Convergence of group actions in metric measure geometry

TL;DR

This paper extends the concepts of box and observable distances to metric measure spaces with group actions, establishing fundamental properties and demonstrating convergence in a sequence of lens spaces to an infinite-dimensional space using mass-transport theory.

Contribution

It introduces a generalization of distances in metric measure geometry to include group actions and applies this to a convergence example involving lens spaces and infinite-dimensional spaces.

Findings

Generalized distances between metric measure spaces with group actions.

Proved fundamental properties of these generalized distances.

Constructed an example of convergence from lens spaces to an infinite-dimensional space.

Abstract

We generalize the box and observable distances to those between metric measure spaces with group actions, and prove some fundamental properties. As an application, we obtain an example of a sequence of lens spaces with unbounded dimension converging to the cone of the infinite-dimensional complex projective space. Our idea is to use the theory of mass-transport.