Metric thickenings and group actions

TL;DR

This paper investigates the properties of Vietoris-Rips and Čech complexes of quotient metric spaces under group actions, focusing on intermediate scale parameters where homotopy types change, especially in projective spaces.

Contribution

It introduces the study of metric thickenings at intermediate scales for quotient spaces, bridging the gap between small-scale homology and large-scale geometric group theory.

Findings

Analysis of homotopy type changes in Vietoris-Rips and Čech complexes

Identification of the first scale parameter where homotopy type changes in projective spaces

Insights into the behavior of complexes under group actions at intermediate scales

Abstract

Let $G$ be a group acting properly and by isometries on a metric space $X$; it follows that the quotient or orbit space $X/G$ is also a metric space. We study the Vietoris-Rips and \v{C}ech complexes of $X/G$. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris-Rips or \v{C}ech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris-Rips and \v{C}ech thickenings of projective spaces at the first scale parameter where the homotopy type changes.