Inheritance of Isomorphism Conjectures under colimits

TL;DR

This paper studies the stability of important isomorphism conjectures in group theory under colimits, showing that some conjectures hold for groups where others have been disproved.

Contribution

It demonstrates the inheritance of the Farrell-Jones and Bost Conjectures under colimits of groups, expanding understanding of their stability beyond previously known cases.

Findings

Farrell-Jones Conjecture holds for certain groups

Bost Conjecture with coefficients holds for those groups

Results extend the applicability of these conjectures to new group classes

Abstract

We investigate when Isomorphism Conjectures, such as the ones due to Baum-Connes, Bost and Farrell-Jones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that both the K-theoretic Farrell-Jones Conjecture and the Bost Conjecture with coefficients hold for those groups for which Higson, Lafforgue and Skandalis have disproved the Baum-Connes Conjecture with coefficients.