details, see [3].

7

The first immediate example of a profinite graph is the following:

Example 2.2.4 ([1], Example 2.1.1(a)). A finite abstract graph Γ with the discrete topology isa profinite graph.

Next we give a less trivial example and that will be useful to show certain unique propertiesthat arise from the profinite definition of connectivity:

Example 2.2.5 ([1], Example 2.1.1(b)). LetN = {0,1,2,⋯} andN = {n ∣ n ∈ N} be two copiesof the set of the natural numbers (each one with the discrete topology). Define Γ = N ⊍N ⊍{∞}to be the one-point compactification of the space N ⊍ N . Hence Γ is a profinite space becauseit is compact, Hausdorff and totally disconnected. We can introduce a profinite graph structureinto Γ by setting:

• V (Γ) = N ⊍∞;

• E(Γ) = N ;

• d0(n) = n for n ∈ E(Γ) and d0(n) = n for n ∈ V (Γ);

• d1(n) = n + 1 for n ∈ E(Γ) and d1(n) = n for n ∈ V (Γ).

0 1 2 3 4 ∞0 1 2 3

. . .

In this case the subset of edges E(Γ) = N = {n ∣ n ∈ N} is open, but not closed in Γ. Indeed,we are taking the discrete topology, so N is open, but it is not compact and therefore not closed.

As now we are working with topological spaces, additional conditions on a subset of aprofinite graph are needed to constitute a profinite subgraph. It has be a closed subset in orderto be profinite. We state this in the following definition:

Definition 2.2.6 (Profinite subgraph). A nonempty closed subset ∆ of a profinite graph Γ iscalled a profinite subgraph of Γ if whenever m ∈ Γ, then dj(m) ∈ Γ (j = 0,1).

The major changes from the abstract world start to appear in the next definition, the q-morphism, where we allow edges go to vertices in a map of profinite graphs. This fits very wellwith inverse limits constructions that are not needed in the abstract case.

Definition 2.2.7 (q-morphism of profinite graphs). Let Γ and ∆ be profinite graphs. A q-morphism or a quasi-morphism of profinite graphs α ∶ Γ → ∆ is a continuous map such thatdj(α(m)) = α(dj(m)), for all m ∈ Γ and j = 0,1. If in addition α(e) ∈ E(∆) for everye ∈ E(Γ), we say that α is a morphism.

8

If α is a surjective q-morphism (respectively injective, bijective), we say that α is anepimorphism (respectively, monomorphism, isomorphism). An isomorphism α ∶ Γ → Γ ofthe graph Γ to itself is called an automorphism.

Remark 2.2.8. The relation dj(α(m)) = α(dj(m)), (j = 0,1;m ∈ Γ) implies that a q-morphism of profinite graphs maps vertices to vertices. Indeed, suppose for contradiction thatgiven m ∈ V (Γ), α(m) ∈ E(∆). Thus, dj(α(m)) ∈ V (∆) because dj ∶ ∆ → V (∆) andα(dj(m)) ∈ E(∆) because dj(m) =m, and dj ∣V (Γ) = idV (Γ). Thus they cannot be equal.

Therefore, a q-morphism can send edges to vertices:

Definition 2.2.9 (Profinite quotient graph, [1], Example 2.1.2). Let Γ be a profinite graph and∆ a profinite subgraph of Γ. We can define a natural continuous map α ∶ Γ → Γ/∆, whereΓ/∆ is endowed with the quotient topology, and a profinite graph structure on the space Γ/∆inherited from Γ as follows:

• V (Γ/∆) = α(V (Γ));

• dj(α(m)) = α(dj(m)) (j = 0,1);

for all m ∈ Γ (note that we only need to define the vertex set and the incidence maps on theedges, because they are trivial on the vertices). Then α is a q-morphism of profinite graphs andwe call Γ/∆ a quotient graph of Γ. We can say that Γ/∆ is obtained from Γ by collapsing ∆ toa point. Observe that α maps any edge of Γ which is in ∆ to a vertex of Γ/∆.

If Γ is a profinite graph and ϕ ∶ Γ → Y is a continuous surjection onto a profinite space Y ,there is no assurance that there exists a profinite graph structure on Y so that ϕ is a q-morphismof graphs. The following construction provides necessary and sufficient conditions for this tohappen.

Construction 2.2.10 ([1], Construction 2.1.3). Let Γ be a profinite graph and let ϕ ∶ Γ → Y bea continuous surjection onto a profinite space Y . Then we construct a quotient q-morphism ofgraphs

ϕ ∶ Γ→ Γϕ

with the following properties:

(a) There is a continuous surjection of topological spaces ψϕ ∶ Γϕ → Y such that the diagram

Γϕ //

ϕ

��

Y

Γϕ

ψϕ

??

9

commutes.

(b) If Y admits a profinite graph structure so that ϕ is a q-morphism, then ψϕ is an isomorphismof profinite graphs.

(c) Consequently, there exists a profinite graph structure on Y such that ϕ is a q-morphism ofgraphs if and only if whenever m,m′ ∈ Γ with ϕ(m) = ϕ(m′), then ϕd0(m) = ϕd0(m′)and ϕd1(m) = ϕd1(m′). If this is the case, then that structure is unique (isomorphic to Γϕ)and the incidence maps of Y are defined by di(ϕ(m)) = ϕ(di(m)) (m ∈ Γ, i = 0,1).

(d) If E(Γ) is a closed subset of Γ and ϕ(E(Γ)) ∩ ϕ(V (Γ)) = ∅, then ϕ is a morphism ofprofinite graphs and ψϕ(E(Γϕ)) ∩ ψϕ(V (Γϕ)) = ∅.

To construct Γϕ define a mapϕ ∶ Γ→ Y × Y × Y

byϕ(m) = (ϕ(m), ϕd0(m), ϕd1(m)) .

Let Γϕ = ϕ(Γ). Then Γϕ admits a unique graph structure such that ϕ ∶ Γ → Γϕ is a q-morphism of graphs, namely one is forced to define the incidence maps d0 and d1 of Γϕ by

d0 (ϕ(m), ϕd0(m), ϕd1(m)) = (ϕd0(m), ϕd0(m), ϕd0(m))

andd1 (ϕ(m), ϕd0(m), ϕd1(m)) = (ϕd1(m), ϕd1(m), ϕd1(m))

where ϕ(V (Γ)) = V (Γϕ) ⊆ Y × Y × Y . We have to check that d0 ∶ Γϕ → V (Γϕ) andd1 ∶ Γϕ → V (Γϕ) are well defined. Indeed,

(ϕd0(m), ϕd0(m), ϕd0(m)) ∈ ϕ(V (Γ)) = V (Γϕ) ⊆ Y × Y × Y

and(ϕd1(m), ϕd1(m), ϕd1(m)) ∈ ϕ(V (Γ)) = V (Γϕ) ⊆ Y × Y × Y.

If m =m′, (m,m′ ∈ Γ) we have that

ϕ(m) = (ϕ(m), ϕd0(m), ϕd1(m))

andϕ(m′) = (ϕ(m′), ϕd0(m′), ϕd1(m′)) .

But as m = m′, d0(m) = d0(m′), d1(m) = d1(m′) and as ϕ is well defined, wehave that ϕ(m′) = ϕ(m), ϕd0(m′) = ϕd0(m), ϕd1(m′) = ϕd1(m). Therefore, d0 andd1 are well defined. The map ϕ is also a q-morphism of graphs, because ϕ(dj(m)) =(ϕ(dj(m)), ϕd0(dj(m)), ϕd1(dj(m))). As d0 and d1 are the identity on V (Γ),

10

ϕ(dj(m)) = (ϕdj(m), ϕdj(m), ϕdj(m))= dj (ϕ(m), ϕd0(m), ϕd1(m))

as desired. Next note that there exists a unique map ψϕ ∶ Γϕ → Y such that ψϕϕ = ϕ, namely,ψϕ (ϕ(m), ϕd0(m), ϕd1(m)) = ϕ(m).

If Y is a profinite graph and ϕ is a q-morphism of profinite graphs, then ψϕ is anisomorphism of graphs because in this case the map ρ ∶ Y → Γϕ given by ρϕ(m) =(ϕ(m), ϕd0(m), ϕd1(m)) is a well defined q-morphism of graphs and it is inverse to ψϕ. Thisproves properties (a) and (b). Property (c) is clear.

Property (d) says that E(Γ) is closed (and so compact) and ϕ(E(Γ)) ∩ ϕ(V (Γ)) = ∅. Bythe continuity of ϕ, ϕ(E(Γ)) is compact in Y and so closed and ϕ(E(Γ)) ∩ ϕ(V (Γ)) = ∅ isthe intersection of two clopen sets. Using this information together with the definition of ϕ,ϕ(m) = (ϕ(m), ϕd0(m), ϕd1(m)), we conclude that ϕ is a morphism of profinite graphs andas ψϕ is an isomorphism, ψϕ(E(Γϕ)) ∩ ψϕ(V (Γϕ)) = ∅, proving (d).

Definition 2.2.11 (Inverse system of profinite graphs). Let (I,⪰) be a directed poset. An inversesystem of profinite graphs {Γi, ϕij, I} over the directed poset I consists of a collection ofprofinite graphs Γi indexed by I and q-morphisms of profinite graphs ϕij ∶ Γi → Γj , wheneveri ⪰ j, in such a way that ϕii = Idi for all i ∈ I and ϕjkϕij = ϕik, whenever i ⪰ j ⪰ k.

The inverse limit of such a system Γ = lim←Ði∈IΓi is the subset of ∏i∈I Γi consisting of those

tuples (mi) with ϕij(mi) = mj , whenever i ⪰ j. Therefore, we can define a natural profinitegraph structure in the inverse limit by defining V (Γ) = lim←Ði∈I

V (Γi) and the incidence maps tobe the composition with the projection maps.

Let Γ be a profinite graph and consider the set R of all open equivalence relations R onthe set Γ (i.e., the equivalence classes xR are open for all x ∈ Γ). For R ∈ R, denote byϕR ∶ Γ → Γ/R the corresponding quotient map as topological spaces. We can define a partialordering ⪰ on R as follows: for R1,R2 ∈ R, we say that R1 ⪰ R2 if there exists a mapϕR1,R2 ∶ Γ/R1 → Γ/R2 such that the diagram:

Γ

Γ/R1

Γ/R2

ϕR1

ϕR2

ϕR1,R2

commutes. Then, givenR1,R2 ∈R there existsR3 ∈R such that ϕR3 ∶ Γ→ Γ/R3 is the quotientmap and R3 ⪰ R1,R2. Hence (R,⪯) is in fact a directed poset and {Γ/R,ϕR1,R2 ,R} is an

11

inverse system overR, and, as topological spaces, the collection of quotient maps {ϕR ∣ R ∈R}induces a homeomorphism from Γ to lim←ÐR∈R

Γ/R; Therefore, we can identify these two spacesby means of this homeomorphism and write

Γ = lim←ÐR∈R

Γ/R. (2.1)

Consider now the subset R′ of R consisting of those R ∈ R such that Γ/R admits a uniquegraph structure by Construction 2.2.10 so that ϕR ∶ Γ → Γ/R is a q-morphism of profinitegraphs. We check next that the poset (R′,⪯) is directed. Indeed, let R1,R2 ∈ R′. Since R isdirected, there exists an R ∈ R such that R ⪰ R1,R2. Let ϕR ∶ Γ → Γ/R be the correspondingquotient map. Let Γϕ

Rand ϕR ∶ Γ→ Γ/R be as in Construction 2.2.10.

ΓϕR //

ϕR

��

Γ/R

ΓϕR= Γ/R

ψϕ

;;(2.2)

Then ΓϕR= Γ/R, where R is the equivalence relation on Γ whose equivalence classes are

{ϕ−1R (x) ∣ x ∈ Γϕ

R}. Clearly R ∈R′ and R ⪰ R; hence R ⪰ R1,R2, as needed.

Remark 2.2.12. If R1,R2 ∈R′ and R1 ⪰ R2, then the map ϕR1,R2 ∶ Γ/R1 → Γ/R2 is in fact a q-morphism of finite graphs. Therefore the collection {Γ/R,ϕR1,R2} of all finite quotient graphsof Γ is an inverse system of finite graphs and q-morphisms over the directed poset R′.

We summarise this construction in the following proposition:

Proposition 2.2.13 ([1], Proposition 2.1.4). Let Γ be a profinite graph.

(a) Γ is the inverse limit of all its finite quotient graphs:

Γ = lim←ÐR∈R′

Γ/R.

ConsequentlyV (Γ) = lim←Ð

R∈R′

V (Γ/R).

(b) If the subset E(Γ) of edges of Γ is closed, then a directed subposetR” ofR′ can be chosenso that whenever R1,R2 ∈ R” with R1 ⪰ R2, then ϕR1,R2 ∶ Γ/R1 → Γ/R2 is a morphism ofgraphs and

Γ = lim←ÐR∈R”

Γ/R.

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Consequently,

V (Γ) = lim←ÐR∈R”V (Γ/R) and E(Γ) = lim←ÐR∈R”

E(Γ/R).


The next proposition gives an important tool to work with inverse limits:

Proposition 2.2.14 ([1], Proposition 2.1.5). Let {Γi, ϕij, I} be an inverse system of profinitegraphs and q-morphisms over a directed poset I , and set

Γ = lim←Ði∈I

Γi. (2.3)

Let ρ ∶ Γ → ∆ be a q-morphism into a finite graph ∆. Then there exists a k ∈ I such thatρ factors through Γk, i.e., there exists a q-morphism ρ′ ∶ Γk → ∆ such that ρ = ρ′ϕk, whereϕk ∶ Γ→ Γk is the projection.


Another very different definition is the one of connectivity. As we will show, there existsprofinite graphs that are connected, but have vertices with no edge beginning or ending at it.

Definition 2.2.15 (Connected profinite graph). A profinite graph Γ is said to be connected ifwhenever ϕ ∶ Γ→ A is a q-morphism of profinite graphs onto a finite graph, thenA is connectedas an abstract graph.

We can now summarise some of the properties related to connected profinite graphs.

Proposition 2.2.16 ([1], Proposition 2.1.6). (a) Every quotient graph of a connected profinitegraph is connected.

(b) IfΓ = lim←Ð

i∈I

Γi

and each Γi is a connected profinite graph, then Γ is a connected profinite graph.

(c) Let Γ be a connected profinite graph. If ∣Γ∣ > 1, then Γ has at least one edge. Furthermore,if the set of edges E(Γ) of Γ is closed in Γ, then for any vertex v ∈ V (Γ), there exists andedge e ∈ E(Γ) such that either v = d0(e) or v = d1(e).

(d) Let Γ be a profinite graph, and let ∆ be a connected profinite subgraph of Γ. Consider thequotient graph Γ/∆ obtained by collapsing ∆ to a point and let α ∶ Γ→ Γ/∆ be the naturalprojection. Then the inverse image Λ = α−1(Λ) in Γ of a connected profinite subgraph Λ ofΓ/∆ is a connected profinite subgraph.


13

Proposition 2.2.17 ([1], Proposition 2.1.7). (a) Let D be an abstract subgraph of a profinitegraph Γ. Then the topological closure D of D in Γ is a profinite graph. If D is connectedas an abstract graph, then D is a connected profinite graph.

(b) Let {∆j ∣ j ∈ J} be a collection of connected profinite subgraphs of a profinite graph Γ. If

⋂j∈J ∆j ≠ ∅, then ∆ = ⋃j∈J ∆j is connected.


The following example shows a connected profinite graph which is not connected as anabstract graph and has a vertex with no edge beginning or ending at it.

Example 2.2.18 ([1], Example 2.1.8). Consider Γ the same graph of Example 2.2.5, Γ =N ⊍ N ⊍ {∞}, the one point compactification of a disjoint union of two copies N andN = {n ∣ n ∈ N} of the natural numbers;

• V (Γ) = N ⊍ {∞};

• E(Γ) = N ;

• d0(n) = n;

• d1(n) = n + 1

for n ∈ E(Γ). We can represent Γ as follows:

0 1 2 3 4 ∞0 1 2 3

. . .

Then Γ is a connected profinite graph; to see this consider the connected finite graphs Gn

0 1 2 3 n − 1 n

0 1 2

. . .

n − 1

with vertices V (Γn) = {0,1,2,3,⋯, n} and edges E(Γn) = {0, 1, 2,⋯, n − 1} such that d0(i) =i, d1(i) = i + 1 (i = 0,1,⋯, n − 1) and dj(i) = i for i ∈ V (Γ), (i = 0,1,⋯, n; j = 0,1). If n ≤m,define ϕm,n ∶ Γm → Γn to be the map of graphs that sends the segment [0, n] identically to[0, n], and the segment [n,m] to the vertex n. Then {Γn, ϕm,n} is an inverse system of finitegraphs, and

Γ = lim←Ðn∈N

Γn,

where ∞ = (n)n∈N (note here the importance of the new definition of q-morphism. Here areallowed to send the edge [n,m] to the vertex n, what would not be possible with the abstractdefinition of a morphism). Hence Γ is a connected profinite graph, because any morphism ofG to a finite graph can be factored through a finite connected graph (cf. Proposition 2.2.14), sothe finite graph has to be connected. We observe that there is no edge e of Γ which has ∞ asone of its vertices; and so Γ is not connected as an abstract graph.

14

The next proposition is another characterisation of connected profinite graphs and it is easilyproved using the quotient profinite graph. We can collapse Γ1 to a point and Γ2 to another pointin order to get a disconnected finite quotient graph with two vertices and no edges. Therefore,it cannot be connected.

Proposition 2.2.19 ([1], Proposition 2.1.9). Let Γ = Γ1 ⊍ Γ2 be a profinite graph which is thedisjoint union of two open profinite subgraphs Γ1 and Γ2; then Γ is not connected. In particular,a profinite graph that contains two different vertices and no edges is not connected.

The following is a natural definition motivated by the topology and the abstract graph theory:

Definition 2.2.20 (Connected profinite component). A maximal connected profinite subgraphof a profinite graph Γ is called a connected profinite component of Γ.

The next proposition characterises these connected profinite components:

Proposition 2.2.21 ([1], Proposition 2.1.10). Let Γ be a profinite graph

(a) Let m ∈ Γ. Then there exists a unique connected profinite component of Γ containing m,which we shall denote by Γ∗(m).

(b) Any two connected profinite components of Γ are either equal or disjoint.

(c) Γ is the union of its connected profinite components.


We finish this section with the definition of a Cayley graph of a profinite group, that will bevery useful through this thesis:

Definition 2.2.22 (Cayley graph). Let G be a profinite group and X a closed subset of G. PutX = X ∪ {1}. We can define the Cayley graph Γ(G,X) of the profinite group G with respectto the subset X as follows:

• Γ(G,X) = G × X

• V (Γ(G,X)) = {(g,1) ∣ g ∈ G}

• dj ∶ Γ(G,X) = G × X → V (Γ(G,X))

– d0(g, x) = g

– d1(g, x) = gx

whereG×X has the product topology and j = 0,1. Note thatG×X is a profinite space, becauseG is a profinite group (and in particular a profinite space), X is a closed subspace of a profinitespace, and so it compact and therefore profinite. Hence, G × X with the product topology isalso profinite.

15

We can identify the space V (Γ(G,X)) with G through the homeomorphism (g,1) ↦ g

(g ∈ G). The incidence maps d0 and d1 are continuous (one is just a projection and the other isa left multiplication) and they are the identity map when restricted to V (Γ(G,X)) = {(g,1) ∣g ∈ G} = G. Therefore Γ(G,X) = G × X is a profinite graph.

Note that the space of edges is E(Γ(G,X)) = Γ(G,X) − V (Γ(G,X)) = G × (X − {1}):

g gx(g, x)

where x ∈ X − {1}. It is already open, because V (Γ(G,X)) is closed in Γ(G,X). It is alsoclosed if and only if 1 is an isolated point of X . Indeed, note that if 1 ∉X , then V (Γ(G,X)) = Gand E(Γ(G,X)) = G ×X , and in this case E(Γ(G,X)) is clopen. On the other hand, if 1 ∈ Xwe have that X = X . If in addition 1 is an isolated point of X (for example, if X is finite),then X − {1} is also a closed subspace and we have Γ(G,X) = Γ(G,X − {1}). Note that theCayley graph Γ(G,X) does not contain loops since the elements of the form (g,1) are verticesby definition.

Let ϕ ∶ G → H be a continuous homomorphism of profinite groups and let X be a closedsubset of G, and so compact. Put Y = ϕ(X) that is a compact subset of H by the continuity ofϕ and therefore closed. So we can define the Cayley graphs of G with respect to the subset X ,denoted by Γ(G,X) and of H with respect to the subset ϕ(X), denoted by Γ(H,ϕ(X)). Thenϕ induces a q-morphism of the corresponding Cayley graphs

ϕ ∶ Γ(G,X)→ Γ(H,Y ).

In particular, if U is an open normal subgroup of G and XU = ϕU(X), where ϕU ∶ G → G/Uis the canonical epimorphism, then ϕU induces a corresponding epimorphism of Cayley graphsϕU ∶ Γ(G,X) → Γ(G/U,XU). So we can construct with the morphisms ϕU ∶ Γ(G,X) →Γ(G/U,XU) and ϕUij

∶ Γ(G/U1,XU1)→ Γ(G/U2,XU2) an inverse system such that

Γ(G,X) = lim←ÐU◁oG

Γ(G/U,XU)

is a decomposition of Γ(G,X) as an inverse limit of finite Cayley graphs.

2.3 Groups acting on profinite graphs

In this section we define an action of a profinite group on a profinite graph and the quotientgraph by an action of a profinite group. These concepts will be useful to define Galois coveringsin the next chapter.

Definition 2.3.1. Let G be a profinite group and let Γ be a profinite graph. We say that theprofinite group G acts on the profinite graph Γ on the left, or that Γ is a G-graph, if

16

(i) G acts continuously on the topological space Γ on the left, i.e., there is a continuous mapG × Γ→ Γ, denoted by (g,m)↦ gm, g ∈ G, m ∈ Γ, such that

(gh)m = g(hm) and 1m =m,

for all g, h ∈ G, m ∈ Γ, where 1 is the identity element of G; and

(ii) dj(gm) = gdj(m), for all g ∈ G, m ∈ Γ, j = 0,1.

Next we define an important topology that will be used on the space of automorphisms ofΓ, denoted Aut(Γ), the compact-open topology.

Definition 2.3.2. The compact-open topology on Aut(Γ) is generated by a sub-base of opensets of the form

B(K,U) = {f ∈ Aut(Γ) ∣ f(K) ⊆ U},

where K ranges over the compact subsets of Γ and U ranges over the open subsets of Γ.

So, if a profinite group G acts on a profinite graph Γ, for a fixed g ∈ G, we can define a mapρg ∶ Γ → Γ given by m ↦ gm (m ∈ Γ). This map is an automorphism of the graph Γ and, by(cf. [2], Remark 5.6.1), G acts on a profinite graph Γ if and only if there exists a continuoushomomorphism

ρ ∶ G→ Aut(Γ)

where Aut(Γ) is the group of automorphisms of Γ as a profinite graph, and where the topologyon Aut(Γ) is induced by the compact-open topology. The kernel of this action is the kernel ofρ, i.e., the closed normal subgroup of G consisting of all elements g ∈ G such that gm =m, forall m ∈ Γ.

We can define actions on the right in a similar manner, but we will only consider left actionsin this thesis.

Definition 2.3.3 (G-map of graphs). Let G be a profinite group that acts continuously on twoprofinite graphs Γ and Γ′. A q-morphism of graphs ϕ ∶ Γ→ Γ′ is called a G-map of graphs if

ϕ(gm) = gϕ(m)

for all m ∈ Γ, g ∈ G.

Definition 2.3.4 (Stabiliser). Assume that a profinite group acts on a profinite graph Γ and letm ∈ Γ. Define Gm = {g ∈ G ∣ gm =m} to be the stabiliser, or G-stabiliser of the element m.

It follows from the continuity of the action and the compactness of G that Gm is a closedsubgroup of G. We have that

Gm ≤ Gdj(m)

for everym ∈ Γ, j = 0,1, becauseGdj(m) = {g ∈ G ∣ gdj(m) = dj(m)} and as gdj(m) = dj(gm),Gdj(m) = {g ∈ G ∣ dj(gm) = dj(m)}.

17

Definition 2.3.5 (Free action). If the stabiliserGm of every elementm ∈ Γ is trivial, i.e.,Gm = 1,we say that G acts freely on Γ.

Definition 2.3.6 (G-orbit). If m ∈ Γ, the G-orbit of m is the closed subset Gm = {gm ∣ g ∈ G}.

The next definition is very important for the Galois coverings, that will be presented in thenext chapter:

Definition 2.3.7 (Quotient graph under the action of a profinite group). Let G be a profinitegroup that acts on a profinite graph Γ. In particular, G acts on V (Γ) (also a profinite space,because it is a closed subset of Γ) and E(Γ). The space

G/Γ = {Gm ∣m ∈ Γ}

ofG-orbits with the quotient topology is a profinite space which admits a natural graph structureas follows:

• V (G/Γ) = G/V (Γ);

• dj(Gm) = Gdj(m) (j = 0,1).

The profinite graph G/Γ is called the quotient graph of Γ under the action of G. Thecorresponding quotient map

ϕ ∶ Γ→ G/Γ

is an epimorphism of profinite graphs given by m↦ Gm (m ∈ Γ, g ∈ G).

Remark 2.3.8. The map ϕ sends edges to edges (it is a morphism). Indeed, given e ∈ E(Γ) theelement ϕ(e) = Ge belongs to G/E(Γ) = G/(Γ − V (Γ)) = G/Γ −G/V (Γ) = G/Γ − V (G/Γ) =E(G/Γ).

Remark 2.3.9. If N ◁c G, there is an induced action of G/N on N/Γ defined by

(gN)(Nm) = N(gm)

for g ∈ G, m ∈ Γ.

Proposition 2.3.10 ([1], Proposition 2.2.1). Let a profinite group G act on a profinite graph Γ.

(a) Let N be a collection of closed normal subgroups of G filtered from below (i.e., theintersection of any two groups in N contains a group in N ) and assume that

G = lim←ÐN∈N

G/N.

Then the collection of graphs {N/Γ ∣ N ∈ N} is an inverse system in a natural way and

Γ = lim←ÐN∈N

N/Γ.

18

(b) Let N ◁c G. For m ∈ Γ, denote by m′ the image of m in N/Γ. Consider the naturalaction of G/N on N/Γ defined above. Then (G/N)m′ is the image of Gm under the naturalepimorphism G → G/N . In particular, if Gm ≤ N , for all m ∈ Γ, then G/N acts freely onN/Γ.

Let Γ be a profinite graph. If {Γi, ϕij, I} is an inverse system of profinite G-graphs andG-maps over the directed poset I , then

Γ = lim←Ði∈I

Γi

is in a natural way a profinite G-graph, defined by V (Γ) = lim←Ði∈IΓi and the incidence maps

as the compositions. As each Γi is a profinite G-graph, we have that V (Γi) = V (G/Γi) =G/V (Γi). Therefore, V (Γ) = lim←Ði∈I

Γi = lim←Ði∈IG/V (Γi) = G/(lim←Ði∈I

V (Γi)) = G/V (Γ),where Γ = lim←Ði∈I

Γi.

Proposition 2.3.11 ([1], Proposition 2.2.2). Let a profinite group G act on a profinite graph Γ.

(a) Then there exists a decompositionΓ = lim←Ð

i∈I

Γi

of Γ as the inverse limit of a system of finite quotientG-graphs Γi andG-maps ϕij ∶ Γi → Γj

(i ⪰ j) over a directed poset (I,⪯).

(b) If G is finite and acts freely on Γ, then the decomposition of part (a) can be chosen so thatG acts freely on each Γi.


Example 2.3.12 (The Cayley graph as a G-graph, [1], Example 2.2.3). Let G be a profinitegroup and X be a closed subset of G and Γ(G,X) the Cayley graph of G with respect to X asdefined in Example 2.2.22. Define a left action of G on Γ(G,X) by setting

g′ ⋅ (g, x) = (g′g, x)

∀x ∈ X =X ∪{1} and g′, g ∈ G. We can see that gdi(m) = di(gm), for all g ∈ G, m ∈ Γ(G,X),(i = 0,1). Indeed, if m ∈ E(Γ(G,X)), we can write m = (g, x) such that d0(g, x) = g andd1(g, x) = gx, so

g′d0(g, x) = gg′ = d0(gg′, x)

andg′d1(g, x) = g′gx = d1(g′g, x).

The incidence maps are trivial on the vertices. Hence, G acts (continuously and freely) on theCayley graph Γ(G,X).

19

Now if N is the collection of all open normal subgroups of G, we have

Γ(G,X) = lim←ÐN∈N

Γ(G/N,XN),

where XN is the image of X in G/N . Note that G/N also acts freely on Γ(G/N,XN).

We finish this chapter with an necessary and sufficient condition for a Cayley graph of aprofinite group to be a connected profinite graph.

Proposition 2.3.13 ([1], Proposition 2.2.4). (a) Let X be a closed subset of a profinite groupG that generates the group topologically, i.e., G = ⟨X⟩. Assume that G acts on a profinitegraph Γ. Let ∆ be a connected profinite subgraph of Γ such that ∆∩x∆ ≠ ∅, for all x ∈X .Then

G∆ = ⋃g∈G

g∆

is a connected profinite subgraph of Γ.

(b) Let G be a profinite group and let X be a closed subset of G. The Cayley graph Γ(G,X) isconnected if and only if G = ⟨X⟩.


20

Chapter 3

The fundamental group of a profinitegraph

In this chapter we define Galois coverings, universal Galois coverings, the fundamentalgroup of a profinite graph, show cases when a q-morphism ϕ ∶ Γ → G/Γ of profinite graphsadmits a continuous section (and when it does not) and give an idea of the construction of theuniversal Galois coverings, with some important examples as Example 3.4.5. We finish thischapter with an original proof of the Nielsen-Schreier Theorem for free profinite groups on afinite space using everything stated so far.

3.1 Galois coverings

In this section we define a Galois covering of a profinite graph, the associated group withthis Galois covering and obtain some properties.

Definition 3.1.1 (Galois covering). Let G be a profinite group that acts freely on a profinitegraph Γ. The natural epimorphism of profinite graphs ζ ∶ Γ → ∆ = G/Γ of Γ onto the quotientgraph by the action of G, ∆ = G/Γ is called a Galois covering of the profinite graph ∆. Theassociated group G is called the group associated with the Galois covering ζ and we denote itby G = G(Γ∣∆). If Γ is finite, one says that the Galois covering ζ is finite. The Galois coveringis said to be connected if Γ is connected.

Remark 3.1.2. The Galois covering ζ is always a morphism, i.e., always send edges to edges(cf. Remark 2.3.8) and if Γ is finite then the associated group G(Γ∣∆) is also finite.

Example 3.1.3 ([1], Example 3.1.1(a)). Let ζ ∶ Γ → ∆ be a Galois covering of the profinitegraph ∆ with associated group G = G(Γ∣∆). Let K ◁c G. Then G/K acts freely on K/Γ andK/Γ→∆ is also a Galois covering of ∆, with associated group G/K (cf. Remark 2.3.9).

Example 3.1.4 ([1], Example 3.1.1(b)). Let (X,∗) be a pointed profinite space (i.e., X is aprofinite space with a distinguished point ∗). Define a profinite graph B = B(X,∗) by B = X ,

21

V (B) = {∗} and di(x) = {∗}, (x ∈ X) for i = 0,1. This graph B(X,∗) is named the bouquetof loops associated to (X,∗).

For example, if X has 7 points, B(X,∗) is the graph

∗

Let G be a profinite group, X a closed subset of G and let Γ(G,X) be the Cayley graph of Gwith respect to X . Then the natural action of G on Γ(G,X) described in Example 2.3.12 isfree. Therefore the natural epimorphism

ζ ∶ Γ(G,X)→ G/Γ(G,X)

is a Galois covering. We can see that G/Γ(G,X) is just the bouquet of loops B(X ∪ {1},1)because all the vertices go to the same orbit with the action defined in Example 2.3.12, that isg′(g, x) = (g′g, x).

Definition 3.1.5 (Morphism of Galois coverings). Let ζ1 ∶ Γ1 → ∆1, ζ2 ∶ Γ2 → ∆2 be Galoiscoverings such that G1 = G(Γ1∣∆1) and G2 = G(Γ2∣∆2). We can define a morphism of Galoiscoverings

ν ∶ ζ1 → ζ2

as a pair ν = (γ, f), where γ ∶ Γ1 → Γ2 is a q-morphism of graphs and f ∶ G1 → G2 is acontinuous homomorphism of groups such that γ(gm) = f(g)γ(m), for all g ∈ G1, m ∈ Γ1.

This morphism ν of Galois coverings induces a unique q-morphism of profinite graphsδ ∶ ∆1 →∆2 such that the diagram

Γ1γ //

ζ1

��

Γ2

ζ2

��

∆1δ // ∆2

commutes. Indeed, for all m ∈ Γ1 and g ∈ G1, gm will be mapped to the same element of ∆1,named a = ζ1(m). By definition of γ, we have that γ(gm) = f(g)γ(m), so as f(g) ∈ G2 andγ(m) ∈ Γ2 it constitutes a unique orbit of the action of G2 on Γ2. Therefore, ζ2(γ(gm)) is aunique element in ∆2, named b = ζ2(γ(gm)). Now we can define δ ∶ ∆1 → ∆2 to map δ(a) = bmaking the diagram commute.

Definition 3.1.6 (Epimorphims). The morphism ν = (γ, f) is called surjective or anepimorphism if γ and f (and hence δ) are epimorphisms.

22

With the morphisms of Galois coverings defined above we can construct an inverse system{ζi ∶ Γi →∆i, νij = (γij, fij), I} indexed by a directed poset I .

We stablish some notation as follows: the associated groups of the Galois coverings ζi,G(Γi∣∆i) will be denoted by

G(Γi∣∆i) = Gi = G(ζi)

(i ∈ I).Then we have corresponding inverse systems {Γi, γij, I} and {Gi, fij, I} of profinite graphs

and profinite groups, respectively, such that the profinite groupG = lim←ÐGi acts continuously andfreely on the profinite graph Γ = lim←ÐΓi; hence, the quotient profinite graph G/Γ is isomorphicwith the profinite graph ∆ = lim←Ð∆i.

Therefore we have established the following proposition:

Proposition 3.1.7 ([1], Proposition 3.1.2). The inverse limit lim←Ð ζi of Galois coverings

ζi ∶ Γi →∆i

with associated group Gi = G(ζi) is a Galois covering with associated group lim←ÐGi.

A useful property for profinite structures is the decomposition as an inverse limit. It doesnot always hold, but the next proposition shows that it is true for Galois coverings.

Proposition 3.1.8 ([1], Proposition 3.1.3). Any Galois covering ζ ∶ Γ → ∆ of profinite graphscan be decomposed as an inverse limit of finite Galois coverings, with surjective projections.


The following proposition gives an equivalent way of viewing morphisms of Galoiscoverings.

Proposition 3.1.9 ([1], Proposition 3.1.4). Let ζ1 ∶ Γ1 → ∆1, ζ2 ∶ Γ2 → ∆2 be connectedGalois coverings with associated groups G1 = G(Γ1∣∆1) and G2 = G(Γ2∣∆2), respectively. Letγ ∶ Γ1 → Γ2 and δ ∶ ∆1 →∆2 be morphisms of graphs such that the diagram

Γ1γ //

ζ1

��

Γ2

ζ2

��

∆1δ // ∆2

commutes. Then there exists a unique continuous homomorphism

f ∶ G1 → G2

which is compatible with γ. Explicitly, f is defined as follows: given g ∈ G1, choose any m ∈ Γ1;then f(g) is the unique element of G2 such that γ(gm) = f(g)γ(m).

23


The next propositions will be useful for further references.

Proposition 3.1.10 ([1], Proposition 3.1.5). Let ζ ∶ Γ → ∆, ξ ∶ Σ → ∆ be connected Galoiscoverings and let γ ∶ Γ→ Σ be a q-morphism such that ζ = ξγ.

Γγ //

ζ ��

Σ

ξ��

∆

Let f ∶ G(Γ∣∆)→ G(Σ∣∆) be the homomorphism constructed in Proposition 3.1.9. Then

(a) The maps γ and f are surjective;

(b) The map γ is a Galois covering with G(Γ∣Σ) =Ker(f), and consequently

G(Σ∣∆) ≅ G(Γ∣∆)/G(Γ∣Σ)


Corollary 3.1.11 ([1], Corollary 3.1.6). Let

Γ1γ //

ζ1

��

Γ2

ζ2

��

∆1δ // ∆2

be a commutative diagram of profinite graphs and q-morphisms such that ζ1 and ζ2 areconnected Galois coverings. Then γ is surjective if and only if δ is surjective.

Proof. See [1], Corollary 3.1.6, page 71.

Proposition 3.1.12 ([1], Proposition 3.1.7). Let ζ ∶ Γ → ∆ be a connected Galois covering andlet Σ be a connected profinite graph. If β1, β2 ∶ Σ → Γ are morphisms of profinite graphs withζβ1 = ζβ2, and β1(m) = β2(m), for some m ∈ Σ, then β1 = β2.


We can show that it is possible to see G(Γ∣∆) as a closed subgroup of Aut(Γ), whereAut(Γ) is endowed with the compact-open topology.

Let ζ ∶ Γ→∆ be a Galois covering of a profinite graph ∆, andG = G(Γ∣∆) be the associatedprofinite group. If we fix an element g of G, it determines a continuous automorphism

νg ∶ Γ→ Γ

24

given by νg(m) = gm (g ∈ G,m ∈ Γ), as stated before. Hence, the map

ν ∶ G→ Aut(G)

which sends g to νg is a homomorphism, and it is injective because G acts freely on Γ.Therefore, the space Aut(Γ) can be endowed with the compact-open topology, in such a waythat G(Γ∣∆) can be seen as a closed subgroup of Aut(Γ), as the next proposition shows:

Proposition 3.1.13 ([1], Proposition 3.2.1). Let ζ ∶ Γ → ∆ be a Galois covering of a profinitegraph ∆, and let G = G(Γ∣∆) be the associated profinite group. Consider the group Aut(Γ)of automorphisms of the profinite graph endowed with the compact- open topology. ThenG is naturally embedded in Aut(Γ) as a topological group, i.e., there exists a topologicalisomorphism of G with a closed subgroup of Aut(Γ).


Let ζ ∶ Γ → ∆ be a connected Galois covering of a profinite graph ∆, and put G = G(Γ∣∆).By the previous proposition, we may think of G as a closed subgroup of Aut(Γ). Let

H = NAut(Γ)(G)

be the normalizer ofG inAut(Γ). ThenH is a closed subgroup ofAut(Γ) becauseG is alreadyseen as a closed subgroup of Aut(Γ) and if f ∈ H, then f induces a map

Φ(f) ∶ ∆ = G/Γ→∆ = G/Γ

defined byΦ(f)(Gm) = Gf(m), (m ∈ Γ).

The next proposition summarises some properties of the induced maps Φ(f) and Φ:

Proposition 3.1.14 ([1], Proposition 3.2.2). (a) If f ∈ H, then Φ(f) is a continuousautomorphism of the profinite graph ∆, i.e., Φ(f) ∈ Aut(∆).

(b) The mapΦ ∶ H → Aut(∆)

is a continuous homomorphism.

(c) Ker(Φ) = G.

(d) If ∆ is finite, then the homomorphism Φ ∶ H → Aut(∆) is open and the groupH = NAut(Γ)(G) is profinite.


25

3.2 Universal Galois coverings and fundamental groups

In this section we define a universal Galois covering and its associated group, thefundamental group of a profinite graph.

Definition 3.2.1 (Galois C-covering). Let C be a pseudovariety of finite groups. A Galoiscovering ζ ∶ Γ → ∆ is said to be a Galois C-covering if its associated group G(Γ∣∆) is apro-C group.

Remark 3.2.2. Since a Galois C-covering is a Galois covering, all the results of the previoussections are valid for Galois C-coverings.

Definition 3.2.3 (Universal Galois C-covering). A universal Galois C covering is a connectedGalois C-covering that respects the following universal property: given any q-morphism β ∶ Γ→∆ to a connected profinite graph ∆, any connected Galois C-covering ξ ∶ Σ→∆, and any pointsm ∈ Γ, s ∈ Σ such that βζ(m) = ξ(s), there exists a q-morphism of profinite graphs α ∶ Γ → Σ,such that βζ = ξα and α(m) = s

Γα //

ζ

��

Σ

ξ

��

Γβ // ∆

We say that α lifts β, or that α is a lifting (q-morphism) of β.

Remark 3.2.4. Once m ∈ Γ and s ∈ Σ with βζ(m) = ξ(s) are given, the lifting q-morphism α

is unique, by Proposition 3.1.12. Indeed, suppose the lifting is not unique. We have then α1 andα2 such that

Γ

α1

$$

α2

::

ζ

��

Σ

ξ

��

Γβ // ∆

and βζ = ξα1,βζ = ξα2. Hence ξα1 = ξα2 and we have, as stated in the lemma, a connectedGalois covering ξ, a connected profinite graph Γ and α1, α2 ∶ Γ → Σ q-morphisms of profinitegraphs with ξα1 = ξα2 and α1(m) = α2(m) (commutativity of the diagram), for some m ∈ Γ,hence α1 = α2 as desired.

Note also that if the map β is surjective, so is α by Corollary 3.1.11. Furthermore, it followsfrom Proposition 3.1.8 that it is sufficient to check the universal property above for finite GaloisC-coverings ξ ∶ Σ→∆.

26

Proposition 3.2.5 (Uniqueness of universal Galois C-coverings, [1], Proposition 3.3.1). Letζ ∶ Γ→ Γ be a universal Galois C-covering of a profinite connected graph Γ.

(a) Assume that α ∶ Γ → Γ is a morphism of profinite graphs such that ζα = ζ . Then α is anautomorphism.

(b) A universal Galois C-covering is unique, if it exists. More precisely, if ζ ′ ∶ Γ′ → Γ is anotheruniversal Galois C-covering of Γ, then there exists an isomorphism α ∶ Γ → Γ′ of profinitegraphs such that ζ ′α = ζ .


Definition 3.2.6 (Fundamental pro-C group of a connected profinite graph). Let ζ ∶ Γ→ Γ be theuniversal Galois covering of the connected profinite graph Γ. The pro-C group πC1 (Γ) = G(Γ∣Γ)is called the fundamental pro-C group of Γ.

Remark 3.2.7. The fundamental pro-C group of Γ is well-defined up to isomorphism. Indeed,suppose for absurd that there exists another fundamental group πC1 (Γ) of Γ, which implies

Γγ //

ζ ��

Γ′

ξ��

Γ

such that G(G∣Γ) = πC1 (Γ) and G(G′∣Γ) = πC1 (Γ). By Proposition 3.1.10, γ is a Galoiscovering (isomorphism by Proposition 3.2.5(b)) with G(Γ∣Γ′) = Ker(f) (the map f beingf ∶ πC1 (Γ)→ πC1 (Γ)) and

{1} = G(Γ∣Γ′) ≅ πC1 (Γ)/πC1 (Γ).

Therefore, πC1 (Γ) ≅ πC1 (Γ).

Definition 3.2.8 (C-simply connected profinite graph). We say that a connected profinite graphΓ is C-simply connected if πC1 (Γ) = 1.

Next we show that universal Galois C-coverings commutes with inverse limits:

Proposition 3.2.9 ([1], Proposition 3.3.2). Let {Γi, ϕij, I} be an inverse system of profiniteconnected graphs Γi over a poset (I,⪯), and let

Γ = lim←Ði∈I

Γi.

For each i ∈ I , let ζi ∶ Γi → Γi be a universal Galois C-covering of Γi. Then

(a) The Galois C-coverings ζi form an inverse system over I and

ζ = lim←Ði∈I

ζi

27

is a universal Galois C-covering ζ ∶ Γ→ Γ of Γ.

(b) The fundamental groups πC1 (Γi) form an inverse system over I and

πC1 = lim←Ði∈I

πC1 (Γi).


The last proposition of this section characterises finite trees and their inverse limits as C-simply connected profinite graphs.

Proposition 3.2.10 ([1], Proposition 3.3.3). (a) Let T be a finite tree. Then T is C-simplyconnected for every pseudovariety of finite groups C.

(b) Let Γ be a profinite graph which is an inverse limit of finite trees. Then for everypseudovariety of finite groups C, Γ is a C-simply connected profinite graph.


3.3 0-transversals and 0-sections

In this section we show that a map ϕ ∶ X → G/X , where G is a profinite group and X is aprofinite G-space does not always have a continuous section. However, if G acts freely on X ,e.g., ϕ is a Galois covering, then Lemma 3.3.9 states that this section always exist. This will bevital for the proof of the Kurosh Subgroup Theorem.

Definition 3.3.1 (Spanning profinite subgraph). Let Γ be a connected profinite graph. Aspanning profinite subgraph of Γ is a profinite subgraph T of Γ with V (T ) = V (Γ).

Another very interesting example is the following: it is well known that every connectedabstract graph has a spanning subtree, but it is not always true for profinite graphs. Indeed, thenext example shows a connected profinite graph with no spanning C-simply connected profinitesubgraph:

Example 3.3.2 ([1], Example 3.4.1). Let N = {0,1,2,⋯} be the set of natural numbers withthe discrete topology and let N = N ⊍ {∞} be the one-point compactification of N . Define aprofinite graph Γ = N × {0,1} with space of vertices and edges

• V (Γ) = {i = (i,0) ∣ i ∈ N};

• E(Γ) = {i = (i,1) ∣ i ∈ N};

• d0(i) = i for n ∈ E(Γ) and d0(i) = i for i ∈ V (Γ);

• d1(i) = n + 1 for i ∈ E(Γ) and d1(i) = i for i ∈ V (Γ).

where ∞+ 1 =∞.

28

So we can represent Γ as follows:

0 1 2 3 4 ∞0 1 2 3

. . .

∞

Observe that V (Γ) and E(Γ) are disjoint and they are both profinite spaces, because theyare both clopen. Note that Γ is the inverse limit of the following finite connected graphs Γ(n)(n ≥ 0)

0 1 2 3 n − 1 ∞0 1 2

. . .

n − 1 ∞

where the canonical map Γ(n + 1) → Γ(n) sends i to i identically, if i ≤ n − 1, and it sends nand ∞ to ∞. Hence Γ is a connected profinite graph. We claim that any connected profinitesubgraph Γ′ of Γ coincides with Γ; First note that the profinite graph ∆

• V (∆) = {i = (i,0) ∣ i ∈ N};

• E(∆) = {i = (i,1) ∣ i ∈ N};

• d0(i) = i for n ∈ E(∆) and d0(i) = i for i ∈ V (∆);

• d1(i) = n + 1 for i ∈ E(∆) and d1(i) = i for i ∈ V (∆).

0 1 2 3 4 ∞0 1 2 3

. . .

is not a spanning profinite subgraph of Γ, because it is not a closed subset of Γ. Indeed,suppose for absurd that ∆ is closed in Γ. Since V (∆) = V (Γ) and it is a closed subset ofΓ = V (Γ)⊍E(Γ), we only have to verify that E(∆) is closed in E(Γ). But E(Γ) is closed in Γ

(because it is compact), so it is profinite. Thus, E(∆) must be compact, an absurd. Therefore,∆ is not a profinite subgraph of Γ.

To prove the claim, since Γ′ is connected and contains all the vertices of Γ, it must containall the edges of the form i (i = 0,1,⋯); therefore since Γ′ is compact, it also contains ∞;this proves the claim. On the other hand, if C is a pseudovariety of finite groups, we see thatπC1 (Γ) ≅ Z

C(cf. Example 3.4.6). Hence Γ does not contain any spanning C-simply connected

profinite subgraph.

Definition 3.3.3 (Lifting). Let G be a profinite group that acts on a connected profinite graphΓ, and let ϕ ∶ Γ → ∆ = G/Γ be the canonical quotient map. Let Λ be a profinite subgraphof ∆; if there is a profinite subgraph Λ′ of Γ such that ϕ∣Λ′ is a monomorphim of graphs withϕ∣Λ′(Λ′) = Λ, we say that Λ′ is a lifting of Λ.

29

Definition 3.3.4 (G-transversal). Let G be a profinite group that acts on a connected profinitegraph Γ, and let ϕ ∶ Γ → ∆ = G/Γ be the canonical quotient map. A G-transversal or atransversal of ϕ is a closed subset J of Γ such that ϕ∣J ∶ J →∆ is a homeomorphism. Associatedwith this transversal there is a continuousG-section or section of ϕ, j ∶ ∆→ Γ, i.e., a continuousmapping such that ϕj = id∆ and j(∆) = J .

Note that, in general, J is not a graph.

Definition 3.3.5 (0-transversal). We say that a transversal J is a 0-transversal if d0(m) ∈ J , foreach m ∈ J ; in this case we refer to j as a 0-section.

Note that if j is a 0-section, then jd0 = d0j.

Definition 3.3.6 (Fundamental 0-transversal). If the quotient graph ∆ = G/Γ admits a spanningC-simply connected profinite subgraph, then we say that a 0-transversal J is a fundamental 0-transversal and the corresponding 0-section j ∶ ∆ → Γ is a fundamental 0-section if for somespanning C-simply connected profinite subgraph T of ∆, T ′ = j(T ) is a lifting of T , i.e., therestriction of j to T is a morphism of graphs (remember that a lifting is a monomorphism whenrestricted to T ′).

The next example shows that given the quotient map π ∶ X → G/X of a profinite group Gacting on a profinite G-space X does not always have a continuous section. This fact is used inExample 3.3.8 below, which shows that G-transversals do not exist in general.

Example 3.3.7 ([2], Example 5.6.8). We construct a profinite G-space X such that the quotientmap

π ∶X → G/X

does not have a continuous section.Let K = {0,1,−1} be the field of integers modulo 3 with the discrete topology, and let

G = {1,−1} be the multiplicative group of K. Let I be an indexing set, and consider the directproduct

X =∏I

K

of copies of K indexed by I . Then X is a profinite space on which G operates continuously ina natural way. Let π ∶ X → G/X be the canonical quotient map. We shall prove that π admitsa continuous section if and only if I is countable. If I is countable, by [2], Lemma 5.6.7, page186, as X is second countable, it admits a continuous section.

Conversely, assume that σ ∶ G/X → X is a continuous section of π and let Z = Im(σ).Hence Z is a compact subset of X because the section σ is continuous and we have that 0 ∈ Zbecause the action of G on X fixes 0. The image of a non zero element can be either 1 or −1, itis, either 1 ∈ Z or −1 ∈ Z (not both, because if it happens, Z would have the same generator setas X , so X = Z and the section would be an inverse map). Let J be a finite subset of I and let

30

u = (ui) ∈X be such that ui = 0 for i ∉ J . Define

B(J, u) = {x ∈∏I

K ∣ xj = uj,∀j ∈ J}.

Then the subsets of X of the form B(J, u) are clopen and constitute a base for the topology ofX . For i ∈ I , write ei for the element of X which has entry 1 at position i and entry 0 elsewhere.Define εi ∈ {1,−1} to be such that εiei ∉ Z. Since Z is closed, for each i ∈ I there exists a finitesubset Ji of I such that i ∈ Ji and B(Ji, εiei) ∩Z = ∅.

Consider now any two distinct indices i, j ∈ I . We claim that either i ∈ Jj or j ∈ Ji (or both).To see this, set x = εiei − εjej . Assume that i ∉ Jj and j ∉ Ji. Then, x ∉ Z (since j ∉ Ji impliesx ∈ B(Ji, εiei)); similarly, −x ∉ Z (since i ∉ Jj implies x ∈ B(Jj, εjej)). This is a contradiction,and so the claim is proved.

Next we show that I is countable. Let N be a countably infinite subset of I and setP = ⋃i∈N Ji. If I were uncountable, there would be some j ∈ I − P , since P is countable.Then, by construction, j ∉ Ji, for any i ∈ N . Therefore, i ∈ Jj by the preceding paragraph. Inparticular, N ⊆ Jj , contradicting the finiteness of Jj .

In contrast with the situation for abstract groups that act on abstract graphs (cf. [3],Proposition I.14), a general subtree of a C-simply connected profinite subgraph of ∆ need nothave a lifting to an isomorphic profinite subgraph of Γ, as the following example shows:

Example 3.3.8 (Quotient G-graph with no lifting of trees or simply connected subgraphs, [1],Example 3.4.2). Let X be a profinite space on which a pro-C group G acts continuously in sucha way that the canonical epimorphism ϕ ∶ X → G/X does not admit a continuous section (seeExample 3.3.7).

Construct a profinite graph C = C(X,P ), where P is a point not in X , as follows:C = V (C) ⊍E(C), where

• V (C) =X ⊍ {P};

• E(C) = {(x,P ) ∣ x ∈X};

• d0(x,P ) = P ;

• d1(x,P ) = x.

We termC the cone ofX , represented by the following diagram, where i ∈ I , not necessarilya countable set.

31

P

xi1xi2

xi3

xi4

xi5xi6

xi7

xi8

xi9

. ..

Extend the action of G on X to an action of G on V (C) by letting every g ∈ G fix P . Define anaction of G on E(C) as follows:

g(x,P ) = (gx,P ),

g ∈ G, x ∈X . One checks that this defines a continuous action of G on the profinite graph C.We claim that C is a C-simply connected graph, for any pseudovariety of finite groups C;

indeed, write X as an inverse limit of finite quotient spaces Xi; then

C = lim←ÐC(Xi, P );

therefore C is the inverse limit of finite trees. The quotient graph of C under the action of G isthe cone of G/X:

G/C = C(G/X,P ).

In particular, G/C is a C-simply connected graph (see Proposition 3.2.10) which does not havea lifting to a profinite subgraph of C(X,P ), i.e., there is no morphism of profinite graphsψ ∶ C(G/X,P )→ C(X,P ) such that ϕψ = idC(G/X,P ).

The following result proves the existence of 0-sections for Galois coverings:

Lemma 3.3.9 ([1], Lemma 3.4.3). Let ζ ∶ Γ→∆ be a Galois C-covering of a profinite graph ∆

with associated group G = G(Γ∣∆). Assume that Λ′ is a lifting of a profinite subgraph Λ of ∆.Then there exists a 0-transversal J ⊆ Γ of ζ such that Λ′ ⊆ J .

Proof. Note that G/V (Γ) = V (∆). Since V (Γ) is a profinite space and G acts on it freely, by[2], Lemma 5.6.5, page 185, there is a continuous section

jV ∶ V (∆)→ V (Γ)

of π = ζ ∣V (Γ) such that, as Y = {d0(m) ∣ m ∈ Λ′} ⊆c V (Γ) and π∣Y is injective, then j can bechosen such that Y is a subset of jV (V (∆)). Thus, let {d0(m) ∣m ∈ Λ′} ⊆ jV (V (∆)). Define

J = d−10 (jV (V (Γ))).

Note that Λ′ ⊆ J and clearly d0(J) ⊆ J . It remains to prove that the restriction ζ ∣J ∶ J → ∆

of ζ to J is a homeomorphism of topological spaces (cf. Definition 3.3.4). Since J is

32

compact, it suffices to prove that ζ ∣J is a bijection. Let m ∈ ∆, and let m ∈ Γ be such thatζ(m) = m. Put v = jV d0(m). Since ζd0(m) = ζ(v) = d0(m), (by q-morphism definition,ζ(dj(m)) = dj(ζ(m)), cf. Definition 2.2.7) there exists some g ∈ G with gd0(m) = v. Putm′ = gm. Then ζ(m′) = m, and d0(m′) = v, i.e., m′ ∈ J . So ζ ∣J is onto. Now, if m1,m2 ∈ Jand ζ(m1) = ζ(m2), then there exists some g ∈ G with gm1 = m2, and so gd0(m1) = d0(m2).Since d0(m1), d0(m2) ∈ jV (V (Γ)), we deduce that d0(m1) = d0(m2); and since the action ofG is free, g = 1. Therefore m1 =m2, proving that ζ ∣J is also an injection.

Definition 3.3.10. Let ζ ∶ Γ → Γ be the universal Galois C-covering of a profinite graphΓ. By Lemma 3.3.9 there exists a continuous 0-section j ∶ Γ → Γ of ζ . Let J = j(Γ) bethe corresponding 0-transversal. Associated with this transversal we are going to define twocontinuous maps

κ = κj ∶ Γ→ πC1 (Γ) and χ = χj ∶ Γ→ πC1 (Γ).

If m ∈ Γ, define κ(m) to be the unique element of πC1 (Γ) such that

κ(m)(jζ(m)) = m. (3.1)

For an arbitrary m ∈ Γ one has ζd1j(m) = d1ζj(m) = d1(m) = ζjd1(m) (by q-morphismdefinition, ζ(dj(m)) = dj(ζ(m)), see Definition 2.2.7). We define χ(m) to be the uniqueelement of πC1 (Γ) such that

χ(m)(jd1(m)) = d1j(m). (3.2)

Lemma 3.3.11 ([1], Lemma 3.4.4). The following properties hold for the functions κ = κj andχ = χj defined above.

(a) κd1j(m) = χ(m),∀m ∈ Γ;

(b) κ(hm) = hκ(m),∀m ∈ Γ, h ∈ πC1 (Γ);

(c) κ(m) = κ(d0(m)),∀m ∈ Γ;

(d) κj(m) = 1, ∀m ∈ Γ;

(e) χ(v) = 1, ∀v ∈ V (Γ);

(f) κ(m)(χζ(m)) = κd1(m),∀m ∈ Γ;

(g) The maps κ and χ are continuous.


33

One can sharpen Lemma 3.3.9 when the quotient graph ∆ = G/Γ is finite.

Lemma 3.3.12 ([1], Lemma 3.4.5). Let a profinite group G act on a connected profinite graphΓ and let ϕ ∶ Γ→∆ = G/Γ be the corresponding projection.

(a) Let T be a finite subtree of the graph ∆ and let T ′ be a finite subtree of Γ that ϕ sendsinjectively into T . Then T lifts to a subtree of Γ containing T ′;

(b) Assume further that the action of G on Γ is free and that the quotient graph ∆ = G/Γ isfinite. Let m0 ∈ Γ. Then there exists a fundamental 0-transversal J in Γ containing m0.

Proof. (a) Let L be the set of finite subtrees of Γ containing T ′ which are sent injectively intoT by means of ϕ. Let T ′

0 be a maximal element of L with respect to inclusion, and let T0 beits image in T . Suppose that T0 ≠ T . Since T is finite and connected, there exists an edgee of T not belonging to T0 such that one of the vertices of e is in T0, say d0(e) ∈ V (T0);then d1(e) ∉ V (t0). Let v′ be a vertex of T ′

0 whose image in T is d0(e). Let e” ∈ Γ withϕ(e”) = e. Since v′ and d0(e”) are in the same G-orbit, there exists some g ∈ G withgd0(e”) = v′. Define e′ = ge”. Then d0(e′) = d0(ge”) = gd0(e”) = v′ and ϕ(e′) = e. SinceT ′

0∪{e′, d1(e′)} ∈ L, this contradict the maximality of T ′

0. Thus ϕ(T ′

0) = T0 = T , as desired.

(b) Since ∆ is finite and connected, it has a subtree T with V (T ) = V (Γ) (i.e., T is a spanningsimply connected profinite subgraph of Γ). By part (a) there exists a lifting T ′ of T suchthat d0(m0) ∈ V (T ′). Define J = d−1

0 (V (T ′)). Note that m0 ∈ J and T ′ ⊆ J . Then arguingas in the proof of Lemma 3.3.9, we see that J is a 0-transversal, and since T ′ is a liftingof a maximal tree of ∆, J is a fundamental 0-transversal. Equivalently, one can describe Jmore explicitly: for each edge e ∈ ∆ − T , choose e′ ∈ Γ such that d0(e′) ∈ T ′ and ϕ(e′) = e(this can be done since d0(e) ∈ T , and every vertex of ∆ is in the G-orbit of a vertex of T ′;furthermore, such e′ is unique because G acts freely on Γ); then J consists of T ′ togetherwith all the chosen edges e′.

3.4 Existence of universal Galois coverings and the Nielsen-Schreier Theorem for pro-C groups

In this section we show the existence of universal Galois coverings in Construction 3.4.1 andTheorem 3.4.3. We finish the section with an original proof of the Nielsen-Schreier theorem forfree profinite groups on a finite space using everything we have stated so far.

Construction 3.4.1 ([1], Construction 3.5.1). Let Γ be a finite connected graph and let T be aconnected subgraph of Γ with V (T ) = V (Γ) (T need not equal Γ). Denote by X = Γ/T thecorresponding quotient space with canonical map

ω ∶ Γ→X = Γ/T.

34

Consider the element ∗ = ω(T ) as a distinguished point of X . Let F = FC(X,∗) be the freepro-C group on the pointed profinite space (X,∗) and think of (X,∗) as being a subspace ofFC(X,∗) in the natural way. Define a profinite graph ΥC(Γ, T ) as follows:

• ΥC(Γ, T ) = FC(X,∗) × Γ;

• V (ΥC(Γ, T )) = FC(X,∗) × V (Γ);

• d0(r,m) = (r, d0(m));

• d1(r,m) = (rω(m), d1(m)),

(r ∈ F , m ∈ Γ). Next define an action of F on the graph ΥC(Γ, T ) by

r′(r,m) = (r′r,m)

(r, r′ ∈ F,m ∈ Γ). Clearly this is a free action (because F is a free group) and F /ΥC(Γ, T ) = Γ.Therefore the natural epimorphism

υ ∶ ΥC(Γ, T )→ Γ

that sends (r,m) to m (r ∈ F,m ∈ Γ) is a Galois C-covering.

Lemma 3.4.2 ([1], Lemma 3.5.2). The Galois covering υ ∶ ΥC(Γ, T )→ Γ is connected.


Theorem 3.4.3 ([1], Theorem 3.5.3). Let Γ be a finite connected graph and let T be a maximalsubtree of Γ (T is a spanning C-simply connected profinite subgraph of Γ by Proposition 3.2.10).Then one has the following properties:

(a) The Galois C-covering υ ∶ ΥC(Γ, T )→ Γ of Construction 3.4.1 is universal.

(b) Let (X,∗) = (Γ/T,∗); thenπC1 (Γ) = FC(X,∗)

is a free pro-C group of finite rank ∣Γ∣ − ∣T ∣;

(c) The universal Galois C-covering υ ∶ ΥC(Γ, T ) → Γ is independent of the maximal subtreeT chosen.


Corollary 3.4.4 ([1], Corollary 3.5.4). Let ζ ∶ Γ → Γ be a universal C-covering of a finiteconnected graph Γ, and let T be a maximal subtree of Γ. Choose a fundamental 0-sectionj ∶ Γ → Γ of ζ lifting T , and let χj ∶ Γ → πC1 (Γ) be the corresponding map (see the definitionof χ). Then the pointed space (χj(Γ),1) with distinguished point 1 is a basis for the free pro-Cgroup πC1 (Γ).

35

Proof. See [1], Corollary 3.5.4, page 86.

Example 3.4.5 ([1], Exercise 3.3.4(a)). Let C be a pseudovariety of finite groups and L(0) bethe loop

0 0

Take a maximal subtree T = {0} of L(0) and ω ∶ L(0)→ L(0)/T = L(0). The universal Galoiscovering of L(0), by Construction 3.4.1 and Theorem 3.4.3, is

ΥC(L(0), T ) = FC(L(0)/T,ω(T )) ×L(0)= FC(L(0),{0}) ×L(0),

Also by Theorem 3.4.3, rank(F ) = ∣L(0)∣ − ∣{0}∣ = 2 − 1 = 1, thus F = ZC

(whereF = FC(L(0),{0})) and

FC(L(0),{0}) ×L(0) = ZC× {0}

= Γ(ZC,{1}).

Thereafter ΥC(L(0), T ) = Γ(ZC,1) is defined by

V (ΥC(L(0),{0})) = V (Γ(ZC,1))

= {(g,1) ∣ g ∈ ZC} = Z

C;

• d0(r,m) = (r, d0(m));

• d1(r,m) = (rω(m), d1(m));

(r ∈ F , m ∈ L(0)), with an action of F = ZC

on the graph ΥC(L(0),{0}) = Γ(ZC,{1})

by r′(r,m) = (r′r,m), (r, r′ ∈ F,m ∈ L(0)). Therefore, ZC/Γ(Z

C,{1}) = L(0) and

πC1 (L(0)) ≅ ZC .

Example 3.4.6 ([1], Exercise 3.3.4(b)). Let C be a pseudovariety of finite groups and n be anatural number. Consider the finite graph L(n)

0 1 2 3 n − 1 n

0 1 2

. . .

n − 1 n

Take a maximal subtree T of L(n),

36

0 1 2 3 n − 1 n

0 1 2

. . .

n − 1

and ω ∶ L(n) → L(n)/T = L(0), ω(T ) = {0}. The universal Galois covering of L(0), byConstruction 3.4.1 and Theorem 3.4.3, is

ΥC(L(n), L(0)) = ΥC(L(0),{0})= FC(L(0),{0}) ×L(n),

By Example 3.4.5, FC(L(0),{0}) ≅ ZC. Therefore, ΥC(L(n), L(0)) = Γ(Z

C,{1}),

ZC/Γ(Z

C,{1}) = L(0) and πC1 (L(n)) ≅ πC1 (L(0)) ≅ ZC .

The next results are needed for the proof of the Nielsen-Schreier Theorem for profinitegroups:

Theorem 3.4.7 ([1], Proposition 3.6.1). Let ζ ∶ Γ → Γ be a universal Galois covering of aconnected profinite graph Γ and let H be a closed subgroup of πC1 (Γ).

(a) The canonical epimorphism ξ ∶ Γ→H/Γ is a universal Galois covering

(b) πC1 (H/Γ) =H .


Theorem 3.4.8 ([1], Proposition 3.7.4). Let Γ be a connected profinite graph having a spanningC-simply connected profinite subgraph T . The following properties hold:

(a) The Galois C-covering υ ∶ Υ(Γ, T )→ T constructed in 3.4.1 is universal.

(b) The fundamental group πC1 (Γ) is a free pro-C group on the pointed profinite space (X,∗) =(Γ/T,∗).

Proof. See [1], Theorem 3.7.4, page 94.

Theorem 3.4.9 ([1], Proposition 3.8.1). The Cayley graph Γ(F (X,∗),X) of a free profinitegroup on a pointed profinite space (X,∗) with respect to X is simply connected. In fact,Γ(F (X,∗),X) is the universal Galois covering space of the bouquet of loops B = B(X,∗)and π1(B) = F (X,∗).


We finish this chapter with an original proof of the Nielsen-Schreier Theorem for profinitegroups on a finite space. This Theorem is proved in [2], from the article of Binz, Neukirchand Wenzel [1971] and uses the abstract version. Ribes and Steinberg (2010) gave a new proofwithout using the abstract version, through wreath products, but it is not so simple as the onepresented in this thesis. The approach is entirely different as well.

37

Theorem 3.4.10 (Profinite version of the Nielsen-Schreier theorem on a finite space). Let F bea free profinite group on a finite space X and H be an open subgroup of F . Then H is a freeprofinite group.

Proof. Let Γ(F,X) be the Cayley graph of F . As F acts freely on Γ(F,X) (cf. Example2.3.12), the Cayley graph Γ(F,X) is a Galois covering of F /Γ(F,X) and by Theorem 3.4.9,Γ(F,X) is the universal Galois covering of F /Γ(F,X) = B(X,∗) and π(Γ(F,X)) = F .Similarly, as H is a closed subgroup of F with finite index and π1(Γ(F,X)) = F , by Theorem3.4.7 the canonical epimorphism ξ ∶ Γ(F,X)→H /Γ(F,X) is a universal Galois covering andπ1(H /Γ(F,X)) =H .

Hence we have the following diagram:

Γ(F,X) π1(H /Γ(F,X))=H //

π1(Γ(F,X))=F

��

H /Γ(F,X)

F /Γ(F,X)

However, the vertices of Γ(F,X) can be mapped homeomorphically to the elements of thefree profinite group F as stated in Example 2.2.22. Thus, as [F ∶ H] is finite, H /Γ(F,X)will have finitely many vertices and as X is finite, E(Γ(F,X)) is finite, which implies thatH /Γ(F,X) is finite. Hence, by Theorem 3.4.3,

H = π1(H /Γ(F,X)) = F ((H /Γ(F,X))/T,∗),

where T is the maximal subtree of H /Γ(F,X). Therefore H is free, as desired.

38

Chapter 4

Graphs of pro-C groups

In this chapter we define a sheaf, free product of a sheaf, a graph, the fundamental group andthe standard graph of pro-C groups. We also show that we can see a free product of a sheaf ofpro-C groups as a fundamental group of a graph of pro-C groups and use it to prove the KuroshSubgroup Theorem in the last section, the main result of this thesis.

4.1 Free pro-C product of a sheaf of pro-C groups

In this section we define a sheaf of pro-C groups and the free pro-C product of a sheaf.

Definition 4.1.1 (Sheaf). Let T be a profinite space. A sheaf of pro-C groups over T is a triple(G, π, T ), where G is a profinite space and π ∶ G → T is a continuous surjection satisfying thefollowing conditions:

(a) For every t ∈ T , the fiber G(t) = π−1(t) over t is a pro-C group whose topology is inducedby the topology of G as the subspace topology;

(b) If we defineG2 = {(g, h) ∈ G × G ∣ π(g) = π(h)},

then the map µ ∶ G2 → G given by µG(g, h) = gh−1 is continuous.

Example 4.1.2 ([1], Example 5.1.1(a)). Let T = {1,⋯, n} be a finite discrete space with n

points and let G1,⋯,Gn be pro-C groups. Define the space G = G1 ⊍⋯⊍Gn to have the disjointunion topology. Let π ∶ G → T be the map that sends Gi to i (i = 1,⋯, n). Then (G, π, T ) is ina natural way a sheaf over the space T with G(i) = Gi (i = 1,⋯, n). For the second condition,G2 = {(g, h) ∈ G × G ∣ g, h ∈ Gi} and we can write G = ⊍i∈T Gi ×Gi, so each Gi ×Gi is clopen.Take U an open subset of Gi. Then the map µ restricted to it is continuous because Gi is atopological group and µ−1(U) is open in Gi ×Gi. Hence, as Gi ×Gi is open in G2, we have thatµ−1(U) is open in G2, as desired.

39

Example 4.1.3 ([1], Example 5.1.1(b)). Let G be a profinite group and let T be a profinitespace. Define the constant sheaf over T with constant fiber G to be the sheaf

KT (G) = (T ×G,π,T )

where π ∶ T ×G→ T is the usual projection map.

Definition 4.1.4 (Morphism of sheaves). A morphism α = (α,α′) ∶ (G, π, T ) → (G′, π′, T ′) ofsheaves of pro-C groups consists of a pair of continuous maps α ∶ G → G′ and α′ ∶ T → T ′ suchthat the diagram

G α //

π

��

G′

π′

��

Tα′ // T ′

commutes and the restriction of α to G(t) is a homomorphism from G(t) into G′(α′(t)), foreach t ∈ T .

Example 4.1.5. Let (G, π, T ) be a sheaf of pro-C groups and let H be a pro-C group. We maythink of H as the fiber of a sheaf over a singleton space.

G α //

π

��

H

π′

��

Tα′ // 1

Hence we have a natural notion of a morphism

α ∶ G →H

from the sheaf G to the group H , namely, α is a continuous map from the space G into H suchthat the restriction of α to each fiber G(t) is a homomorphism.

Definition 4.1.6 (Monomorphism). A morphism α is said to be a monomorphism if α and α′

are injective.

The definition of epimorphism is analogue.

Definition 4.1.7 (Subsheaf). The image of a monomorphism G → G′ is called a subsheaf of thesheaf G′.

Example 4.1.8. If (G, π, T ) is a sheaf and T ′ is a closed subspace of T , then the triple

(π−1(T ′), π∣π−1(T ′), T ′)

40

is a subsheaf of (G, π, T ). Note that this is well defined, because a closed subspace of a profinitespace is profinite and the map is just the restriction.

Definition 4.1.9 (Free pro-C product of a sheaf). A free pro-C product of the sheaf (G, π, T )is defined to be a pro-C group G together with a morphism ω ∶ G → G (see Example 4.1.5),that respects the following universal property: for every morphism β of the sheaf G into a pro-Cgroup H there exists a unique continuous homomorphim β ∶ G → H such that the followingdiagram:

G ω //

β

��

G

β

��

H

commutes, that is: βω = β.

Remark 4.1.10. Since a pro-C group H is an inverse limit of groups in C, it is sufficient tocheck the above universal property for groups H ∈ C only. Indeed, let H = lim←Ði∈I

Hi and Y atopological space. By using the universal property of the inverse limit (cf. Definition 2.1.4), wehave the following

G ω //

β

""

G

β

��

H = lim←ÐHi

ϕi

��

Yψoo

ψi

||Hi

And we can rewrite it as

G ω //

βϕi

��

G

βϕi

��

Hi

Proposition 4.1.11 (Existence and uniqueness of the free pro-C product, [1], Proposition 5.1.2).Let (G, π, T ) be a sheaf of pro-C groups. Then there exists a free pro-C product of G and it isunique up to isomorphism.

41

Proof. The uniqueness follows from the universal property. We shall give an explicitconstruction of the free pro-C product. Let

L = ˚t∈TG(t)

be the free product of the groups G(t), t ∈ T , considered as abstract groups (cf. [3], Example1, I.2) and ρ ∶ G → L be defined on each G(t) as the inclusion map. Consider the set N of allnormal subgroups N of L with L/N ∈ C such that the composite map

G ρ // L // L/N ;

is continuous. If N1,N2 ∈ N , then N1 ∩N2 ∈ N , because the composition map is continuous.So, L can be made into a topological group by considering N as a fundamental system ofneighbourhoods of the identity element of L. Denote by KN (L) the corresponding completionof L with respect to this topology, that is

KN (L) = lim←ÐN∈N

L/N.

Then KN (L) is a pro-C group. Letı ∶ L→ KN (L)

be the natural map. Put ω = ıϕ; then ω is continuous because the composite

G → KN (L)→ L/N

is continuous for each N ∈ N . Since the restriction of ω to each G(t) (t ∈ T ) is ahomomorphism, the map ω ∶ G → KN (L) is a morphism from the sheaf G to the pro-C groupH (cf. Example 4.1.5). We claim that (KN (L), ω) is the free pro-C product of G. To see thiswe check the corresponding universal property. By Remark 4.1.10, we only need to check theuniversal property for groups H ∈ C. Hence, let H ∈ C and let β ∶ G →H be a morphism.

L

KN (L)

G

H

ı

ρ

ωβ

β

β

By the universal property of abstract free products, there exists a unique homomorphismβ ∶ L → H with β = βρ. Since βρ is continuous, it follows from the definition of N thatKer(β) ∈ N ; so there exists a continuous homomorphism β ∶ KN (L)→H with β = βı. Hence

42

β = βω. The uniqueness of β follows from the fact KN (L) = ⟨ω(G)⟩.

The free pro-C product of a sheaf of pro-C groups (G, π, T ) will be denoted by ∐T G.

Example 4.1.12 ([1], Example 5.1.3(a)). Let T = {1,⋯, n} be a finite discrete space with npoints and let G1,⋯,Gn be pro-C groups. Consider the sheaf defined in Example 4.1.2: G isjust the disjoint union of the groups G1,⋯Gn. The corresponding free pro-C product G = ∐Gcoincides with the standard concept of a free pro-C product of the groups Gi (cf. [2], Section9.1, page 353) and it is usually written

G = G1 ∐⋯ ∐Gn =n

∐i=1

Gi.

Indeed, if ω ∶ G → G is the canonical morphism, define ωi ∶ Gi → G to be the restriction of ω toGi (i = 1,⋯, n); then the corresponding universal property defining G = ∐G is the following:for any given pro-C group H and continuous homomorphims ϕi ∶ Gi → H (i = 1,⋯, n), thereexists a unique continuous homomorphism ϕ ∶ G→H such that ϕωi = ϕi (i = 1,⋯, n).

Example 4.1.13 ([1], Example 5.1.3(b)). Let T be a profinite space. Consider the constantsheaf KT (ZC) = (T × Z

C, π, T ) (see Example 4.1.3). Then the free pro-C product ∐T KT (ZC)

is isomorphic to the free pro-C group FC(T ) on the profinite space T . Indeed, by the definitionof the free pro-C product we have the following

T ×ZC

ω //

β

$$

∐T KT (ZC)

β

��

H

Adding the projection map π ∶ T ×ZC→ T map, we have:

T ×ZC

ω //

β

$$

π

��

∐T KT (ZC)

β

��

T H

Now take the compositions ωı ∶ T → ∐T KT (ZC) and βı ∶ T → H , (where ı ∶ T → T × ZC

43

defined by t↦ (t,0) is the inclusion map) that produces the diagram:

∐T KT (ZC)β // H

T

ωı

OO

βı

;;

Therefore, as they respect the same universal property, the free pro-C product ∐T KT (ZC) isisomorphic to the free pro-C group FC(T ) on the profinite space T , as desired.

Proposition 4.1.14 ([1], Example 5.1.6). Let (G, π, T ) be a sheaf of pro-C groups, G = ∐CT G,and let ω ∶ G → G be the canonical morphism. Then:

(a) The group G is generated by its subgroups Gt = ω(G(t)), t ∈ T ;

(b) If s ≠ t, then Gs ∩Gt = 1;

(c) The morphism ω maps G(t) isomorphically onto Gt, for all t ∈ T .


Definition 4.1.15. Let T be a profinite space and let G be a profinite group. A collection ofclosed subgroups Gt of G (t ∈ T ) is said to be continuously indexed by T if whenever U is anopen subset of G then the subset T (U) = {t ∈ T ∣ Gt ⊆ U} of T is open in T .

Lemma 4.1.16 ([1], Example 5.2.1). Let G be a profinite group and let {Gt ∣ t ∈ T} be acollection of closed subgroups of G indexed by a profinite space T . The family {Gt ∣ t ∈ T} iscontinuously indexed by T if and only if the set

E = {(t, g) ∈ T ×G ∣ t ∈ T, g ∈ Gt}

is a closed subset of T ×G.


Lemma 4.1.17 ([1], Example 5.2.2). Let G be a profinite group that acts continuously on aprofinite space T . Then the collection {Gt ∣ t ∈ T} of the G-stabilisers Gt of the points t of T iscontinuously indexed by T .

Proof. Consider the mapα ∶ T ×G→ T × T

defined by α(t, g) = (t, gt) (g ∈ G, t ∈ T ). Since α is continuous and the diagonal subset∆ = {(t, t) ∣ t ∈ T} of T × T is closed because the space T × T with the product topology isprofinite, it follows that

E = {(t, g) ∈ T ×G ∣ g ∈ Gt} = α−1(∆)

44

is closed in T × G by the continuity of the map α. The result now follows from Lemma4.1.16.

Proposition 4.1.18 ([1], Example 5.2.3). Let G be a profinite group and let

F = {Gr ∣ r ∈ T}

be a continuously indexed family of closed subgroups of G, where T is a profinite space.Consider the equivalence relation on the product T ×G defined by

(t, g) ∼ (t′, g′) if t = t′ and g−1g′ ∈ Gt (t, t′ ∈ T ; g, g′ ∈ G)

Then the quotient space G/F = T ×G/ ∼ is a profinite space.


4.2 Graphs of pro-C groups and specialisations

In this section we define a graph of pro-C groups and specialisations and give someexamples.

Definition 4.2.1 (Profinite graph of pro-C groups, [1], Definition 6.1.1). Let Γ be a connectedprofinite graph with incidence maps d0, d1 ∶ Γ→ V (Γ). We can define a profinite graph of pro-Cgroups over Γ as a sheaf (G, π,Γ) of pro-C groups together with two morphisms of sheaves

(B0, d0), (B1, d1) ∶ (G, π,Γ)→ (GV , π∣π−1(V (Γ)), V (Γ))

[here GV denotes the restriction subsheaf of G to the space V (Γ), that we term the ’vertexsubsheaf of G’. Note that it is indeed a sheaf, because the vertex set V (Γ) is closed in Γ, soit is also a profinite space and the vertex sheaf is well defined (cf. Example 4.1.8)], wherethe restriction of Bi to GV is the identity map idGV (i = 0,1); in addition, we assume that therestriction of Bi to each fiber G(m) is an injection (m ∈ Γ), (i = 0,1).

Definition 4.2.2. The vertex groups of a graph of pro-C groups (G, π,Γ) are the groups G(v)with v ∈ V (Γ), and the edge groups are the groups G(e), with e ∈ E(Γ).

Let (G, π,Γ) be a graph of pro-C groups over Γ, and let

ζ ∶ Γ→ Γ

be a universal Galois C-covering of the profinite graph Γ. Choose a continuous 0-section j

of ζ , and denote by J = j(Γ) the corresponding 0-transversal (that exists by Lemma 3.3.9).Associated with j there is a continuous function

χ ∶ Γ→ πC1 (Γ)

45

from Γ into the fundamental group of Γ defined by

χ(m)(jd1(m)) = d1j(m)

(See Equation 3.2 on Section 3.3)

Definition 4.2.3 (J-specialisation). Given a pro-C group H , define a J-specialisation of thegraph of pro-C groups (G, π,Γ) in H to consist of a pair (β, β′), where

β ∶ (G, π,Γ)→H

is a morphism from the sheaf (G, π,Γ) to H , and where

β′ ∶ πC1 (Γ)→H

is a continuous homomorphism satisfying the following conditions:

β(x) = βB0(x) = (β′χ(m))(βB1(x))(β′χ(m))−1 (4.1)

for all x ∈ G, where m = π(x). Note that the definition of the map χ depends uniquely onthe 0-section j and the corresponding G-transversal J (cf. Definition 3.3.10). The followingdiagram shows the compositions:

G H

πC1 (Γ)

GV Γ

β

Bi

π

β′β∣GV

χ

(4.2)

Example 4.2.4 ([1], Example 6.1.2(a)). Assume that the graph of pro-C groups (G, π,Γ) hastrivial edge groups, i.e., G(e) = 1, for every edge e ∈ Γ. Then we may think of a J-specialisation(β, β′) of (G, π,Γ) in a pro-C group H as simply a morphism β from the sheaf (G, π,Γ) to H ,since conditions (4.1) are automatic in this case.

Indeed, (4.1) gives us that:

β(G(m)) = βB0(G(m)) = (β′χ(m))(βB1(G(m)))(β′χ(m))−1

So, if m ∈ E(Γ), we have that:

β(G(e)) = βB0(G(e))β(1) = βB0(1).

46

By diagram 4.2,

G β //

B0

��

H

GV

β∣GV

>>

so this diagram commutes for every e ∈ E(Γ). If v ∈ V (Γ), B0(G(v)) = idG(V ) = G(v), then

β(G(v)) = βB0(G(v))β(G(v)) = β(G(v)).

and we have nothing to show. Thus, for B1 and e ∈ E(Γ), we have:

β(G(e)) = (β′χπ(G(e)))(βB1(G(e)))(β′χπ(G(e)))−1

β(1) = (β′χπ(1))(βB1(1))(β′χπ(1))−1

By diagram 4.2,

G H

πC1 (Γ)

GV Γ

β

Bi

π

β′β∣GV

χ

and the condition of Equation (4.1) is automatically satisfied.

Example 4.2.5 ([1], Example 6.1.2(b)). If Γ is C-simply connected, then πC1 = 1 and Γ = Γ = J .Then we can refer to a ’specialisation’ rather than ’J-specialisation’: it is just a morphismβ ∶ G →H such that

β(x) = βB0(x) = βB1(x)

for all x ∈ G. It happens for example if Γ is a finite tree or, more generally, an inverse limit offinite trees (see Proposition 3.2.10).

4.3 The fundamental group of a graph of pro-C groups

In this section we define the fundamental group of a graph of pro-C groups through auniversal property and finish the section with Example 4.3.6, which shows the free pro-Cproduct of a sheaf as the fundamental group of a graph of pro-C groups.

Definition 4.3.1 (The fundamental group of a graph of pro-C groups). Choose a continuous 0-section j of the universal Galois C-covering ζ ∶ Γ→ Γ and denote by J = j(Γ) the corresponding

47

0-transversal (see Lemma 3.3.9). We define a fundamental pro-C group of the graph of groups(G, π,Γ) with respect to the 0-transversal J to be a pro-C group ΠC1(G,Γ), together with aJ-specialisation (ν, ν′) of (G, π,Γ) in ΠC1(G,Γ) satisfying the following universal property:

H

G πC1 (Γ)

ΠC1(G,Γ)

δ

β β′

ν ν′

whenever H is a pro-C group and (β, β′) a J-specialisation of (G, π,Γ) in H , there exists aunique continuous homomorphism

δ ∶ ΠC1(G,Γ)→H

such that δν = β and δν′ = β′. We refer to (ν, ν′) as a universal J-specialisation of (G, π,Γ).

Remark 4.3.2. Observe that to check the above universal property it suffices to consider onlyfinite groups H in the pseudovariety C, since every pro-C group is an inverse limit of groups inC (cf. Remark 4.1.10).

Proposition 4.3.3 (Existence of fundamental groups, [1], Proposition 6.2.1). Let (G, π,Γ) bea graph of pro-C groups over a connected profinite graph Γ, and let J be a 0-transversal ofζ ∶ Γ→ Γ. Then

(a) There exists a fundamental pro-C group ΠC1(G,Γ) of (G, π,Γ) with a universal J-specialisation (ν, ν′);

(b) (uniqueness for a fixed J) The fundamental pro-C group ΠC1(G,Γ) is unique in the sense thatif Π is another fundamental pro-C group of (G, π,Γ), with respect to the same 0-transversalJ , and (µ,µ′) is a universal J-specialisation of (G, π,Γ) in Π, then there exists a uniquecontinuous isomorphism ξ ∶ ΠC1(G,Γ)→ Π such that ξν = µ and ξν′ = µ′;

(c) ΠC1(G,Γ) is topologically generated by {ν(G(v)) ∣ v ∈ V = V (Γ)} and ν′(πC1 (Γ)).

Proof. Part (b) follows from the universal property. Consider the free pro-C product of thevertex sheaf (see Section 4.1),

W =∐v∈V

G(v) =∐GV .

We can do an embedding of each vertex group each vertex group G(v) in W under the naturalmorphism

ω ∶ Gv →W

48

(cf. Proposition 4.1.14(c)). Define

ΠC1(G,Γ) = (W ∐ πC1 (Γ))/N

where πC1 (Γ) denotes the free pro-C product of pro-C groups, and where N is the topologicalclosure of the normal subgroup of the group W ∐ πC1 (Γ) generated by the set

{B0(x)−1(χπ(x))B1(x)(χπ(x))−1 ∣ x ∈ G}.

Define ν ∶ G → ΠC1(G,Γ) to be the composition of the natural continuous maps

G B0 // GV ω // Wı //W ∐ πC1 (Γ) α // ΠC1(G,Γ) = (W ∐ πC1 (Γ))/N,

where ı is the inclusion and α is the quotient map and ν′ ∶ πC1 (Γ)→ ΠC1(G,Γ) is the compositionof the natural continuous homomorphisms

πC1 (Γ) ı′ //W ∐ πC1 (Γ) α // ΠC1(G,Γ) = (W ∐ πC1 (Γ))/N

where ı′ is the inclusion. We want to verify that (ν, ν′) is indeed a J-specialisation of the graphof pro-C groups (G, π,Γ) in ΠC1(G,Γ).

By the definition of N , it is the closure of the normal subgroup of the group W ∐ πC1 (Γ)generated by the set

{B0(x)−1(χπ(x))B1(x)(χπ(x))−1 ∣ x ∈ G},

and it means that B0(x) = (χπ(x))B1(x)(χπ(x))−1 in ΠC1(G,Γ). Therefore, as ν = αıωB0 andν′ = αı′, we have that

νB0(x) = (ν′χπ(x))(νB1(x))(ν′χπ(x))−1

and as J is a 0-transversal, ν(x) = νB0(x) for all x ∈ G as desired.It follows that the pro-C group ΠC1(G,Γ) thus constructed, together with (ν, ν′), satisfies the

universal property of a fundamental pro-C group of the graph of pro-C groups (G, π,Γ). Thisproves (a). Part (c) is clear from the construction above.

The special case when πC1 (Γ) = 1 is very important and will be used in the proof of theKurosh Subgroup Theorem (KST). To say that πC1 (Γ) = 1 is the same as consider Γ a C-simplyconnected profinite graph. Then there is only one 0-transversal J of ζ ∶ Γ→ Γ, that is J = Γ = Γ.In this case we can refer just to ΠC1(G,Γ) (without considering the transversal) because there isonly one choice for J . The following corollary is an easy consequence of Proposition 4.3.3(c).

Corollary 4.3.4 ([1], Corollary 6.2.2). Let (G, π,Γ) be a graph of pro-C groups over a C-simplyconnected profinite graph Γ. Then ΠC1(G,Γ) is generated as a topological group by the imagesν(G(v)) (v ∈ V (Γ)) of the vertex groups.

49

Example 4.3.5 ([1], Example 6.2.3(a)). Assume that the graph of pro-C groups (G, π,Γ)satisfies G(e) = 1, for all edges e ∈ Γ.

As in Example 4.2.4, we can think of a J-specialisation (β, β′) in a pro-C group H as amorphism β from G to H , that is, J plays no role. In this case the fundamental pro-C group ofthis graph of pro-C groups is just the free pro-C product

ΠC1(G,Γ) = ∐v∈V (Γ)

G(v) ∐ πC1 (Γ),

because the subgroup N of W ∐πC1 (Γ) is trivial. We may take the universal specialisation to be(ν, ν′) as constructed in Proposition 4.3.3, where ν is the composition

G ∐G ∐v∈V (Γ) G(v) ∐v∈V (Γ) G(v) ∐ πC1 (Γ).≅

[the homomorphism ∐G → ∐v∈V (Γ) G(v) is induced by B0 ∶ G → GV , and in this case it is anisomorphism], while ν′ is the natural inclusion

πC1 (Γ) ∐v∈V (Γ) G(v) ∐ πC1 (Γ).

Example 4.3.6 ([1], Example 6.2.3(b)). Let X be a profinite space and G = ∐X Gx be a freepro-C product, where the set {Gx ∣ x ∈ X} is continuously indexed by the profinite space X(cf. Definition 4.1.15). Then G can be viewed as the fundamental group of a graph of groups.To see this, first take a single point space {ω} disjoint from X and construct a profinite graphT = T (X) as follows:

• V (T ) =X ⊍ {ω};

• E(T ) = {ω} ×X = {(ω,x) ∣ x ∈X};

• d0(ω,x) = ω;

• d1(ω,x) = x (x ∈X),

where V (Γ) and T = V (T ) ⊍ E(T ) are endowed with the disjoint union topology and E(Γ)with the product topology. Observe that T is an inverse limit of finite trees T (Xi). Define G tobe the subset of T ×G consisting of those elements (m,y) ∈ T ×G such that

⎧⎪⎪⎨⎪⎪⎩

y ∈ Gm, if m ∈X;

y = 1, if m ∈ {ω} ⊍E(T ).

From Lemma 4.1.16 we have that G is a profinite space, and so (G, π, T ) is a subsheaf overT of the constant sheaf T ×G. In fact, (G, π, T ) has the structure of a graph of pro-C groupswith obvious morphisms B0 and B1; they are the identity maps on vertex groups and trivialhomomorphisms otherwise.

50

1

Gx1

Gx2

Gx3

Gx

1 1

11

. ..

Since T is an inverse limit of finite trees, by Proposition 3.2.10(b), πC1 (T ) = 1; hence, as aparticular instance of Example 4.3.5, we have

G =∐X

GX = ΠC1(G, T ).

Example 4.3.7. Let (G, π,Γ) be a graph of pro-C groups over a finite graph Γ and T amaximal subtree of Γ. Using the idea of abstract Bass-Serre theory, we can think of the subset{te ∣ e ∈ E(Γ) −E(T )} as a basis for the free pro-C group πC1 (Γ). Then, by the construction ofProposition 4.3.3, we have that

Π = ΠC1(G,Γ) =⎛⎝⎛⎝ ∐v∈V (Γ)

G(v)⎞⎠∐ FC

⎞⎠/N,

where FC is the free pro-C group with basis {te ∣ e ∈ E(Γ)} and N is the smallest closed normalsubgroup of ((∐v∈V (Γ) G(v)) ∐ FC) containing the set

{te ∣ e ∈ E(T )} ∪ {B0(x)−1teB1(x)t−1e ∣ x ∈ G(e), e ∈ E(Γ)}.

For each v ∈ V (Γ), νv ∶ G(v)→ Π is the natural continuous homomorphism x↦ xN .

4.4 The standard graph of a graph of pro-C groups

In this section we define the standard graph of a graph of pro-C groups, which can be seenas the C-universal covering of a graph of pro-C groups with respect to a pro-C group H , byTheorem 4.4.5.

Definition 4.4.1 (Standard graph of a graph of pro-C groups). Let (G, π,Γ) be a graph of pro-Cgroups over a connected profinite graph Γ, j ∶ Γ → Γ be a continuous 0-section of the universalGalois C-covering ζ ∶ Γ → Γ of Γ (that exists by Lemma 3.3.9), and let J = j(Γ) be thecorresponding 0-transversal (see Section 4.1). Let (γ, γ′) be a J-specialisation of (G, π,Γ) in apro-C group H (cf. Definition 4.2.3). Then we can define a profinite graph

SC(G,Γ,H)

51

which is canonically associated to the graph of groups (G, π,Γ) and H . The followingconstruction is done in a way that the quotient of the action of the fundamental group on thisgraph of pro-C groups is Γ.

For m ∈ Γ, define H(m) = γ(G(m)). Then the collection

{H(m) ∣m ∈ Γ}

of closed subgroups of H is continuously indexed by Γ (cf. Definition 4.1.15). Indeed, if U isan open subset of H , then γ−1(H −U) is a closed and therefore a compact subset of G; since

Γ − {m ∈ Γ ∣H(m) ⊆ U} = π(γ−1(H −U)),

it follows that Γ(U) = {m ∈ Γ ∣ H(m) ⊆ U} is open in Γ, because π(γ−1(H − U)) is compact(by the continuity of π), and therefore closed.

As a topological space, SC(G,Γ,H) is defined to be the quotient space of Γ×H modulo theequivalence relation ∼ given by

(m,h) ∼ (m′, h′) if m =m′, h−1h′ ∈H(m) (m,m′ ∈ Γ, h, h′ ∈H).

So, as a set, SC(G,Γ,H) is the disjoint union

SC(G,Γ,H) = ⊍m∈Γ

H/H(m).

By Proposition 4.1.18, SC(G,Γ,H) is a profinite space. Denote by

α ∶ Γ ×H → SC(G,Γ,H)

the quotient map. The projection p′ ∶ Γ ×H → Γ induces a continuous epimorphism

p ∶ SC(G,Γ,H)→ Γ,

such that p−1(m) =H/H(m) and p′ = pα.

Γ ×H p′ //

α��

Γ

SC(G,Γ,H)p

99

To make SC(G,Γ,H) into a profinite graph we define the subspace of vertices ofSC(G,Γ,H) by

V (SC(G,Γ,H)) = p−1(V (Γ)).

The incidence mapsdi ∶ SC(G,Γ,H)→ V (SC(G,Γ,H))

52

(i = 0,1) are defined as follows:

d0(hH(m)) = hH(d0(m)) (4.3)

d1(hH(m)) = h(γ′χ(m))H(d1(m)), (4.4)

(h ∈H,m ∈ Γ), where χ ∶ Γ→ πC1 (Γ) is the continuous map considered in Equation 3.2, Section3.3. Note that (γ′χ(m)) ∈H , as the following composition diagram shows:

Γ πC1 (Γ) H.χ γ′

We must check that these maps are well-defined and continuous. The map d0 is well definedsince H(m) ≤ H(d0(m)), for all m ∈ Γ. Indeed, by the definition of H(m) we have thatH(m) ≤H(d0(m)) if and only if

γ(G(m)) ≤ γ(G(d0(m))).

But it happens because G(m)↪ G(d0(m)) is an embedding. Let y ∈H(m) and h ∈H; then

d1(hyH(m)) = hy(γ′χ(m))H(d1(m)).

To see that d1 is well-defined we need to check that

h(γ′χ(m))H(d1(m)) = hy(γ′χ(m))H(d1(m)),

or equivalently that

y ∈ (γ′χ(m))H(d1(m))(γ′χ(m))−1 = (γ′χ(m))γ(G(d1(m)))(γ′χ(m))−1.

But according to Equation (4.1), this is true since y = γ(x), for some x ∈ G(m), andB1(x) ∈ G(d1(m)).

To verify that the maps d0 and d1 are continuous, observe first that since V (Γ)×H is closedin Γ×H , the restriction αV of α to V (Γ)×H is the quotient map αV ∶ V (Γ)×H → V (Γ) withrespect to the (restriction of the) equivalence relation ∼. Consider the commutative diagram

Γ ×H di //

α

��

V (Γ) ×H

αV

��

SC(G,Γ,H)di

// V (SC(G,Γ,H))

(i = 0,1), where d0(m,h) = (d0(m), h) and d1(m,h) = (d1(m), (γ′χ(m)h) (m ∈ Γ, h ∈ H).Since the topologies of SC(G,Γ,H) and V (SC(G,Γ,H)) are quotient topologies, to check thecontinuity of d0 and d1 it suffices to prove that the maps d0 and d1 are continuous. For d0 this is

53

obvious, and for d1 it follows from the continuity of γ′, χ and of the multiplication in H . Thiscompletes the definition of SC(G,Γ,H).

The profinite graph SC(G,Γ,H) is called the profinite C-standard graph of the graph of pro-C groups (G, π,Γ) with respect to H or the C-universal covering of the graph of pro-C groups(G, π,Γ) with respect to H .

We can define a natural continuous action of H on the graph SC(G,Γ,H) given by

h(h′H(m)) = hh′H(m)

(h,h′ ∈H,m ∈ Γ).

Definition 4.4.2 (C-standard graph). For a technical reason we will assume that C is extensionclosed (cf. Definition 2.1.14), so the definition of ΠC1(G,Γ) is independent of the G-transversalJ (cf. [1], Theorem 6.2.4, page 189). If

H = Π = ΠC1(G,Γ)

and (γ, γ′) = (ν, ν′) is the universal J-specialisation of the graph of pro-C groups (G, π,Γ) inΠ = ΠC1(G,Γ), we use instead the notation S = SC(G,Γ) rather than SC(G,Γ,Π), and refer to

S = SC(G,Γ) = ⊍m∈Γ

Π/Π(m)

(Π(m) = ν(G(m)),m ∈ Γ) as the C-standard graph (or C-universal graph) of the graph of pro-Cgroups (G, π,Γ).

Example 4.4.3 ([1], Example 6.3.1). Let Γ be a finite graph and (G,Γ) a graph of pro-C groupsover Γ. Using the notation of Example 4.3.7, the C-standard graph S = SC(G,Γ) has verticesand edges

V (S) = ⋃v∈V (Γ)Π/Π(v) and E(S) = ⋃e∈E(Γ)Π/Π(e)

and incidence maps

d0(gΠ(e)) = gΠ(d0(e)), d1(gΠ(e)) = gteΠ(d1(e)),

(g ∈ Π, e ∈ E(Γ)).

Before stating Lemma 4.4.5, we have a technical result:

Lemma 4.4.4. Let G be a profinite group, X a profinite G-space and σ ∶ G ×X → X be anaction of G on X with a ∈ G, s ∈X such that s′ = as = σ(a, s). Then Gs′ = aGsa−1.

Proof. Take an element g ∈ Gs′ . So g stabilises s′ and we have that gs′ = s′. By definition,s′ = as, so g(as) = as. The associativity of the action gives to us that g(as) = (ga)s, so(ga)s = as. Multiplying both sides by a−1 we have that (a−1ga)s = s. Therefore a−1ga stabilisess, as desired.

54

Lemma 4.4.5 ([1], Example 6.3.2). Assume that C is extension-closed. Let Π = ΠC1(G,Γ) andS = SC(G,Γ) be as above.

(a) The quotient space Π/S is Γ. Furthermore, the Π-stabiliser of a point gΠ(m) of S isgΠ(m)g−1.

(b) The map σ ∶ Γ→ S given byσ(m) = 1Π(m),

(m ∈ Γ) is a continuous section of p ∶ S → Γ, as a map of topological spaces.

(c) Assume that Γ is C-simply connected. Then the section σ defined in (b) is a monomorphismof profinite graphs; in particular, in this case Γ is embedded as a profinite subgraph of S.

(d) Assume that Γ admits a spanning profinite subgraph T which is C-simply connected. Thenthe map σ ∶ Γ→ S given by

σ(m) = 1Π(m),

(m ∈ Γ) is a fundamental 0-section of p ∶ S → Γ lifting T .

Proof. By the natural action, given g ∈ Π, we have that g(g′Π(m)) = gg′Π(m), and as Π(m) iscontinuously indexed by Γ, Π/S = Γ. The other statement of (a) follows from Lemma 4.4.4. Toprove (b), note that the map σ1 ∶ Γ → Γ ×Π given by σ1(m) = (m,1) (m ∈ Γ) is a continuoussection of the natural projection pr1 ∶ Γ × Π → Π. Let σ be the composition of σ1 and thequotient map

α ∶ Γ ×Π→ S,

i.e., σ(m) = 1Π(m) (m ∈ Γ).

Γ ×H pr1//

α

��

Γ

σ1{{

σ

}}S

Then σ is a continuous section of p, as maps of topological spaces, proving (b).Part (c) is a consequence of (d) when T = Γ. We next prove (d). To show that σ is a

fundamental 0-section of p lifting T , observe first that d0(1Π(m)) = 1Π(d0(m)) ∈ σ(T ) forevery m ∈ T , by Equation (4.3). Hence it suffices to show that the injection σ∣T ∶ T → S is amorphism of graphs. This follows from the fact that the function χ used in formula 4.4 can bechosen so that χ(m) = 1 if m ∈ T (see Theorem 3.4.8 and the definition of χ).

Let Σ be a profinite graph, G a profinite group, Γ = G/Σ be a finite graph and T a maximalsubtree of Γ . Construct a fundamental 0-section j ∶ Γ → Γ of the quotient morphism Γ → Γ

55

lifting T , and a fundamental 0-section δ ∶ Γ → Σ of the quotient morphism Σ → Γ lifting T .Observe that d1δ(m) and δd1(m) are in the same G-orbit; choose tm ∈ G such that

tmδd1(m) = d1δ(m), (4.5)

and note that tm = 1, form ∈ T . Similarly, for eachm ∈ Γ, we can take the map χ (cf. Definition3.3.10) and define a graph of pro-C groups (G,Γ) as follows: for m ∈ Γ, put

G(m) = Gδ(m),

the G-stabiliser of the element δ(m) under the action of G. The incidence morphisms B0 and B1

are defined as follows: on each fiber G(m), B0 is just the inclusion map

G(m)↪ G(d0(m)),

and B1 is the composition of an inclusion and conjugation by tm:

G(m) = Gδ(m) → G(d1(m)) = Gδd1(m) = Gt−1m δd1(m) = t−1mGδd1(m)tm,

x↦ t−1m xtm, (x ∈ G(m)) by Lemma 4.4.4 and Equation (4.5).

Letγ′ ∶ πC1 (Γ)→ G

be the unique continuous homomorphism such that γ′χ(m) = tm (m ∈ Γ) (see Corollary 3.4.4).On the other hand, the inclusion maps

G(m) = Gδ(m) ↪ G

(m ∈ Γ) determine a morphismγ ∶ G → G.

So (γ, γ′) is a J-specialisation of G in G, where J = j(Γ). Hence it induces a continuoushomomorphism

ϕ ∶ ΠC1(G,Γ)→ G,

such that ϕν = γ and ϕν′ = γ′, where (ν, ν′) is the universal J-specialisation of (G, π,Γ) inΠC1(G,Γ). We remark that this implies that ν is an injection on each G(m) (m ∈ Γ), i.e., that(G, π,Γ) is an injective graph of groups.

Theorem 4.4.6 ([1], Theorem 6.6.1). Let C be extension-closed. Suppose that a pro-C groupG acts on a C-simply connected profinite graph Σ so that the quotient graph Γ = G/Σ is finite.Construct a graph of pro-C groups (G,Γ) over Γ as above. Then the homomorphism

ϕ ∶ ΠC1(G,Γ)→ G

56

defined above is an isomorphism of profinite groups. Moreover, Σ is isomorphic to the standardgraph SC(G,Γ) of the graph of pro-C groups (G,Γ).

Proof. Define a mapΨ ∶ Γ ×G→ Σ

by Ψ(m,g) = gδ(m) (m ∈ Γ). Clearly Ψ is continuous and onto. It induces a continuous map

Ψ′ ∶ SC(G,Γ,G) = ⊍m∈Γ

G/G(m)→ Σ,

given by Ψ′(gG(m)) = gδ(m), which is a bijection, and so a homeomorphism. We claim thatΨ′ is an isomorphism of profinite graphs. To see this it remains to check that it is a morphismof graphs: if g ∈ G and m ∈ Γ, we have

Ψ′d0(gG(m)) = Ψ′(gG(d0(m))) = gδ(d0(m)) = d0(gδ(m)) = d0Ψ′(gG(m)),

and

Ψ′d1(gG(m)) = Ψ′(gγ′χ(m)G(d1(m))) = Ψ′(gtmG(d1(m))) = gtm(δd1(m))

= g(d1δ(m)) = d1(gδ(m)) = d1Ψ′(gG(m)).

This proves the claim.Since Σ is C-simply connected, so is SC(G,Γ,G). Therefore by [1], Theorem 6.3.7, the

homomorphism ϕ ∶ ΠC1(G,Γ) → G is an isomorphism, and SC(G,Γ,G) is isomorphic toSC(G,Γ). It follows that Σ and SC(G,Γ) are isomorphic.

4.5 The Kurosh Theorem for free pro-C products

In this last section we will prove the pro-C version of the Kurosh subgroup theorem, themain result of this thesis, by seeing the free pro-C product of a sheaf as a fundamental group ofa graph of pro-C groups (see Example 4.3.6).

Definition 4.5.1 (Double cosets). If H and K are subgroups and x is an element of a group G,the subset

HxK = {hxk ∣ h ∈H,k ∈K}

is called an (H,K)-double coset.

There is a partition of the group into double cosets:

Proposition 4.5.2. Let H and K be subgroups of a group G.

(a) The group G is a union of (H,K)-double cosets;

(b) Two (H,K)-double cosets are either equal or disjoint;

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(c) The double coset HxK is a union of the right cosets of H and a union of left cosets of K.

Proof. Define x ∼ y to mean that x = hyk for some h ∈ H and k ∈ K. It is easy to check that ∼is an equivalence relation on G, the equivalence class containing x being HxK. Thus (a) and(b) follow at once, (c) is clear.

Theorem 4.5.3 (Kurosh, [1], Theorem 7.3.1). Let C be an extension-closed pseudovariety offinite groups. Let G =∐n

i=1Gi be a free pro-C product of a finite number of pro-C groups Gi. IfH is an open subgroup of G, then

H =n

∐i=1

∐τ∈H/G/Gi


is a free pro-C product of groups H ∩ gi,τGig−1i,τ , where, for each i = 1,⋯, n, gi,τ ranges over

a system of representatives of the double cosets H/G/Gi, and F is a free pro-C group of finiterank rF ,

rF = 1 − t +n

∑i=1

(t − ti),

where t = [G ∶H] and ti = ∣H/G/Gi∣.

Proof. By Example 4.3.6 we can see G =∐ni=1Gi as the fundamental group of a finite graph of

pro-C groups (G,Γ). Indeed, let X be the finite set {1,2,⋯, n} and Γ be defined as follows:

• V (Γ) =X ⊍ {ω};

• E(Γ) = {ω} ×X = {(ω,x) ∣ x ∈X};

• d0(ω,x) = ω;

• d1(ω,x) = x (x ∈X),

where {ω} is a single point space disjoint from X and V (Γ), Γ = V (Γ) ⊍ E(Γ) are endowedwith the disjoint union topology and E(Γ) with the product topology. Hence we define thegraph of pro-C groups by G as the subset of Γ ×G consisting of those elements (m,y) ∈ T ×Gsuch that

⎧⎪⎪⎨⎪⎪⎩

y ∈ Gm, if m ∈X;

y = 1, if m ∈ {ω} ⊍E(Γ).

From Lemma 4.1.16 we have that G is a profinite space, and so (G, π,Γ) is a subsheaf over Γ

of the constant sheaf Γ ×G. In fact, (G, π,Γ) has the structure of a graph of pro-C groups withobvious morphisms B0 and B1.

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The following diagram represents this graph of groups:

1

Gx1

Gx2

Gx3

Gx

1 1

11

. ..

We denote the vertices of Γ by ω,1,⋯, n and its edges by (ω,1),⋯, (ω,n). The vertexgroups of (G,Γ) in this case are G(ω) = 1,G(1) = G1,⋯,G(n) = Gn; and the edge groupsG(ω, i) are all trivial.

Denote by ν ∶ G → G the canonical map. In this case ν is injective when restricted to eachG(m) (m ∈ Γ) because it is a morphism of sheaves over a singleton (cf. Example 4.1.5). Hencewe can make the identifications ν(G(i)) = G(i) = Gi (i = 1,⋯, n).

LetS = SC(G,Γ) = ⊍

m∈Γ

G/G(m)

be the standard graph of groups (G,Γ) (cf. Definition 4.4.2). Since Γ is finite and H is open inG (and so a closed subgroup of finite index in G), the quotient graph

Γ′ =H/S = ⊍m∈Γ

H/G/G(m)

is finite.Hence, we have the following diagram:

S

Γ = G/S Γ′ =H/S

δ

Let X = {1G(m) ∣ m ∈ Γ} be a subgraph of S. According to Lemma 4.4.5, X is a subtreeof S isomorphic to Γ. Hence X is isomorphic to the subtree

Y = {H1G(m) ∣m ∈ Γ}

of Γ′. Therefore, by Proposition 3.3.12 and Lemma 3.3.9, there exists a fundamental 0-sectionδ ∶ Γ′ → S such that

δ(H1G(m)) = 1G(m), (4.6)

for every m ∈ Γ. By Theorem 4.4.6, H = ΠC1(H,Γ′), where H(m′) = Hδ(m′), for m′ ∈ Γ′.Observe that in this case the edge groups of (H,Γ′) are trivial. Indeed, by Equation (4.6), if

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m′ ∈ E(Γ′) then H(m′) = Hδ(m′) and δ(m′) = 1G(m). But G(e) = 1 for all e ∈ E(Γ) and δ isa morphism by the definition of the fundamental 0-transversal; so (see Example 4.3.5)

H = ΠC1(H,Γ′) = ∐v∈V (Γ′)

H(v) ∐ πC1 (Γ′).

Fix m ∈ Γ; thenδ(H/G/G(m)) = {gm,τG(m) ∣ τ ∈ Im},

and {gm,τ ∣ τ ∈ Im} is a system of representatives of the double cosetsH/G/G(m); furthermore,by our choice of δ, one has that

1 ∈ {gm,τ ∣ τ ∈ Im}.

If m′ ∈H/G/G(m) ⊆ Γ′, then δ(m′) = gG(m), where g ∈ {gm,τ ∣ τ ∈ Im}, and

H(m′) =Hδ(m′) =HgG(m) =H ∩GgG(m).

By Proposition 4.4.4,H(m′) =H ∩GgG(m) =H ∩ gG(m)g−1,

since G(ω) = 1 and we are identifying G(i) with Gi when i = 1,⋯, n, we have

H =n

∐i=1

∐τ∈Ii

(H ∩ gi,τGig−1i,τ) ∐ F,

where F = πC1 (Γ′). Then F is a free pro-C group by Proposition 3.4.3, whose rank isrF = ∣Γ′∣ − ∣T ∣, where T is the maximal tree of Γ′. To compute this number, put t = [G ∶H] andti = ∣H/G/Gi∣. Then

∣Γ′∣ = (n + 1)t +n

∑i=1

ti.

Since T is a finite tree ∣T ∣ = 2∣V (T )∣ − 1 = 2∣V (Γ′)∣ − 1; so

∣T ∣ = 2(t +n

∑i=1

ti) − 1.

Therefore,

rF = 1 − t +n

∑i=1

(t − ti).

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Bibliography

[1] RIBES, L., ‘Profinite graphs and groups’, Ergebnisse der Mathematik undihrer Grenzgebiete. 3. Folge, A Series of Modern Surveys in Mathematics 66(Springer, Berlin, 2017).

[2] RIBES, L. and ZALESSKII, P. A., ’Profinite groups’, 2nd edn, Ergebnisse derMathematik und ihrer Grenzgebiete (3) 40 (Springer, Berlin-Heidelberg, NewYork, 2010).

[3] SERRE, J.-P., ‘Trees’, 1st english edition, (Springer, Berlin, 1980).

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