On Sharp Bounds for the Dynamic Asymptotic Dimension

TL;DR

This paper establishes sharp bounds for the dynamic asymptotic dimension in free isometric actions, linking it to the group's asymptotic dimension and providing a detailed analysis for actions on compact Lie groups.

Contribution

It characterizes the dynamic asymptotic dimension for free isometric actions and translation actions on compact Lie groups, connecting it to group properties like amenability.

Findings

Dynamic asymptotic dimension is either infinite or equals the group's asymptotic dimension.

Full description of dynamic asymptotic dimension for translation actions on compact Lie groups.

Connection between dynamic asymptotic dimension, amenability, and asymptotic dimension of groups.

Abstract

We prove the dynamic asymptotic dimension of a free isometric action on a space of finite doubling dimension is either infinite or equal to the asymptotic dimension of the acting group; and give a full description of the dynamic asymptotic dimension of translation actions on compact Lie groups in terms of the amenability and asymptotic dimension of the acting group.