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Profinite perspectives on l^2-cohomology

How much information on a residually finite group can one recover from its finite quotients? This question has been raised in diverse contexts but it is particularly intriguing for lattices in simple Lie groups.  In this research project, we want to investigate to what extend $$\ell^2$$-cohomological properties of groups, in particular lattices, are determined by their profinite completions. The latter will also be investigated as dynamical systems when endowed with translation actions. As geometric outcomes, we will identify situations in which the sign of the Euler characteristic or the volume of a locally symmetric space are determined by the deck transformation groups of its finite sheeted coverings. We will investigate algebraic approximation properties of $$\ell^2$$-cohomology, which should give new insights in the algebraic eigenvalue property. One goal is to establish the Atiyah conjecture for left-orderable groups.

We will start to investigate profinite aspects of $$\ell^2$$-cohomology using rigidity properties of lattices. In the course of this project we further plan to investigate in how far results for lattices extend to larger classes of residually finite groups. For instance, while $$\ell^2$$-acyclicity is a profinite invariant of higher rank lattices, it is unknown whether this holds true for arbitrary residually finite groups.  This line of investigations links our project to group theoretic problems which arise in the attempt to construct counterexamples.

## Publications

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.

 Journal Glasgow Math. Journal Link to preprint version Link to published version

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.

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.

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## Team Members

Jun.-Prof. Dr. Holger Kammeyer
Heinrich-Heine-Universität Düsseldorf
holger.kammeyer(at)hhu.de

Jun.-Prof. Dr. Steffen Kionke