S-arithmetic spinor groups with the same finite quotients and distinct $\ell^2$-cohomology

TL;DR

This paper demonstrates that the vanishing of the $\ell^2$-Betti number in certain groups is not preserved under profinite completions, refining previous examples from arithmetic to S-arithmetic groups.

Contribution

It extends Aka's examples to S-arithmetic groups, showing that $\ell^2$-Betti number vanishing is not a profinite invariant for all $i \ge 2$.

Findings

Vanishing of $\ell^2$-Betti numbers is not a profinite invariant.

Refinement of Aka's examples from arithmetic to S-arithmetic groups.

$\ell^2$-cohomology behavior varies under profinite completions.

Abstract

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