Related papers: Profinite genus of free products with finite amalg…

Let $G$ be the fundamental group of a graph of finitely generated virtually free groups with virtually cyclic edge groups. We shaw that $G$ is cohomologically good if $G$ is residually finite. If $G$ is LERF, we prove that G splits…

We study the profinite genus of HNN-extensions whose associated subgroups are finite. We give precise formulas for the number of isomorphism classes of HNN(G,H,K,t,f) and of its profinite completion and compute the profinite genus of such…

Given an arbitrary, finitely presented, residually finite group $\Gamma$, one can construct a finitely generated, residually finite, free-by-free group $M_\Gamma = F_\infty\rtimes F_4$ and an embedding $M_\Gamma \hookrightarrow (F_4\ast…

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…

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…

We investigate accessible subgroups of a profinite group $G$, i.e. subgroups $H$ appearing as vertex groups in a graph of profinite groups decomposition of $G$ with finite edge groups. We prove that any accessible subgroup $H \leq G$ arises…

Can one detect free products of groups via their profinite completions? We answer positively among virtually free groups. More precisely, we prove that a subgroup of a finitely generated virtually free group $G$ is a free factor if and only…

Let $S$ be either a free group or the fundamental group of a closed hyperbolic surface. We show that if $G$ is a finitely generated residually-$p$ group with the same pro-$p$ completion as $S$, then two-generated subgroups of $G$ are free.…

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…

Let $(\mathcal{G},\Gamma)$ be an abstract graph of finite groups. If $\Gamma$ is finite, we can construct a profinite graph of groups in a natural way $(\hat{\mathcal{G}},\Gamma)$, where $\hat{\mathcal{G}}(m)$ is the profinite completion of…

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…

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 $\Ga_2$…

In this article we calculate the profinite genus of the fundamental group of a $n$-dimensional compact flat manifold $X$ with holonomy group of prime order. As consequence, we prove that if $n\leq 21$, then $X$ is determined among all…

We prove that there exist finitely presented, residually finite groups that are profinitely rigid in the class of all finitely presented groups but not in the class of all finitely generated groups. These groups are of the form $\Gamma…

A formula is given for the profinite genus of groups of the form $\mathbb{Z}^n \rtimes C_{p^2}$, completing the calculation of the size of the genus of semidirect products of the form $\mathbb{Z}^n \rtimes G$ where $G$ is a finite $p$-group…

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…

We study in detail the profinite group G arising as geometric \'etale iterated monodromy group of an arbitrary quadratic polynomial over a field of characteristic different from two. This is a self-similar closed subgroup of the group of…

For $S=S_{g,n}$ a closed orientable differentiable surface of genus $g$ from which $n$ points have been removed, such that $\chi(S)=2-2g-n<0$, let $\mathrm{P}\Gamma(S)$ be the pure mapping class group of $S$ and…

We present a construction that yields infinite families of non-isomorphic semidirect products $N \rtimes F_m$ sharing a specified profinite completion. Within each family, $m \ge 2$ is constant and $N$ is a fixed group. For $m=2$ we can…

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).…