## Jun.-Prof. Dr. Holger Kammeyer

### Project leader

Heinrich-Heine-Universität Düsseldorf

E-mail: holger.kammeyer(at)hhu.de

Telephone: +49 211 81 13709

Homepage: https://topology.math.kit.edu/21_375.php

## Project

**58**Profinite perspectives on l2-cohomology

## Publications within SPP2026

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):****58**Profinite 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):****58**Profinite perspectives on l2-cohomology

For every Lie group *G*, we compute the maximal *n* such that an *n*-fold product of nonabelian free groups embeds into *G*.

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

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):****18**Analytic L2-invariants of non-positively curved spaces**58**Profinite 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):****18**Analytic 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):****18**Analytic L2-invariants of non-positively curved spaces