The G-Gromov-Hausdorff Distance and Equivariant Topology

TL;DR

This paper introduces a new Gromov-Hausdorff distance concept for finite group actions on compact metric spaces, using equivariant topology to establish bounds, rigidity, and finiteness results, including sharp sphere distance bounds.

Contribution

It defines a G-Gromov-Hausdorff distance for finite group actions and applies equivariant topology to derive new bounds and rigidity theorems.

Findings

Established lower bounds for G-Gromov-Hausdorff distance

Proved equivariant rigidity and finiteness theorems

Derived sharp bounds on distances between spheres

Abstract

For each arbitrary finite group $G$, we consider a suitable notion of Gromov Hausdorff distance between compact $G$ metric spaces and derive lower bounds based on equivariant topology methods. As applications, we prove equivariant rigidity and finiteness theorems, and obtain sharp bounds on the Gromov Hausdorff distance between spheres.