We consider finitely presented,residually finite groups $G$ and finitely generated normal subgroups $A$ such that the inclusion $A\hookrightarrow G$ induces an isomorphism from the profinite completion of $A$ to a direct factor of the profinite completion of $G$. We explain why $A$ need not be a direct factor of a subgroup of finite index in $G$; indeed $G$ need not have a subgroup of finite index that splits as a non-trivial direct product. We prove that there is no algorithm that can determine whether $A$ is a direct factor of a subgroup of finite index in $G$.
We fix a finitely presented group $Q$ and consider short exact sequences $1\to N\to G\to Q\to 1$ with $G$ finitely generated. The inclusion $N\to G$ induces a morphism of profinite completions $\hat N\to \hat G$. We prove that this is an isomorphism for all $N$ and $G$ if and only if $Q$ is super-perfect and has no proper subgroups of finite index. We prove that there is no algorithm that, given a finitely presented, residually finite group $G$ and a finitely presentable subgroup $P\subset G$, can determine whether or not $\hat P\to\hat G$ is an isomorphism.
We introduce and investigate a class of profinite groups defined via extensions of centralizers analogous to the extensively studied class of finitely generated fully residually free groups, that is, limit groups (in the sense of Z. Sela). From the fact that the profinite completion of limit groups belong to this class, results on their group-theoretical structure and homological properties are obtained.
We construct an uncountable family of 3-generated residually finite just-infinite groups with isomorphic profinite completions. We also show that word growth rate is not a profinite property.
We present two uncountable families of finitely generated residually finite groups all having the same profinite completion. One consists of soluble groups, the other of branch groups.
We show that property (T) is not profinite, that is, we construct two finitely generated residually finite groups which have isomorphic profinite completions while one admits property (T) and the other does not. This settles a question raised by M. Kassabov.
In this note we show that various (geometric/homological) finiteness properties are not profinite properties. For example for every $1 \le k, \ell \le \bbn$, there exist two finitely generated residually finite groups $\Ga_1$ and $\Ga_2$ with isomorphic profinite completions, such that $\Ga_1$ is strictly of type $F_k$ and $\Ga_2$ of type $F_\ell$.
In this paper we prove that the profinite completion $\mathcal{\hat G}$ of the Grigorchuk group $\mathcal{G}$ is not finitely presented as a profinite group. We obtain this result by showing that $H^2(\mathcal{\hat G},\field{F}_2)$ is infinite dimensional. Also several results are proven about the finite quotients $\mathcal{G}/ St_{\mathcal{G}}(n)$ including minimal presentations and Schur Multipliers.
We investigate the ability of a free pro-$\CC$ group of infinite rank to abstractly solve abstract embedding problems, and conclude that for some varieties $\CC$, the profinite completion of any order, of a free pro-$\CC$ group of infinite rank, is a free pro-$\CC$ group as well, of the corresponding rank.
We investigate the profinite completions of a certain family of groups acting on trees. It turns out that for some of the groups considered, the completions coincide with the closures of the groups in the full group of tree automorphisms. However, we introduce an infinite series of groups for which that is not so, and describe the kernels of natural homomorphisms of the profinite completions onto the aforementioned closures of respective groups.
We introduce a class $\A$ of finitely generated residually finite accessible groups with some natural restriction on one-ended vertex groups in their JSJ-decompositions. We prove that the profinite completion of groups in $\A$ almost detects its JSJ-decomposition and compute the genus of free products of groups in $\A$.
The class of multi-EGS groups is a generalisation of the well-known Grigorchuk-Gupta-Sidki (GGS-)groups. Here we classify branch multi-EGS groups with the congruence subgroup property and determine the profinite completion of all branch multi-EGS groups. Additionally our results show that branch multi-EGS groups are just infinite.
We define, for any group $G$, finite approximations ; with this tool, we give a new presentation of the profinite completion $\hatπ : G \to \hat{G}$ of an abtract group $G$. We then prove the following theorem : if $k$ is a finite prime field and if $V$ is a $k$-vector space, then, there is a natural isomorphism between $\hat{V}$ (for the underlying additive group structure) and the additive group of the double-dual $V^{**}$. This theorem gives counter-examples concerning the iterated profinite completions of a group. These phenomena don't occur in the topological case.
This article explores the interplay between the finite quotients of finitely generated residually finite groups and the concept of amenability. We construct a finitely generated, residually finite, amenable group $A$ and an uncountable family of finitely generated, residually finite non-amenable groups all of which are profinitely isomorphic to $A$. All of these groups are branch groups. Moreover, picking up Grothendieck's problem, the group $A$ embeds in these groups such that the inclusion induces an isomorphism of profinite completions. In addition, we review the concept of uniform amenabil…
We study homological approximations of the profinite completion of a limit group (see Thm.~A) and obtain the analogous of Bridson and Howie's Theorem for the profinite completion of a non-abelian limit group (see Thm.~B).
Let $G$ be a group and let $K$ be a commensurated subgroup of $G$. Then there is a totally disconnected, locally compact (t.d.l.c.) group $\hat{G}_K$ that contains the profinite completion of $K$ as an open compact subgroup and also contains $G$ (modulo the finite residual of $K$) as a dense subgroup. Moreover, given an arbitrary group $G$, then every t.d.l.c. group containing an image of $G$ as a dense subgroup can be realised as a quotient of $\hat{G}_K$ for some commensurated subgroup $K$.
We consider coincidence Reidemeister zeta functions for tame endomorphism pairs of nilpotent groups of finite rank, shedding new light on the subject by means of profinite completion techniques. In particular, we provide a closed formula for coincidence Reidemeister numbers for iterations of endomorphism pairs of torsion-free nilpotent groups of finite rank, based on a weak commutativity condition, which derives from simultaneous triangularisability on abelian sections. Furthermore, we present results in support of a Pólya-Carlson dichotomy between rationality and a natural boundary for the an…
Let $(\mathcal{G},Γ)$ be an abstract graph of finite groups. If $Γ$ is finite, we can construct a profinite graph of groups in a natural way $(\hat{\mathcal{G}},Γ)$, where $\hat{\mathcal{G}}(m)$ is the profinite completion of $\mathcal{G}(m)$ for all $m \in Γ$. The main reason for this is that $Γ$ is finite, so it is already profinite. In this paper we deal with the infinite case, by constructing a profinite graph $\overlineΓ$ where $Γ$ is densely embedded and then defining a profinite graph of groups $(\widehat{\mathcal{G}},\overlineΓ)$. We also prove that the fundamental group $Π_1(\widehat{…
Let $\C$ be a variety of finite groups. We use profinite Bass--Serre theory to show that if $u:H\hookrightarrow G$ is a map of finitely generated residually $\C$ groups such that the induced map $\hat{u}:\hat{H}\rightarrow\hat{G}$ is a surjection of the pro-$\C$ completions, and $G$ has more than one end, then $H$ has the same number of ends as $G$. However if $G$ has one end the number of ends of $H$ may be larger; we observe cases where this occurs for $\C$ the class of finite $p$-groups. We produce a monomorphism of groups $u:H\hookrightarrow G$ such that: either $G$ is hyperbolic but not r…
Profinite genus of HNN-extensions with finite associated subgroups
Federal University of Viçosa（维索萨联邦大学）；ICT, Federal University of the Jequitinhonha and Mucuri Valleys（Jequitinhonha和Mucuri山谷联邦大学ICT学院）；Department of Mathematics, University of Brasilia（巴西利亚大学数学系）
We prove that if a Cartesian product of alternating groups is topologically finitely generated, then it is the profinite completion of a finitely generated residually finite group. The same holds for Cartesian producs of other simple groups under some natural restrictions.
We present novel constructions concerning the homology of finitely generated groups. Each construction draws on ideas of Gilbert Baumslag. There is a finitely presented acyclic group $U$ such that $U$ has no proper subgroups of finite index and every finitely presented group can be embedded in $U$. There is no algorithm that can determine whether or not a finitely presentable subgroup of a residually finite, biautomatic group is perfect. For every recursively presented abelian group $A$ there exists a pair of groups $i:P_A\hookrightarrow G_A$ such that $i$ induces an isomorphism of profinite c…
Using quantum representations of mapping class groups we prove that profinite completions of Burnside-type surface group quotients are not virtually prosolvable, in general. Further, we construct infinitely many finite simple characteristic quotients of surface groups.
We consider pairs of finitely presented, residually finite groups $P\hookrightarrow\G$ for which the induced map of profinite completions $\hat P\to \hat\G$ is an isomorphism. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not $P$ is isomorphic to $\G$. We construct pairs for which the conjugacy problem in $\G$ can be solved in quadratic time but the conjugacy problem in $P$ is unsolvable. Let $\mathcal J$ be the class of super-perfect groups that have a compact classifying space and no proper subgroups of finite index. We prove that there does…
We completely describe the finitely generated pro-$p$ subgroups of the profinite completion of the fundamental group of an arbitrary $3$-manifold. We also prove a pro-$p$ analogue of the main theorem of Bass--Serre theory for finitely generated pro-$p$ groups.
Let R be any ring (with 1), Γa group and RΓthe corresponding group ring. Let H be a subgroup of Γof finite index. Let M be an RΓ-module, whose restriction to RH is projective. Moore's conjecture: Assume for every nontrivial element x in Γ, at least one of the following two conditions holds: M1) the subgroup generated by x intersects H non-trivially (in particular this holds if Γis torsion free). M2) ord(x) is finite and invertible in R. Then M is projective as an RΓ-module. More generally, the conjecture has been formulated for crossed products R*Γand even for strongly graded rings R(Γ). We pr…
We consider pairs of finitely presented, residually finite groups $u:P\hookrightarrow Γ$. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not the associated map of profinite completions $\hat{u}: \widehat{P} \to \widehatΓ$ is an isomorphism. Nor do there exist algorithms that can decide whether $\hat{u}$ is surjective, or whether $\widehat{P}$ is isomorphic to $\widehatΓ$.
We show that a regular isomorphism of profinite completion of the fundamental groups of two 3-manifolds $N_1$ and $N_2$ induces an isometry of the Thurston norms and a bijection between the fibered classes. We study to what extent does the profinite completion of knot groups distinguish knots and show that it distinguishes each torus knot and the figure eight knot among all knots. We show also that it distinguishes between hyperbolic knots with cyclically commensurable complements under the assumption that their Alexander polynomials have at least one zero which is not a root of unity.
We define cofinite graphs and cofinite groupoids in a unified way that extends the notion of cofinite groups introduced by Hartley. The common underlying structure of all these objects is that they are directed graphs endowed with a certain type of uniform structure, that we call a cofinite uniformity. We begin by exploring the fundamental theory of cofinite directed graphs in full generality. The general theory turns out to be almost completely analogous to that of cofinite groups. For instance, every cofinite directed graph has a unique completion which is a com- pact cofinite direct graph. …
We construct the first example of a finitely-presented, residually-finite group that contains an infinite sequence of non-isomorphic finitely-presented subgroups such that each of the inclusion maps induces an isomorphism of profinite completions.
Suzuki recently gave constructions of non-discrete examples of locally compact C*-simple groups and Raum showed C*-simplicity of the relative profinite completions of the Baumslag-Solitar groups by using Suzuki's results. We extend this result to some fundamental groups of graphs of groups called generalized Baumslag-Solitar groups. In this article, we focus on some sufficient condition to show that these locally compact groups are C*-simple and that KMS-weights of these reduced group C*-algebras are unique. This condition is an analogue of the Powers averaging property of discrete groups and …
The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser--Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco--Shalen--Johannson decomposition of $M$.
In this paper we prove that RAAGs are distinguished from each other by their pro-$p$ completions for any choice of prime $p$, and that RACGs are distinguished from each other by their pro-2 completions. We also give a new proof that hyperbolic virtually special groups are good in the sense of Serre. Furthermore we give an example of a property of the underlying graph of a RAAG that translates to a property of the profinite completion.
Using conjugation of Shimura varieties, we produce nonisomorphic, cocompact, torsion-free lattices in $\mathrm{PU}(n,1)$ with isomorphic profinite completions for all $n \ge 2$. This disproves a conjecture of D. Kazhdan and gives the first examples nonisomorphic lattices in a semisimple Lie group of real rank one with isomorphic profinite completions, answering two questions of A. Reid.
We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C_1,...,C_n are finite cyclic groups then there exists a finite group G which is generated by isomorphic copies of C_1,...,C_n and the minimal number of generators of G is n.
Assume $G$ is a polycyclic group and $ϕ:G\to G$ an endomorphism. Let $G\ast_ϕ$ be the ascending HNN extension of $G$ with respect to $ϕ$; that is, $G\ast_ϕ$ is given by the presentation $$G\ast_ϕ= < G, t \ |\ t^{-1}gt = ϕ(g)\ \{for all}\ g\in G >.$$ Furthermore, let $\hat{G\ast_ϕ}$ be the profinite completion of $G\ast_ϕ$. We prove that, for any finite discrete $\hat{G\ast_ϕ}$-module $A$, the map $H^*(\hat{G\ast_ϕ}, A)\to H^*(G\ast_ϕ,A)$ induced by the canonical map $G\ast_ϕ\to \hat{G\ast_ϕ}$ is an isomorphism.
Rheinische Friedrich-Wilhelms-Universität Bonn（波恩大学）；Heinrich-Heine-University Düsseldorf（杜塞尔多夫大学）；University of Münster（明斯特大学）；University of Oxford（牛津大学）
There has been much recent interest into those properties of a 3-manifold determined by the profinite completion of its fundamental group. In this paper we give readily computable criteria specifying precisely when two orientable graph manifold groups have isomorphic profinite completions. Our results also distinguish graph manifolds among the class of all 3-manifolds and give information about the structure of totally hyperbolic manifolds, and give control over the pro-$p$ completion of certain graph manifold groups.
For any positive integer $n$, $\mathcal{A}_n$ is the class of all groups $G$ such that, for $0\leq i\leq n$, $H^i(\hat{G},A)\cong H^i(G,A)$ for every finite discrete $\hat{G}$-module $A$. We describe certain types of free products with amalgam and HNN extensions that are in some of the classes $\mathcal{A}_n$. In addition, we investigate the residually finite groups in the class $\mathcal{A}_2$.
We construct metric spaces that do not have property A yet are coarsely embeddable into the Hilbert space. Our examples are so called warped cones, which were introduced by J. Roe to serve as examples of spaces non-embeddable into a Hilbert space and with or without property A. The construction provides the first examples of warped cones combining coarse embeddability and lack of property A. We also construct warped cones over manifolds with isometrically embedded expanders and generalise Roe's criteria for the lack of property A or coarse embeddability of a warped cone. Along the way, it is p…
Profinite detection of free products and free factors
Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM（数学科学研究所）；Departamento de Matemáticas, Universidad Autónoma de Madrid（马德里自治大学数学系）；Departamento de Matemática, Universidade de Brasília（巴西利亚大学数学系）
Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its profinite completion. We show that for a wide class of S-arithmetic groups, this map is finite to one, while the the fibers are of unbounded size.
Surface groups are determined among limit groups by their profinite completions. As a corollary, the set of surface words in a free group is closed in the profinite topology.
We exhibit infinitely many pairs of non-isomorphic finitely presented, residually finite groups $Δ$ and $Γ$ with $Δ$ having Property FA, $Γ$ having a non-trivial action on a tree and $Δ$ and $Γ$ having isomorphic profinite completions.
We prove that the profinite completion of the fundamental group of a compact 3-manifold $M$ satisfies a Tits alternative: if a closed subgroup $H$ does not contain a free pro-$p$ subgroup for any $p$, then $H$ is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of the fundamental group of a closed, hyperbolic 3-manifold does not contain a subgroup isomorphic to $\hat{\mathbb{Z}}^2$. This gives a profinite characterization of hyperbolicity among irreducible 3-manifolds. We also characterize Seifert fibred 3-manifolds as precisely those for wh…
