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For any orientable finite-volume hyperbolic 3-manifold, this paper proves that the profinite isomorphism type of the fundamental group uniquely determines the isomorphism type of the first integral cohomology, as marked with the Thurston norm and the fibered classes; moreover, up to finite ambiguity, the profinite isomorphism type determines the isomorphism type of the fundamental group, among the class of finitely generated 3-manifold groups. [ABSTRACT FROM AUTHOR]

We study a natural generalization of inverse systems of finite regular covering spaces. A limit of such a system is a fibration whose fibres are profinite topological groups. However, as shown in Conner et al. (Topol Appl 239:234–243, 2018), there are many fibrations whose fibres are profinite groups, which are far from being inverse limits of coverings. We characterize profinite fibrations among a large class of fibrations and relate the profinite topology on the fundamental group of the base with the action of the fundamental group on the fibre, and develop a version of the Borel construction for fibrations whose fibres are profinite groups. [ABSTRACT FROM AUTHOR]

We define unimodular measures on the space of rooted simplicial complexes and associate to each measure a chain complex and a trace function. As a consequence, we can define ℓ 2 -Betti numbers of unimodular random rooted simplicial complexes and show that they are continuous under Benjamini-Schramm convergence. [ABSTRACT FROM AUTHOR]

The interaction between the study of three-dimensional manifolds and a particular stream of group theory has often been fruitful. In the realm of this, we recall that a finitely presented group G is properly 3-realizable if for some finite 2-dimensional CW-complex X with $${\pi_1(X) \cong G}$$ , the universal cover of X has the proper homotopy type of a 3-manifold. In this paper, we generalize a previous result on the direct products of groups; more precisely, we show that if $${N \rightarrow G \rightarrow Q}$$ is a short exact sequence of infinite finitely presented groups, then G is properly 3-realizable. In particular, any semidirect product of two infinite finitely presented groups is properly 3-realizable. As an application, we show proper 3-realizability for certain classes of groups. [ABSTRACT FROM AUTHOR]