(Self-)similar groups and the Farrell-Jones conjectures

TL;DR

This paper demonstrates that contracting self-similar groups satisfy the Farrell-Jones conjectures if their universal contracting cover is non-positively curved, extending to bounded self-similar groups and introducing a broader contraction notion.

Contribution

It establishes a link between contracting self-similar groups and the Farrell-Jones conjectures, including a new general contraction concept for groups acting on rooted trees.

Findings

Contracting self-similar groups satisfy Farrell-Jones conjectures under certain curvature conditions.

Introduces a general notion of contraction for groups acting on rooted trees.

Applies to bounded self-similar groups.

Abstract

We show that contracting self-similar groups satisfy the Farrell-Jones conjectures as soon as their universal contracting cover is non-positively curved. This applies in particular to bounded self-similar groups. We define, along the way, a general notion of contraction for groups acting on a rooted tree in a not necessarily self-similar manner.