Thom, Andreas; Wilson, John Some geometric properties of metric ultraproducts of finite simple groups. (English) Zbl 1469.20015 Isr. J. Math. 227, No. 1, 113-129 (2018). Summary: In this article we prove some previously announced results about metric ultraproducts of finite simple groups [C. R., Math., Acad. Sci. Paris 352, No. 6, 463–466 (2014; Zbl 1323.22003)]. 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. Cited in 2 Documents MSC: 20D06 Simple groups: alternating groups and groups of Lie type 22E40 Discrete subgroups of Lie groups Citations:Zbl 1323.22003 PDFBibTeX XMLCite \textit{A. Thom} and \textit{J. Wilson}, Isr. J. Math. 227, No. 1, 113--129 (2018; Zbl 1469.20015) Full Text: DOI arXiv References: [1] Albert, A. A., Symmetric and alternate matrices in an arbitrary field. I., Transactions of the American Mathematical Society, 43, 386-436, (1938) · Zbl 0018.34202 [2] Ben Yaacov, I.; Berenstein, A.; Henson, C. W.; Usvyatsov, A., Model theory for metric structures, Model Theory with Applications to Algebra and Analysis. Vol. 2, 350, 315-427, (2008) · Zbl 1233.03045 · doi:10.1017/CBO9780511735219.011 [3] R. W. Carter, Simple Groups of Lie Yype, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. · Zbl 0248.20015 [4] Elek, G.; Szabó, E., Hyperlinearity, essentially free actions and L2-invariants. the sofic property, Mathematische Annalen, 332, 421-441, (2005) · Zbl 1070.43002 · doi:10.1007/s00208-005-0640-8 [5] P. Kleidman and M. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Mathematical Society Lecture Note Series, Vol. 129, Cambridge University Press, Cambridge, 1990. · Zbl 0697.20004 [6] Liebeck, M. W.; Shalev, A., Diameters of finite simple groups: sharp bounds and applications, Annals of Mathematics, 154, 383-406, (2001) · Zbl 1003.20014 · doi:10.2307/3062101 [7] N. Nikolov, Strange images of profinite groups, arXiv. [8] Point, F., Ultraproducts and Chevalley groups, Archive for Mathematical Logic, 38, 355-372, (1999) · Zbl 0921.03008 · doi:10.1007/s001530050131 [9] Stolz, A.; Thom, A., On the lattice of normal subgroups in ultraproducts of compact simple groups, Proceedings of the London Mathematical Society, 108, 73-102, (2014) · Zbl 1349.20017 · doi:10.1112/plms/pdt027 [10] Thom, A.; Wilson, J. S., Metric ultraproducts of finite simple groups, Comptes Rendus Mathématique. Académie des Sciences. Paris, 352, 463-466, (2014) · Zbl 1323.22003 · doi:10.1016/j.crma.2014.03.015 [11] Wantiez, P., Limites d’espaces métriques et ultraproduits, Méthodes et analyse non standard, 9, 141-168, (1996) · Zbl 0864.54045 [12] Wilson, J. S., On simple pseudofinite groups, Journal of the London Mathematical Society, 51, 471-490, (1995) · Zbl 0847.20001 · doi:10.1112/jlms/51.3.471 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.