## Jun.-Prof. Dr. Holger Kammeyer

### Project leader

Heinrich Heine University Düsseldorf

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

Telephone: +49 211 81 13709

Homepage: http://reh.math.uni-duesseldorf.de/~kamm…

## Project

**58**Profinite perspectives on l2-cohomology

## Publications within SPP2026

We show that the sign of the Euler characteristic of an S-arithmetic subgroup of a simple k-group over a number field k depends on the S-congruence completion only. Consequently, the sign is a profinite invariant for such S-arithmetic groups with the congruence subgroup property. This generalizes previous work of the first author with Kionke-Raimbault-Sauer.

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

We construct pairs of residually finite groups with isomorphic profinite completions such that one has non-vanishing and the other has vanishing real second bounded cohomology. The examples are lattices in different higher rank simple Lie groups. Using Galois cohomology, we actually show that \(\operatorname{SO}^0(n,2)\) for \(n \ge 6\) and the exceptional groups \(E_{6(-14)}\) and \(E_{7(-25)}\) constitute the complete list of higher rank Lie groups admitting such examples.

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

For every number field and every Cartan Killing type, there is an associated split simple algebraic group. We examine whether the corresponding arithmetic subgroups are profinitely solitary so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. Assuming Grothendieck rigidity, we essentially solve the problem by Galois cohomological means.

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

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.

**Related project(s):****58**Profinite 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):****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