Some geometric properties of metric ultraproducts of finite simple groups

TL;DR

This paper investigates the geometric structure of metric ultraproducts of finite simple groups, revealing that they are geodesic or path-connected, which reflects their asymptotic properties.

Contribution

It proves that non-discrete metric ultraproducts of finite simple groups are geodesic or path-connected, advancing understanding of their geometric and asymptotic properties.

Findings

Ultraproducts of alternating or special linear groups are geodesic spaces.

General ultraproducts of finite simple groups are path-connected.

Global properties mirror asymptotic behaviors of finite simple groups.

Abstract

In this article we prove some previously announced results about metric ultraproducts of finite simple groups. We show that any non-discrete metric ultraproduct of alternating or special linear groups is a geodesic metric space. For more general non-discrete metric ultraproducts of finite simple groups, we are able to establish path-connectedness. As expected, these global properties reflect asymptotic properties of various families of finite simple groups.