On the Farrell-Jones conjecture for localising invariants

TL;DR

This paper proves the Farrell-Jones conjecture for a broad class of groups and coefficients, unifying and extending previous results in algebraic K-theory and categories of modules over ring spectra.

Contribution

It establishes the Farrell-Jones conjecture with coefficients in left-exact $$-categories for finitely $$-amenable and Dress-Farrell-Hsiang-Jones groups, extending prior work to new categories.

Findings

Proves the Farrell-Jones conjecture for specific group classes.

Unifies existing proofs for K-theory of additive categories.

Extends results to categories of perfect modules over $ ext{E}_1$-ring spectra.

Abstract

We show the Farrell-Jones conjecture with coefficients in left-exact $\infty$-categories for finitely $\mathcal{F}$-amenable groups and, more generally, Dress-Farrell-Hsiang-Jones groups. Our result subsumes and unifies arguments for the K-theory of additive categories and spherical group rings and extends it for example to categories of perfect modules over $\mathbb{E}_{1}$-ring spectra.