Groups acting amenably on their Higson corona

TL;DR

This paper studies groups acting amenably on their Higson corona, exploring their properties, reformulations, and implications for the Baum-Connes conjecture, including results on Gromov hyperbolic groups.

Contribution

It provides new reformulations of amenable actions on the Higson corona and establishes isomorphisms in K-theory for Gromov hyperbolic groups.

Findings

Gromov hyperbolic groups have isomorphic equivariant K-theories of their boundary and stable Higson corona.

Reformulations relate amenable actions to nuclearity and positive type kernels.

Implications for the Baum-Connes conjecture are discussed.

Abstract

We investigate groups that act amenably on their Higson corona (also known as bi-exact groups) and we provide reformulations of this in relation to the stable Higson corona, nuclearity of crossed products and to positive type kernels. We further investigate implications of this in relation to the Baum-Connes conjecture, and prove that Gromov hyperbolic groups have isomorphic equivariant K-theories of their Gromov boundary and their stable Higson corona.