Groups Acting on Metric Spaces with Asymptotic Property C

TL;DR

This paper proves that if a group acts on a metric space with asymptotic property C and the stabilizers have bounded asymptotic dimension, then the group inherits asymptotic property C.

Contribution

It establishes a new inheritance result for asymptotic property C under group actions with controlled stabilizers.

Findings

Groups acting on spaces with asymptotic property C inherit the property under certain conditions.

Quasi-stabilizers with bounded asymptotic dimension ensure the group also has asymptotic property C.

Provides a framework for analyzing group actions on metric spaces with geometric properties.

Abstract

We show that if a group $G$ acts by isometries on a metric space $M$ which has asymptotic property C, such that the quasi-stabilizers of a point $x \in M$ have asymptotic dimension less than or equal to $n$, then $G$ itself has asymptotic property C.