# Profinite invariants of arithmetic groups

## TL;DR

This paper demonstrates that the sign of the Euler characteristic of certain arithmetic groups is determined by their profinite completion, but the Euler characteristic itself is not, highlighting nuanced relationships between group invariants and profinite structures.

## Contribution

It establishes the profinite invariance of the sign of the Euler characteristic for arithmetic groups with CSP and provides counterexamples for the Euler characteristic itself.

## Key findings

- Sign of Euler characteristic is profinite invariant for groups with CSP.
- Counterexamples show Euler characteristic is not profinite invariant in general.
- Results extend to $\, ext{ extltilde} ext{}^2$-torsion and Novikov-Shubin invariants.

## Abstract

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 $\ell^2$-torsion as well as a strong profiniteness statement for Novikov-Shubin invariants.