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We present a new method to construct finitely generated, residually finite, infinite torsion groups. In contrast to known constructions, a profinite perspective enables us to control finite quotients and normal subgroups of these torsion groups. As an application, we describe the first examples of residually finite, hereditarily just-infinite groups with positive first ℓ2-Betti-number. In addition, we show that these groups have polynomial normal subgroup growth, which answers a question of Barnea and Schlage-Puchta.

Related project(s):
58Profinite perspectives on l2-cohomology

We prove that every finitely generated, residually finite group G embeds into a finitely generated perfect branch group such that many properties of G are preserved under this embedding. Among those are the properties of being torsion, being amenable and not containing a non-abelian free group. As an application, we construct a finitely generated, non-amenable torsion branch group.

Related project(s):
58Profinite perspectives on l2-cohomology

We prove that the minimal representation dimension of a direct product G of non-abelian groups G1,…,Gn is bounded below by n+1 and thereby answer a question of Abért. If each Gi is moreover non-solvable, then this lower bound can be improved to be 2n. By combining this with results of Pyber, Segal, and Shusterman on the structure of boundedly generated groups we show that branch groups cannot be boundedly generated.

Related project(s):
58Profinite perspectives on l2-cohomology

We introduce the notion of a telescope of groups. Very roughly a telescope is a directed system of groups that contains various commuting images of some fixed group B. Telescopes are inspired from the theory of groups acting on rooted trees. Imitating known constructions of branch groups, we obtain a number of examples of B-telescopes and discuss several applications. We give examples of 2-generated infinite amenable simple groups. We show that every finitely generated residually finite (amenable) group embeds into a finitely generated (amenable) LEF simple group. We construct 2-generated frames in products of finite simple groups and show that there are Grothendieck pairs consisting of amenable groups and groups with property (τ). We give examples of automorphisms of finitely generated, residually finite, amenable groups that are not inner, but become inner in the profinite completion. We describe non-elementary amenable examples of finitely generated, residually finite groups all of whose finitely generated subnormal subgroups are direct factors.

Related project(s):
58Profinite perspectives on l2-cohomology

We show that if one of various cycle types occurs in the permutation action of a finite group on the cosets of a given subgroup, then every almost conjugate subgroup is conjugate. As a number theoretic application, corresponding decomposition types of primes effect that a number field is determined by the Dedekind zeta function. As a geometric application, coverings of Riemannian manifolds with certain geodesic lifting behaviors must be isometric.

Related project(s):
58Profinite perspectives on l2-cohomology

This article presents a method for proving upper bounds for the first \(\ell^2\)-Betti number of groups using only the geometry of the Cayley graph. As an application we prove that Burnside groups of large prime exponent have vanishing first \(\ell^2\)-Betti number.

Our approach extends to generalizations of \(\ell^2\)-Betti numbers, that are defined using characters. We illustrate this flexibility by generalizing results of Thom-Peterson on q-normal subgroups to this setting.

Related project(s):
58Profinite perspectives on l2-cohomology

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 70's, and we prove that uniform amenability indeed is detectable from the profinite completion.

Related project(s):
58Profinite perspectives on l2-cohomology

By arithmeticity and superrigidity, a commensurability class of lattices in a higher rank Lie group is defined by a unique algebraic group over a unique number subfield of \(\mathbb{R}\) or \(\mathbb{C}\). We prove an adelic version of superrigidity which implies that two such commensurability classes define the same profinite commensurability class if and only if the algebraic groups are adelically isomorphic. We discuss noteworthy consequences on profinite rigidity questions.

Related project(s):
58Profinite perspectives on l2-cohomology

We investigate which higher rank simple Lie groups admit profinitely but not abstractly commensurable lattices.  We show that no such examples exist for the complex forms of type \(E_8\), \(F_4\), and \(G_2\).  In contrast, there are arbitrarily many such examples in all other higher rank Lie groups, except possibly \(\mathrm{SL}_{2n+1}(\mathbb{R})\), \(\mathrm{SL}_{2n+1}(\mathbb{C})\), \(\mathrm{SL}_n(\mathbb{H})\), or groups of type~\(E_6\).

Related project(s):
58Profinite perspectives on l2-cohomology

We define and study generalizations of simplicial volume over arbitrary seminormed rings with a focus on p-adic simplicial volumes. We investigate the dependence on the prime and establish homology bounds in terms of p-adic simplicial volumes. As the main examples, we compute the weightless and p-adic simplicial volumes of surfaces. This is based on an alternative way to calculate classical simplicial volume of surfaces without hyperbolic straightening and shows that surfaces satisfy mod p and p-adic approximation of simplicial volume.

Related project(s):
58Profinite perspectives on l2-cohomology

We prove that the sign of the Euler characteristic of arithmetic groups with CSP is determined by the profinite completion.  In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type F.  Our methods imply similar results for L2-torsion as well as a strong profiniteness statement for Novikov--Shubin invariants.

Related project(s):
18Analytic L2-invariants of non-positively curved spaces58Profinite perspectives on l2-cohomology

We explain how the construction of the real numbers using quasimorphisms can be transformed into a general method to construct the completion of a field with respect to an absolute value.

Related project(s):
18Analytic L2-invariants of non-positively curved spaces

We define a variant of Benjamini-Schramm convergence for finite simplicial complexes with the action of a fixed finite group G which leads to the notion of random rooted simplicial G-complexes. For every random rooted simplicial G-complex we define a corresponding ℓ2-homology and the ℓ2-multiplicity of an irreducible representation of G in the homology. The ℓ2-multiplicities generalize the ℓ2-Betti numbers and we show that they are continuous on the space of sofic random rooted simplicial G-complexes. In addition, we study induction of random rooted complexes and discuss the effect on ℓ2-multiplicities.

Related project(s):
18Analytic L2-invariants of non-positively curved spaces

The purpose of this article is to define and study new invariants of topological spaces: the p-adic Betti numbers and the p-adic torsion. These invariants take values in the p-adic numbers and are constructed from a virtual pro-p completion of the fundamental group. The key result of the article is an approximation theorem which shows that the p-adic invariants are limits of their classical analogues. This is reminiscent of Lück's approximation theorem for L2-Betti numbers.

After an investigation of basic properties and examples we discuss the p-adic analog of the Atiyah conjecture: When do the p-adic Betti numbers take integer values? We establish this property for a class of spaces and discuss applications to cohomology growth.

Related project(s):
18Analytic L2-invariants of non-positively curved spaces58Profinite perspectives on l2-cohomology

In this note we refine examples by Aka from arithmetic to S-arithmetic groups to show that the vanishing of the i-th ℓ²-Betti number is not a profinite invariant for all i≥2.

Related project(s):
18Analytic L2-invariants of non-positively curved spaces

Given an S-arithmetic group, we ask how much information on the ambient algebraic group, number field of definition, and set of places S is encoded in the commensurability class of the profinite completion. As a first step, we show that the profinite commensurability class of an S-arithmetic group with CSP determines the number field up to arithmetical equivalence and the places in S above unramified primes. We include some applications to profiniteness questions of group invariants.

Related project(s):
18Analytic L2-invariants of non-positively curved spaces