Recipes to compute the algebraic K-theory of Hecke algebras of reductive p-adic groups

Bartels, Arthur; Lück, Wolfgang

doi:10.2140/agt.2025.25.1133

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Recipes to compute the algebraic $K$-theory of Hecke algebras of reductive $p$-adic groups

Arthur Bartels and Wolfgang Lück

Algebraic & Geometric Topology 25 (2025) 1133–1154

DOI: 10.2140/agt.2025.25.1133

Abstract

We compute the algebraic $K$ -theory of the Hecke algebra of a reductive $p$ -adic group $G$ using the fact that the Farrell–Jones conjecture is known in this context. The main tools will be the properties of the associated Bruhat–Tits building and an equivariant Atiyah–Hirzebruch spectral sequence. In particular, the projective class group can be written as the colimit of the projective class groups of the compact open subgroups of $G$ .

Keywords

algebraic $K$-theory of Hecke algebras, reductive $p$-adic groups, Farrell–Jones conjecture

Mathematical Subject Classification

Primary: 55P91

Secondary: 19D50, 20C08

References

Publication

Received: 3 July 2023

Revised: 6 January 2024

Accepted: 4 March 2024

Published: 16 May 2025

Authors

Arthur Bartels
Mathematisches Institut Westfälische Wilhelms-Universität Münster Münster Germany
http://www.math.uni-muenster.de/u/bartelsa
Wolfgang Lück
Mathematisches Institut Universität Bonn Bonn Germany
http://www.him.uni-bonn.de/lueck

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Volume 25, issue 2 (2025)