Homology, Homotopy and Applications

Volume 20 (2018)

Number 2

The algebraic and topological $K$-theory of the Hilbert modular group

Pages: 377 – 402

DOI: https://dx.doi.org/10.4310/HHA.2018.v20.n2.a19

Authors

Luis Jorge Sánchez Saldaña (Unidad Cuernavaca del Instituto de Matemáticas, National University of Mexico, Cuernavaca, Morelos, Mexico)

Mario Velásquez (Departamento de Matemáticas, Pontificia Universidad Javeriana, Bogotá, Colombia)

Abstract

In this paper we provide descriptions of the Whitehead groups with coefficients in a ring for the Hilbert modular group (and its reduced version).We also compute the rational topological $K$-theory of their reduced $C^*$-algebras. This is done by computing the source of the assembly maps in the Farrell–Jones and the Baum–Connes conjecture respectively. We also construct a model for the classifying space of the Hilbert modular group for the family of virtually cyclic subgroups.

Keywords

$K$- and $L$-theory, Farrell–Jones conjecture, Baum–Connes conjecture, classifying space, equivariant homology theory

2010 Mathematics Subject Classification

Primary 19B99, 19D35. Secondary 11F41.

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The first author was supported by the FORDECYT postdoctoral grant and the UNAM-DGAPA postdoctoral grant.

The second author was supported by the project “Index morphism in twisted K-theory” (no. 00008165) of the Faculty of Sciences at Pontificia Universidad Javeriana.

Received 21 August 2017

Received revised 6 May 2018

Published 18 July 2018