Algebraic K-theory of reductive p-adic groups

Abstract

Motivated by the Farrell-Jones Conjecture for group rings, we formulate the $\mathcal{C}$op-Farrell-Jones Conjecture for the K-theory of Hecke algebras of td-groups. We prove this conjecture for (closed subgroups of) reductive p-adic groups G. In particular, the projective class group $K_0(\mathcal{H}(G))$ for a (closed subgroup) of a reductive p-adic group G can be computed as a colimit of projective class groups $K_0(\mathcal{H}(U))$ where U varies over the compact open subgroups of G. This implies that all finitely generated smooth complex representations of a reductive p-adic G admit finite projective resolutions by compactly induced representations. For SL$_n(F)$ we translate the colimit formula for $K_0(\mathcal{H}(G))$ to a more concrete cokernel description in terms of stabilizers for the action on the Bruhat-Tits building. For negative K-theory we obtain vanishing results, while we identify the higher K-groups $K_n(\mathcal{H}(G))$ with the value of G-homology theory on the extended Bruhat-Tits building. Our considerations apply to general Hecke algebras of the form $\mathcal{H}(G;R,\rho,\omega)$, where we allow a central character $\omega$ and a twist by an action $\rho$ of G on R. For the $\mathcal{C}$op-Farrell-Jones Conjecture we need to assume $\mathbb{Q} \subseteq R$ and a regularity assumption. As a key intermediate step we introduce the $\mathcal{C}vcy-Farrell-Jones conjecture. For the latter no regularity assumptions on R are needed.