An infinitely generated virtual cohomology group for noncocompact arithmetic groups over function fields

Abstract

Let G(O S ) be a noncocompact irreducible arithmetic group over a global function field K of characteristic p, and let Γ be a finite-index, residually p-finite subgroup of G(O S ). We show that the cohomology of Γ in the dimension of its associated Euclidean building with coefficients in the field of p elements is infinite.

Infinite-dimensional cohomology of SL2(Z[t,

“…The methods used are based on those of Cesa-Kelly in [CK15], where they are used to show that H 2 SL 3 (Z[t]); Q is infinite-dimensional. Wortman follows a similar outline in [Wor13].…”

$H^2(SL_3(\mathbb{Z}[t]); \mathbb{Q})$ is infinite dimensional

“…In [Wor13], Wortman exhibits a finite index subgroup Γ SL n (F q [t]) such that H n−1 (Γ; F p ) is infinite dimensional. In [Cob15], Cobb shows that H 2 (SL 2 (Z[t, t −1 ]); Q) is infinite dimensional.…”