On the algebraic K-theory of Hecke algebras

TL;DR

This paper proves the algebraic K-theory of Hecke algebras for certain totally disconnected groups, confirming the Farrell-Jones Conjecture in this case and establishing a computational Wang sequence.

Contribution

It introduces a Wang sequence for computing K-groups of Hecke algebras of covirtually cyclic groups and proves the vanishing of negative K-groups, supporting the Farrell-Jones Conjecture.

Findings

Wang sequence for algebraic K-theory of Hecke algebras

Negative K-groups of these algebras vanish

Confirms Farrell-Jones Conjecture for covirtually cyclic groups

Abstract

Consider a totally disconnected group G, which is covirtually cyclic, i.e., contains a normal compact open subgroup L such that G/L is infinite cyclic. We establish a Wang sequence, which computes the algebraic K-groups of the Hecke algebra of G in terms of the one of L, and show that all negative K-groups vanish. This confirms the K-theoretic Farrell-Jones Conjecture for the Hecke algebra of G in this special case. Our ultimate long term goal is to prove it for any closed subgroup of any reductive p-adic group. The results of this paper will play a role in the final proof.