Related papers: Absolute profinite rigidity, direct products, and …
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…
We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group…
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…
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…
We initiate the study of profinite rigidity for modules over a Noetherian domain: to what extent are these objects determined by their finite images? We establish foundational statements in analogy to classical results in the category of…
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…
We prove that any compact, orientable 3-manifold with empty or toral boundary is profinitely almost rigid among all compact, orientable 3-manifolds. In other words, the profinite completion of its fundamental group determines its…
We construct finitely generated groups with strong fixed point properties. Let $\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first…
A profinite group equipped with an expansive endomorphism is equivalent to a one-sided group shift. We show that these groups have a very restricted structure. More precisely, we show that any such group can be decomposed into a finite…
We introduce a class of countable groups by some abstract group-theoretic conditions. It includes linear groups with finite amenable radical and finitely generated residually finite groups with some non-vanishing $\ell^2$-Betti numbers that…
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).…
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…
We introduce the classes of TAP groups, in which various types of algebraic fibring are detected by the non-vanishing of twisted Alexander polynomials. We show that finitely presented LERF groups lie in the class $\mathsf{TAP}_1(R)$ for…
We prove that every finite direct product of crystallographic groups arising from an irreducible root system (in the sense of Lie theory) is profinitely rigid (equiv. first-order rigid). This is a generalization of recent proofs of…
We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite…
A locally compact group $G$ is a cocompact envelope of a group $\Gamma$ if $G$ contains a copy of $\Gamma$ as a discrete and cocompact subgroup. We study the problem that takes two finitely generated groups $\Gamma,\Lambda$ having a common…
In this paper we study obstructions to presentability by products for finitely generated groups. Along the way we develop both the concept of acentral subgroups, and the relations between presentability by products on the one hand, and…
We prove that finitely generated free metabelian groups $\Psi_n$ are profinitely rigid in the absolute sense: they are distinguished by their finite quotients among all finitely generated residually finite groups. The proof is based on a…
For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we…
We say that a group $G$ is of \textit{profinite type} if it can be realized as a Galois group of some field extension. Using Krull's theory, this is equivalent to the ability of $G$ to be equipped with a profinite topology. We also say that…