Inheritance of isomorphism conjectures under colimits

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.