On the Farrell-Jones Conjecture for Waldhausen's $A$-theory

TL;DR

This paper proves the Farrell-Jones Conjecture for Waldhausen's A-theory across various group classes, establishing inheritance properties and extending results to Whitehead spectra and pseudo-isotopies in multiple categories.

Contribution

It establishes the Farrell-Jones Conjecture for A-theory with coefficients for several important classes of groups and demonstrates inheritance properties, extending the scope of the conjecture.

Findings

Proves the Farrell-Jones Conjecture for A-theory for hyperbolic and CAT(0) groups.

Establishes inheritance properties such as subgroup and colimit stability.

Extends results to Whitehead spectra and pseudo-isotopies in different categories.

Abstract

We prove the Farrell-Jones Conjecture for (non-connective) $A$-theory with coefficients and finite wreath products for hyperbolic groups, CAT(0)-groups, cocompact lattices in almost connected Lie groups and fundamental groups of manifolds of dimension less or equal to three. Moreover, we prove inheritance properties such as passing to subgroups, colimits of direct systems of groups, finite direct products and finite free products. These results hold also for Whitehead spectra and spectra of stable pseudo-isotopies in the topological, piecewise linear and smooth category.