The Farrell-Jones Conjecture for some nearly crystallographic groups

TL;DR

This paper proves the Farrell-Jones Conjecture for a class of nearly crystallographic groups involving semi-direct products of rational vector spaces and integers, expanding the conjecture's verified cases.

Contribution

It establishes the Farrell-Jones Conjecture for groups of the form d times , where  acts irreducibly with determinant greater than one.

Findings

Proves the K- and L-theoretic Farrell-Jones Conjecture for these groups.

Extends the class of groups for which the conjecture is verified.

Uses algebraic and geometric techniques specific to nearly crystallographic groups.

Abstract

In this paper, we prove the K-theoretical and L-theoretical Farrell-Jones Conjecture with coefficients in an additive category for nearly crystallographic groups of the form $\mathbb{Q}^n \rtimes \mathbb{Z}$, where $\mathbb{Z}$ acts on $\mathbb{Q}^n$ as an irreducible integer matrix with determinant $d$, $|d |>1$.