Isomorphism questions for metric ultraproducts of finite quasisimple groups

TL;DR

This paper investigates the structure of metric ultraproducts of finite simple groups, showing that their isomorphism types determine the group type and field size, extending previous results in the field.

Contribution

The paper establishes new results on the isomorphism classification of metric ultraproducts of finite simple groups, identifying conditions under which the group type and parameters are uniquely determined.

Findings

Isomorphism type determines group type and field size.

Extends previous results of Thom and Wilson.

Identifies cases of isomorphism between different group ultraproducts.

Abstract

New results on metric ultraproducts of finite simple groups are established. We show that the isomorphism type of a simple metric ultraproduct of groups $X_{n_i}(q)$ ($i\in I$) for $X\in\{{\rm PGL},{\rm PSp},{\rm PGO}^{(\varepsilon)},{\rm PGU}\}$ ($\varepsilon=\pm$) along an ultrafilter $\mathcal{U}$ on the index set $I$ for which $n_i\to_{\mathcal{U}}\infty$ determines the type $X$ and the field size $q$ up to the possible isomorphism of a metric ultraproduct of groups ${\rm PSp}_{n_i}(q)$ and a metric ultraproduct of groups ${\rm PGO}_{n_i}^{(\varepsilon)}(q)$. This extends results of Thom and Wilson.