Permanence properties of the Baum-Connes conjecture

Abstract

In this paper we study the stability of the Baum–Connes conjecture with coefficients under various natural operations on the groups. We show that the class of groups satisfying this conjecture is stable under taking subgroups, Cartesian products, and more generally, under certain group extensions. In particular, we show that a group satisfies the conjecture if it has an amenable normal subgroup such that the associated quotient group satisfies the conjecture. We also study a natural induction homomorphism between the topological K-theory of a subgroup $H$ of $G$ and the topological K-theory of $G$ with induced coefficient algebra, and show that this map is always bijective. Using this, we are also able to present new examples of groups which satisfy the conjecture with trivial coefficients.