The Bost conjecture, open subgroups and groups acting on trees

TL;DR

This paper investigates the Bost conjecture for locally compact groups, demonstrating its inheritance by open subgroups and establishing an equivalence for groups acting on trees based on their vertex stabilisers.

Contribution

It proves the Bost conjecture passes to open subgroups and relates its validity for groups acting on trees to that of their vertex stabilisers.

Findings

Bost conjecture passes to open subgroups.

Validity for groups acting on trees depends on vertex stabilisers.

Provides new insights into the structure of the Bost conjecture in geometric group actions.

Abstract

The Bost conjecture with C*-algebra coefficients for locally compact Hausdorff groups passes to open subgroups. We also prove that if a locally compact Hausdorff group acts on a tree, then the Bost conjecture with C*-coefficients is true for the group if and only if it is true for the stabilisers of the vertices.