The coarse Baum-Connes conjecture for relatively hyperbolic groups

TL;DR

This paper proves that relatively hyperbolic groups satisfy the coarse Baum-Connes conjecture if their peripheral subgroups do, leading to implications for the analytic Novikov conjecture, thus advancing understanding in geometric group theory.

Contribution

It establishes a new criterion for relatively hyperbolic groups to satisfy the coarse Baum-Connes conjecture based on properties of their subgroups.

Findings

Relatively hyperbolic groups satisfy the coarse Baum-Connes conjecture under certain subgroup conditions.

The group satisfies the analytic Novikov conjecture as a consequence.

The result links subgroup properties to large-scale geometric conjectures.

Abstract

We study a group which is hyperbolic relative to a finite family of infinite subgroups. We show that the group satisfies the coarse Baum-Connes conjecture if each subgroup belonging to the family satisfies the coarse Baum-Connes conjecture and admits a finite universal space for proper actions. Especially, the group satisfies the analytic Novikov conjecture.