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2017 | Buch

Kapitel lesen Erstes Kapitel lesen

Profinite Graphs and Groups

verfasst von: Prof. Luis Ribes

Verlag: Springer International Publishing

Buchreihe : Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book offers a detailed introduction to graph theoretic methods in profinite groups and applications to abstract groups. It is the first to provide a comprehensive treatment of the subject.

The author begins by carefully developing relevant notions in topology, profinite groups and homology, including free products of profinite groups, cohomological methods in profinite groups, and fixed points of automorphisms of free pro-p groups. The final part of the book is dedicated to applications of the profinite theory to abstract groups, with sections on finitely generated subgroups of free groups, separability conditions in free and amalgamated products, and algorithms in free groups and finite monoids.

Profinite Graphs and Groups will appeal to students and researchers interested in profinite groups, geometric group theory, graphs and connections with the theory of formal languages. A complete reference on the subject, the book includes historical and bibliographical notes as well as a discussion of open questions and suggestions for further reading.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries

Abstract

For the reader’s convenience, this chapter contains a review of basic concepts and results that are frequently used throughout the book: inverse systems and inverse limits; profinite spaces and profinite groups; profinite free groups; free and amalgamated products; profinite rings and modules; the complete group algebra and complete tensor product; Ext and Tor functors; homology and cohomology of profinite groups; cohomological dimension and homological characterization of projective profinite groups and free pro-$p$ groups. This chapter does not contain proofs. These can be found, for example, in the monograph ‘Profinite Groups’ by L. Ribes and P. Zalesskii, 2nd edition, Springer 2010, which is cited as RZ in the book.

Luis Ribes

Basic Theory

Frontmatter

Chapter 2. Profinite Graphs

Abstract

This chapter contains the first notions about profinite graphs and their basic properties. A profinite graph is a graph in the usual sense, but it also has the structure of a topological space (compact, Hausdorff and totally disconnected). A finite graph with the discrete topology is a basic example of a profinite graph, and it is shown that every profinite graph can be expressed as an inverse limit of finite graphs. Most of the profinite graphs of interest in this book arise in connection with profinite groups (i.e., Galois groups), and they are usually infinite. One way of obtaining a profinite graph is by constructing the Cayley graph of a profinite group with respect to a compact subset.

Associated with a profinite graph there is a short sequence of profinite modules over a profinite ring, which depends on a class $\mathcal {C}$ of finite groups. Using such a sequence one develops the notion of ‘tree’ ($\mathcal {C}$-tree or $\pi$-tree, where $\pi$ is a set of prime numbers). It is proved that, for certain pseudovarieties $\mathcal {C}$, the Cayley graph of a free pro-$\mathcal {C}$ group is a $\mathcal {C}$-tree.

The chapter contains many examples that illustrate concepts and properties in profinite graphs.

Luis Ribes

Chapter 3. The Fundamental Group of a Profinite Graph

Abstract

This chapter contains the definition of the profinite fundamental group of a profinite graph and its properties. To make this precise there is a study of free actions of profinite groups on profinite graphs and their quotients (called Galois coverings). This leads to the concept of universal Galois $\mathcal{C}$-covering and $\mathcal{C}$-simply connected profinite graphs. The concept of $\mathcal{C}$-simple connectivity permits a characterization of freeness of a pro-$\mathcal{C}$ group in terms of Cayley graphs.

It is also shown that the class of the profinite fundamental groups of profinite graphs is precisely the class of projective profinite groups (the closed subgroups of free profinite groups).

The case of a finite graph $\varGamma$ is considered separately and more explicitly; in this case, it is shown that the profinite fundamental group of $\varGamma$ is the profinite completion of its fundamental group as an abstract graph; furthermore, it is proved that the universal covering graph of $\varGamma$ (as an abstract graph) is densely embedded in its universal Galois $\mathcal{C}$-covering graph (as a profinite graph).

Luis Ribes

Chapter 4. Profinite Groups Acting on -Trees

Abstract

The first section of this chapter is concerned with fixed points under the action of a pro-$\pi$ group acting on $\pi$-tree. In particular, it is proved that if a pro-$\pi$ group acts on a $\pi$-tree, the subset of fixed points is a $\pi$-subtree (if it is not empty) and that a finite $\pi$-group acting on a $\pi$-tree fixes a vertex. As a consequence it is shown that the smallest $\pi$-subtree $[v,w]$ containing two distinct vertices $v,w$ of a $\pi$-tree must contain edges. One also deduces that under some mild conditions, if a profinite group $G$ acts on a $\pi$-tree, then this tree contains a unique minimal $G$-invariant $\pi$-subtree. This is a very useful tool in many applications.

The second section contains a description of the structure of a pro-$\pi$ group that acts faithfully and irreducibly on a $\pi$-tree: it must have a nonabelian free pro-$p$ subgroup with an induced free action, or solvable of a very specific form. More generally, one has a description of the possible alternative structures of a pro-$\pi$ group that acts on a $\pi$-tree without fixed points: it contains a nonabelian pro-$p$ subgroup that acts freely or the quotient modulo the stabilizer of some edge is solvable of a special type.

Luis Ribes

Chapter 5. Free Products of Pro- Groups

Abstract

This is a central chapter in this book, and its results are frequently used throughout. The first section of the chapter contains a description a ‘sheaf of pro-$\mathcal{C}$ groups’. Using this one defines a pro-$\mathcal{C}$ group which is the free pro-$\mathcal{C}$ product of the (fibers of the) sheaf. For a more internal viewpoint, one introduces the concept of ‘a collection of subgroups of a pro-$\mathcal{C}$ group continuously indexed by a topological space (a profinite space)’: a prime example arises when one considers the stabilizers of a profinite group that acts on a profinite space. This allows us to describe when a pro-$\mathcal{C}$ group is the free pro-$\mathcal{C}$ product of some of its closed subgroups. After establishing the equivalence between the two viewpoints, external and internal, the chapter contains a large collection of basic properties of free products of pro-$\mathcal{C}$ groups.

The case when all the factors in the free product are isomorphic to each other (corresponding to ‘constant sheaves’) is studied separately.

One section of the chapter explores the relationship between the topological weight of a profinite group $G$ and the weight of a profinite space on which it acts, under appropriate conditions.

Luis Ribes

Chapter 6. Graphs of Pro- Groups

Abstract

This chapter contains a complete treatment of graphs of profinite groups $(\mathcal{G}, \varGamma)$ over profinite graphs $\varGamma$ (this is a certain way of associating pro-$\mathcal{C}$ groups $\mathcal{G} (m)$ to vertices and edges $m$ of $\varGamma$), their fundamental pro-$\mathcal{C}$ groups $\varPi(\mathcal{G}, \varGamma)$ and their standard (or universal covering) profinite graphs $S(\mathcal{G}, \varGamma)$. It is proved that the standard graph of a graph of pro-$\mathcal{C}$ groups is a $\mathcal{C}$-simply connected profinite graph. There are many examples dealing with the special cases of free pro-$\mathcal{C}$ products, amalgamated products of profinite groups, HNN extensions, etc.

For applications to properties in abstract groups, in this chapter there is a study of the connections between a graph of abstract groups $(\mathcal{G}, \varGamma)$ over a finite graph $\varGamma$ and a corresponding graph $(\bar{\mathcal{G}}, \varGamma)$ of profinite completions $\bar{\mathcal{G}}(m)$, for every $m\in \varGamma$. In some cases one can show that $\varPi(\bar{\mathcal{G}}, \varGamma)$ is a profinite completion of the abstract fundamental group $\varPi^{\mathrm{abs}}(\mathcal{G}, \varGamma)$ of $(\mathcal{G}, \varGamma)$, and that the universal covering tree $S^{\mathrm{abs}}(\mathcal{G}, \varGamma)$ is densely embedded in $S(\mathcal{G}, \varGamma)$. This is the case, for example, when dealing with free products of abstract residually finite groups, for graphs of finite groups or for certain types of amalgamated products, and then these connections can be used fruitfully in the study of some properties of the abstract fundamental groups $\varPi^{\mathrm{abs}}(\mathcal{G}, \varGamma)$.

Luis Ribes

Applications to Profinite Groups

Frontmatter

Chapter 7. Subgroups of Fundamental Groups of Graphs of Groups

Abstract

This chapter contains structural results about subgroups of fundamental groups $\varPi(\mathcal{G}, \varGamma)$ of graphs of profinite groups $(\mathcal{G}, \varGamma)$; these include free and amalgamated products of profinite groups, HNN extensions, etc. Unlike fundamental groups of graphs of abstract groups, where often one can use a combinatorial description of the elements of the fundamental group, to obtain structural results in the profinite case one relies exclusively on geometric methods, primarily analysis of actions of profinite groups on profinite trees. In particular there are results on finite subgroups, normalizers, normal subgroups, etc.

As an application, this chapter contains an analogue of the classical Kurosh subgroup theorem for the free product of abstract groups. It describes the structure of an open subgroup $H$ of a free pro-$\mathcal{C}$ product of pro-$\mathcal{C}$ groups as a free pro-$\mathcal{C}$ product of a free pro-$\mathcal{C}$ group and intersections of $H$ with conjugates of the free factors. The original free product may have a finite or infinite number of free factors.

Luis Ribes

Chapter 8. Minimal Subtrees

Abstract

Let $G$ be a group and let $T$ be a tree on which $G$ acts. This chapter deals with minimal $G$-invariant subtrees of $T$. This is done both when $G$ is an abstract group and $T$ an abstract tree, and when $G$ is a pro-$\mathcal{C}$ group and $T$ a $\mathcal{C}$-tree; in fact the interest lies in the study of both cases together and the relationship with each other. For example, attached to a finitely generated free-by-finite group $R$, there is a graph of finite groups $(\mathcal{G}, \Delta)$ over a finite graph $\Delta$, so that $R$ is its fundamental group; moreover $R$ acts naturally on the universal covering graph $S^{\mathrm{abs}}$ of this graph of groups. Let $b\in R$ act freely on $S^{\mathrm{abs}}$ (a hyperbolic element); then there is a unique minimal $\langle b\rangle$-invariant subtree $L_{b}$ of $S^{\mathrm{abs}}$ (called the ‘Tits line’ or the ‘axis’ of $b$). In parallel, the profinite completion $\hat{R}$ of $R$ is the profinite fundamental group of $(\mathcal{G}, \Delta)$, and $S^{\mathrm{abs}}$ is naturally densely embedded in the standard profinite tree $S$ of $(\mathcal{G}, \Delta)$; then it is proved that the closure $\bar{L}_{b}$ of $L_{b}$ in $S$ is precisely the unique minimal $\overline{\langle b\rangle}$-invariant subtree of $S$. Analogous results can be obtained for different types of groups $R$, such as free or amalgamated products of abstract groups. Knowledge about these sort of minimal subtrees can be used to obtain information about properties of $R$, such as conjugacy of elements or subgroups.

Luis Ribes

Chapter 9. Homology and Graphs of Pro- Groups

Abstract

In the first part of the chapter it is shown that if $\varLambda$ is a profinite ring and $M$ is a profinite $\varLambda$-module, then each of the functors $\mathrm{Tor}^{\varLambda}_{n}(M, -)$ commutes with the direct sum of any sheaf of $\varLambda$-modules. In particular, if $G$ is a pro-$\mathcal{C}$ group, each of its homology group functors $H_{n}(G, -)$ commutes with any direct sum $\bigoplus_{t}B_{t}$ of submodules of a $[\![ \varLambda G]\!]$-module $B$ indexed continuously by a profinite space, where $[\![ \varLambda G]\!]$ denotes the complete group algebra and $\varLambda$ is assumed to be commutative. On the other hand, if $\mathcal{F}= \{G_{t}\mid t\in T\}$ is a continuously indexed family of closed subgroups of $G$, there is a corestriction map of profinite abelian groups

$$\mathrm{Cor}^{\mathcal{F}}_{G}: \bigoplus_{t\in T} H_{n}(G_{t} ,B) \longrightarrow H_{n}(G, B), $$

for all profinite modules $B$ over $G$. Using this map one obtains a Mayer-Vietoris exact sequence associated with the action of a pro-$\mathcal{C}$ group $G$ on a $\mathcal{C}$-tree.

When $G$ is a pro-$p$ group, this chapter contains a theorem characterizing in terms of the corestriction map when $G$ is the free pro-$p$ product of a family of closed subgroups continuously indexed by a profinite space. Using this characterization one proves a Kurosh-type theorem describing the structure of second-countable pro-$p$ subgroups of a free pro-$\mathcal{C}$ product $H = \coprod_{z\in Z} H_{z}$, where $H$ is a pro-$\mathcal{C}$ group, and $\{H_{z}\mid z\in Z\}$ is a family of closed subgroups of $H$ continuously indexed by a profinite space $Z$.

Luis Ribes

Chapter 10. The Virtual Cohomological Dimension of Profinite Groups

Abstract

The chapter begins with a detailed description of tensor products of complexes of modules and the tensor product induction for a complex. It contains a theorem of Serre that asserts that torsion-free virtually free pro-$p$ groups are free pro-$p$, as well as an extension due to Scheiderer of this result when the group contains torsion. A second-countable pro-$p$ group with a free pro-$p$ subgroup of index $p$ is described as a free pro-$p$ product of a free pro-$p$ group and a continuously indexed family of groups of the form $H_{\tau}\times T_{\tau}$, where $H_{\tau}$ is free pro-$p$ and $T_{\tau}$ has order $p$.

The chapter also includes an example of a subgroup of a free product of pro-$p$ groups which does not admit a description along the lines of the classical Kurosh subgroup theorem. The last part of this chapter deals with the subgroup of fixed points $\mathrm{Fix}_{F}(\psi)$ of an automorphism $\psi$ of a free pro-$p$ group $F$: if the order of $\psi$ is a finite power of $p$, the rank of that subgroup is finite, and if the order of $\psi$ is prime to $p$, its rank is infinite.

Luis Ribes

Applications to Abstract Groups

Frontmatter

Chapter 11. Separability Conditions in Free and Polycyclic Groups

Abstract

This chapter and the next ones deal with abstract groups. The properties that are studied are stated in the language of the natural profinite (or, more generally, pro-$\mathcal{C}$) topology on an abstract group. For example subgroup separability (i.e., a finitely generated subgroup is the intersections of the subgroups of finite index that contain it) or conjugacy separability. The methods of proof use the geometric techniques developed in the previous chapters.

The second section of this chapter contains a classical theorem of Marshall Hall that says that if $H$ is a finitely generated subgroup of a free abstract group $\varPhi$, then $U= H*L$, where $U$ is a subgroup of finite index in $\varPhi$ and $L$ is some subgroup of $U$. It is shown that this is in fact equivalent to saying that $H$ is closed in the profinite topology of $\varPhi$. A corresponding result holds for other pro-$\mathcal{C}$ topologies, when $\mathcal{C}$ is an extension-closed pseudovariety of finite groups. One can then deduce that the profinite topology of a finitely generated subgroup $H$ of a free-by-finite abstract group $R$ is precisely the topology induced from the profinite topology of $R$. In Sect. 11.3 a more general result is proved: if $H_{1}, \dots, H_{n}$ is a finite collection of finitely generated closed subgroups of a free abstract group $\varPhi$ endowed with the pro-$\mathcal{C}$ topology, then the product $H_{1}\cdots H_{n}$ is a closed subset of $\varPhi$. The last section records properties of abstract polycyclic-by-finite groups; these groups serve as basic building blocks for the free constructions of abstract groups studied in later chapters.

Luis Ribes

Chapter 12. Algorithms in Abstract Free Groups and Monoids

Abstract

The first section of this chapter contains algorithms about subgroups of finite index of an abstract free group of finite rank, e.g., how to decide whether an element of the free group belongs to a given subgroup of finite index or whether a subgroup of finite index is open in a certain topology.

Given a prime number $p$, a free abstract group $\varPhi$ (with an explicit basis $\mathbf{B}=\{b_{1}, \dots, b_{n}\}$) endowed with the pro-$p$ topology, and a finitely generated subgroup $H$ (whose generators are given in terms of $\mathbf{B}$), the second section of this chapter describes an algorithm to find a finite set of generators (written in terms of $\mathbf{B}$) of the closure of $H$ in that topology. The main result in that section presents a general approach to describing the closure of $H$ in a pro-$\mathcal{C}$ topology of $\varPhi$ (when $\mathcal{C}$ is an extension-closed pseudovariety of finite groups); one deduces from this theorem what are the main difficulties that arise when trying to make such a description algorithmic.

The third part of the chapter contains several algorithms of interest in the theory of formal languages on a finite alphabet and in finite monoids; in particular, there is an algorithm that describes how to construct the kernel of a finite monoid, a problem posed by J. Rhodes.

Luis Ribes

Chapter 13. Abstract Groups vs Their Profinite Completions

Abstract

For an abstract group $R$ that is either free-by-finite or polycyclic-by-finite, in Chap. 13 one studies the relationship between certain constructions in $R$ (normalizers and centralizers of a finitely generated subgroup or the intersection of finitely generated subgroups) and corresponding constructions in the profinite completion $\hat{R}$ of $R$.

It is proved, for example, that if $H$ is a finitely generated subgroup of $R$, the topological closure $\overline{\mathrm{N}_{R}(H)}$ (in $\hat{R}$) of the normalizer $\mathrm{N}_{R}(H)$ of $H$ in $R$ coincides with $\mathrm{N}_{\hat{R}}(\bar{H})$, the normalizer in $\hat{R}$ of the closure of $H$ in $\hat{R}$. For finitely generated subgroups $H_{1}$ and $H_{2}$ of $R$, it is proved that $\overline {H_{1}\cap H_{2}}= \overline {H_{1}}\cap \overline {H_{2}}$.

In fact the results are obtained in greater generality for a free-by-$\mathcal{C}$ group, i.e., an extension of a free abstract group by a group in a pseudovariety of finite groups $\mathcal{C}$ (a collection of finite groups closed under subgroups, quotients and finite direct products), and instead of the profinite completion, one considers the pro-$\mathcal{C}$ completion.

Luis Ribes

Chapter 14. Conjugacy in Free Products and in Free-by-Finite Groups

Abstract

Let $\mathcal{C}$ be an extension-closed pseudovariety of finite groups (i.e, a class of finite groups closed under subgroups, quotients and extensions; e.g., the class of all finite groups). An abstract group $R$ is ‘conjugacy $\mathcal{C}$-separable’ if for any pair of elements $x,y\in R$, these elements are conjugate in $R$ if and only if their images in every finite quotient of $R$ which is in $\mathcal{C}$ are conjugate (there is an analogous property of ‘subgroup conjugacy $\mathcal{C}$-separability’, if one replaces elements with finitely generated subgroups). A subgroup $H$ of $R$ is said to be ‘conjugacy $\mathcal{C}$-distinguished’ if whenever $y\in R$, then $y$ has a conjugate in $H$ if and only if the same holds for the images of $y$ and $H$ in every quotient group $R/N\in \mathcal{C}$ of $R$.

In Chap. 14 it is shown that the properties of conjugacy $\mathcal{C}$-separability and subgroup conjugacy $\mathcal{C}$-separability are preserved by taking free products of abstract groups. It is also shown that an abstract free-by-$\mathcal{C}$ group (an extension of a free abstract group by a group in $\mathcal{C}$) is both conjugacy $\mathcal{C}$-separable and subgroup conjugacy $\mathcal{C}$-separable; in these groups every finitely generated pro-$\mathcal{C}$ closed subgroup is conjugacy $\mathcal{C}$-distinguished. The basic tools for proving these results are related to the study of minimal invariant subtrees developed in Chap. 8 for the actions of groups on trees.

Luis Ribes

Chapter 15. Conjugacy Separability in Amalgamated Products

Abstract

In this chapter one studies how conjugacy separability in abstract groups is preserved under the formation of certain free products with amalgamation. The main result shows that one can construct conjugacy separable groups by forming a free product amalgamating a cyclic subgroup of groups which are either finitely generated free-by-finite or polycyclic-by-finite; in fact one can iterate this process to obtain new conjugacy separable groups; in particular residually finite groups.

In addition to conjugacy separability one considers in this chapter a whole array of other properties that are preserved by constructing amalgamated free products of abstract groups with cyclic amalgamation, if one makes certain basic assumptions on the factors of the amalgamated free product.

The main tools in most results in this chapter are related to the action of certain abstract groups on abstract trees and the action of certain profinite groups on profinite trees, and their inter-connections. In most cases in this chapter the pertinent groups are amalgamated free products and their profinite completions, and the pertinent trees and profinite trees are those canonically associated with amalgamated free products.

Luis Ribes

Backmatter

Titel: Profinite Graphs and Groups
verfasst von: Prof. Luis Ribes
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-61199-0
Print ISBN: 978-3-319-61041-2
DOI: https://doi.org/10.1007/978-3-319-61199-0