An Infinitely Generated Virtual Cohomology Group for Noncocompact Arithmetic Groups over Function Fields

Let G(OS) 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(OS). 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. Let K be a global function field that contains the field with p elements, Fp. We let S be a finite nonempty set of inequivalent valuations of K. The ring OS ⊆ K will denote the corresponding ring of Sintegers. For any v ∈ S, we let Kv be the completion of K with respect to v so that Kv is a locally compact field. We denote by G a connected noncommutative absolutely almost simple K-group, and we let