On the Euler characteristic of $S$-arithmetic groups

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.