Topological models of finite type for tree almost automorphism groups

TL;DR

This paper demonstrates that tree almost automorphism groups, including Neretin groups, have a finiteness property analogous to $F_$ in totally disconnected groups, via a cellular action on a contractible complex.

Contribution

It establishes an $F_$-type finiteness condition for tree almost automorphism groups, expanding understanding of their geometric and algebraic properties.

Findings

Tree almost automorphism groups satisfy an $F_$-finiteness condition.

They admit a cellular action on a contractible complex with open, compact stabilizers.

The action is cocompact on each skeleton of the complex.

Abstract

We show that tree almost automorphism groups, including Neretin groups, satisfy the analogue of the $F_\infty$-finiteness condition in the world of totally disconnected groups: They possess a cellular action on a contractible cellular complex such that the stabilizers are open and compact and the restriction of the action on each $n$-skeleton is cocompact.