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 ) {X_{n_{i}}(q)} ( i ∈ I {i\in I} ) for X ∈ { PGL , PSp , PGO ( ε ) , PGU } {X\in\{\operatorname{PGL},\operatorname{PSp},\operatorname{PGO}^{(\varepsilon)% },\operatorname{PGU}\}} ( ε = ± {\varepsilon=\pm} ) along an ultrafilter 𝒰 {\mathcal{U}} on the index set I for which n i → 𝒰 ∞ {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 PSp n i ⁡ ( q ) {\operatorname{PSp}_{n_{i}}(q)} and a metric ultraproduct of groups PGO n i ( ε ) ⁡ ( q ) {\operatorname{PGO}_{n_{i}}^{(\varepsilon)}(q)} . This extends results of [A. Thom and J. Wilson, Metric ultraproducts of finite simple groups, Comp. Rend. Math. 352 2014, 6, 463–466].

中文翻译：

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有限拟单群的度量超积的同构问题

摘要 建立了有限单群度量超积的新结果。我们证明了群 X ni ⁢ ( q ) {X_{n_{i}}(q)} ( i ∈ I {i\in I} ) 对于 X ∈ { PGL , PSp , PGO ( ε ) , PGU } {X\in\{\operatorname{PGL},\operatorname{PSp},\operatorname{PGO}^{(\varepsilon)% },\operatorname{PGU}\}} ( ε = ± {\varepsilon=\pm} ) 沿着索引集 I 上的超滤器 𝒰 {\mathcal{U}}，其中 ni → 𝒰 ∞ {n_{i}\to_{\mathcal{U}}\infty} 确定类型 X 和域大小 q 直到群 PSp ni ⁡ ( q ) {\operatorname{PSp}_{n_{i}}(q)} 和群 PGO ni ( ε ) ⁡ ( q ) {\operatorname{PGO}_{n_{i}}^{(\varepsilon)}(q)} . 这扩展了 [A. Thom 和 J. Wilson，有限单群的度量超积，比较。撕裂。数学。352 2014, 6, 463–466]。