# The algebraic and topological K-theory of the Hilbert Modular Group

## TL;DR

This paper computes algebraic and topological K-theory for the Hilbert modular group and its reduced version, providing explicit descriptions of Whitehead groups and classifying spaces, advancing understanding in algebraic topology and K-theory.

## Contribution

It offers new descriptions of Whitehead groups and models for classifying spaces for the Hilbert modular group, linking algebraic and topological K-theory via assembly maps.

## Key findings

- Computed Whitehead groups with coefficients for the Hilbert modular group
- Described the topological K-theory of associated C*-algebras after tensoring with Q
- Constructed a model for the classifying space for virtually cyclic subgroups

## Abstract

In this paper we provide descriptions of the Whitehead groups with coefficients in a ring of the Hilbert modular group and its reduced version, as well as for the topological K-theory of $C^*$-algebras, after tensoring with $\mathbb{Q}$, 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.