Profinite completions of products

We show that the profinite completions and canonical extensions of bounded distributive lattices and of Boolean algebras coincide. We characterize dual spaces of canonical extensions of bounded distributive lattices and Heyting algebras in terms of Nachbin order-compactifications. We give the dual description of the profinite completion $\widehat{H}$ of a Heyting algebra H, and characterize the dual space of $\widehat{H}$ . We also give a necessary and sufficient condition for the profinite completion of H to coincide with its canonical extension, and provide a new criterion for a variety V of Heyting algebras to be finitely generated by showing that V is finitely generated if and only if the profinite completion of every member of V coincides with its canonical extension. From this we obtain a new proof of a well-known theorem that every finitely generated variety of Heyting algebras is canonical.

In this paper, we show that any finitely generated abelian-by-finite group is an elementary submodel of its profinite completion. It follows that two finitely generated abelian-by-finite groups are elementarily equivalent if and only if they have the same finite images. We give an example of two finitely generated abelian-by-finite groups G , H G,H which satisfy these properties while G × Z G \times {\bf {Z}} and G × Z G \times {\bf {Z}} are not isomorphic. We also prove that a finitely generated nilpotent-by-finite group is elementarily equivalent to its profinite completion if and only if it is abelian-by-finite.

We consider the problem of extending maps from algebras to their profinite completions in finitely generated quasivarieties. Our developments are based on the construction of the profinite completion of an algebra as its natural extension. We provide an extension which is a multi-map and we study its continuity properties, and the conditions under which it is a map.

Arithmetic groups with isomorphic finite quotients

Profinite groups, profinite completions and a conjecture of Moore

An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this question completely for closed orientable Seifert fibre spaces. In particular, all Seifert fibre spaces are distinguished from each other by their profinite completions apart from some previously-known examples due to Hempel. We also characterize when bounded Seifert fibre space groups have isomorphic profinite completions, given some conditions on the boundary.

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 if its closure in the profinite completion of $G$ is a profinite free factor. This generalises results by Parzanchevski and Puder for free groups that were later also proved by Wilton. Our methods are entirely different to theirs, combining homological properties of profinite groups and the decomposition theory of Dicks and Dunwoody.

We prove that the profinite completion construction is a covariant functor from the category of (universal) algebras of a given type into the category of the corresponding Stone algebras. A Grothendieck problem for finitely presented MV-algebras is also formulated and solved. Finally, we characterize finitely presented MV-algebras for which profinite completions and MacNeille completions coincide.

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 amenability, a strengthening of amenability introduced in the 70s, and we prove that uniform amenability is indeed detectable from the profinite completion.

Given an arbitrary, finitely presented, residually finite group Γ, one can construct a finitely generated, residually finite, free-by-free group $M_\Gamma = F_\infty\rtimes F_4$ and an embedding $M_\Gamma \hookrightarrow (F_4\ast \Gamma)\times F_4$ that induces an isomorphism of profinite completions. In particular, there is a free-by-free group whose profinite completion contains $\widehat{\Gamma}$ as a retract.

Canonical extensions of (bounded) lattices have been extensively studied, and the basic existence and uniqueness theorems for these have been extended to general posets. This paper focuses on the intermediate class ${{\boldsymbol{\mathcal{S}}}}_{\wedge}$ of (unital) meet semilattices. Any ${\mathbf S}\in {{\boldsymbol{\mathcal{S}}}}_{\wedge}$ embeds into the algebraic closure system Filt(Filt(S)). This iterated filter completion, denoted Filt2(S), is a compact and ${\textstyle{\bigvee}\,}{\textstyle{\bigwedge}\,}$ -dense extension of S. The complete meet-subsemilattice S δ of Filt2(S) consisting of those elements which satisfy the condition of ${\textstyle{\bigwedge}\,}{\textstyle{\bigvee}\,}$ -density is shown to provide a realisation of the canonical extension of S. The easy validation of the construction is independent of the theory of Galois connections. Canonical extensions of bounded lattices are brought within this framework by considering semilattice reducts. Any S in ${{\boldsymbol{\mathcal{S}}}}_{\wedge}$ has a profinite completion, ${\rm Pro}_{{{\boldsymbol{\mathcal{S}}}}_{\wedge}}({\mathbf S})$ . Via the duality theory available for semilattices, ${\rm Pro}_{{{\boldsymbol{\mathcal{S}}}}_{\wedge}}({\mathbf S})$ can be identified with Filt2(S), or, if an abstract approach is adopted, with ${\mathbb F_{\sqcup}}({\mathbb F_{\sqcap}}({\mathbf S}))$ , the free join completion of the free meet completion of S. Lifting of semilattice morphisms can be considered in any of these settings. This leads, inter alia, to a very transparent proof that a homomorphism between bounded lattices lifts to a complete lattice homomorphism between the canonical extensions. Finally, we demonstrate, with examples, that the profinite completion of S, for ${\mathbf S} \in {{\boldsymbol{\mathcal{S}}}}_{\wedge}$ , need not be a canonical extension. This contrasts with the situation for the variety of bounded distributive lattices, within which profinite completion and canonical extension coincide.

It is known that every morphism $\varphi : F \rightarrow F'$ between free groups of pseudovariety $ \mathbf{V}$ of finite groups is uniformly continuous when both groups are equipped with their respective pro-$ \mathbf{V}$ topologies. In this paper we prove that this morphism can be uniquely extended to a continuous morphism between their pro-$ \mathbf{V}$ completions $\hat{\varphi} : \hat{F} \rightarrow \hat{F'}$? 2000 Mathematics Subject Classification. 20E18; 20E05

This paper is based on a series of 4 lectures delivered at Groups St Andrews 2013. The main theme of the lectures was distinguishing finitely generated residually finite groups by their finite quotients. The purpose of this paper is to expand and develop the lectures. The paper is organized as follows. In §2 we collect some questions that motivated the lectures and this article, and in §3 discuss some examples related to these questions. In §4 we recall profinite groups, profinite completions and the formulation of the questions in the language of the profinite completion. In §5, we recall a particular case of the question of when groups have the same profinite completion, namely Grothendieck’s question. In §6 we discuss how the methods of L 2 -cohomology can be brought to bear on the questions in §2, and in §7, we give a similar discussion using the methods of the cohomology of profinite groups. In §8 we discuss the questions in §2 in the context of groups arising naturally in low-dimensional topology and geometry, and in §9 discuss parafree groups. Finally in §10 we collect a list of open problems that may be of interest.

Profinite genus of free products with finite amalgamation

Torsion in profinite completions