We establish conditions under which lattices in certain simple Lie groups are profinitely solitary in the absolute sense, so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. While cocompact lattices are typically not absolutely solitary, we show that noncocompact lattices in Sp(n,R), G2(2), E8(C), F4(C), and G2(C) are absolutely solitary if a well-known conjecture on Grothendieck rigidity is true.

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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.

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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.

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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.

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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.

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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.

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18Analytic L2-invariants of non-positively curved spaces58Profinite perspectives on l2-cohomology

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