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 $X\in \{\mathrm{PGL},\mathrm{PSp},{\mathrm{PGO}}^{(\epsilon )},\mathrm{PGU}\}$ ($\epsilon =\pm$) 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 ${\mathrm{PSp}}_{n_i}(q)$ and a metric ultraproduct of groups ${\mathrm{PGO}}_{n_i}^{(\epsilon )}(q)$. This extends results of Thom and Wilson.

25 pages, no figures