On the Euler characteristic of $S$-arithmetic groups

Holger Kammeyer, Giada Serafini

Published 2024-05-08, updated 2024-05-29Version 2

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.