Related papers: Decision problems and profinite completions of gro…

We address the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative…

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…

We show that the conjugacy problem is solvable in [finitely generated free]-by-cyclic groups, by using a result of O. Maslakova that one can algorithmically find generating sets for the fixed subgroups of free group automorphisms, and one…

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…

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…

In this article, we study connections between representation theory and efficient solutions to the conjugacy problem on finitely generated groups. The main focus is on the conjugacy problem in conjugacy separable groups, where we measure…

Let $G$ be a classical group defined over a finite field. We consider the following fundamental problems concerning conjugacy in $G$: 1. List a representative for each conjugacy class of $G$. 2. Given $x \in G$, describe the centralizer of…

As it follows from G\"odel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually, this fundamental principle can be efficiently applicable in…

For a group $G$ and $R=\mathbb Z,\mathbb Z/p,\mathbb Q$ we denote by $\hat G_R$ the $R$-completion of $G.$ We study the map $H_n(G,K)\to H_n(\hat G_R,K),$ where $(R,K)=(\mathbb Z,\mathbb Z/p),(\mathbb Z/p,\mathbb Z/p),(\mathbb Q,\mathbb…

Let $\mathcal G$ denote the space of finitely generated marked groups. For any finitely generated group $G$, we construct a continuous, injective map $f$ from the space of subgroups $Sub(G)$ to $\mathcal G$ that sends conjugate subgroups to…

We exhibit classes of groups in which the word problem is uniformly solvable but in which there is no algorithm that can compute finite presentations for finitely presentable subgroups. Direct products of hyperbolic groups, groups of…

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…

In this paper we study the conjugacy problem in polycyclic groups. Our main result is that we construct polycyclic groups $G_n$ whose conjugacy problem is at least as hard as the subset sum problem with $n$ indeterminates. As such, the…

A topological group G is profinite if it is compact and totally disconnected. Equivalently, G is the inverse limit of a surjective system of finite groups carrying the discrete topology. We discuss how to represent a countably based…

We construct an infinite finitely generated recursively presented residually finite algorithmically finite group $G$ answering thereby a question of Myasnikov and Osin. Moreover, $G$ is "very infinite" and "very algorithmically finite" in…

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…

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…

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 study the realization problem of finite groups as the group of homotopy classes of self-homotopy equivalences of finite spaces. Let $G$ be a finite group. Using an infinite family of pairwise non weakly homotopic asymmetric spaces we…

We construct a class of finitely generated groups which have arbitrarily large conjugacy separability function, but in which the conjugacy problem can be solved in polynomial time, demonstrating that the McKinsey algorithm for the conjugacy…