On the Euler characteristic of $S$-arithmetic groups

TL;DR

This paper investigates the sign of the Euler characteristic of $S$-arithmetic groups, showing it depends solely on the $S$-congruence completion, thus establishing it as a profinite invariant under certain conditions.

Contribution

It demonstrates that the sign of the Euler characteristic is determined by the $S$-congruence completion, extending previous results to a broader class of groups.

Findings

Sign of Euler characteristic depends only on $S$-congruence completion

Sign is a profinite invariant for groups with the congruence subgroup property

Results generalize previous work by the first author with Kionke, Raimbault, and Sauer

Abstract

We show that the sign of the Euler characteristic of an $S$-arithmetic subgroup of a simple algebraic group depends on the $S$-congruence completion only, except possibly in type ${}^6 D_4$. 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.