Isomorphism questions for metric ultraproducts of finite quasisimple groups

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.