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Simplicity of the automorphism groups of some Hrushovski constructions. (English) Zbl 1432.03050

Summary: We show that the automorphism groups of certain countable structures obtained using the Hrushovski amalgamation method are simple groups. The structures we consider are the ‘uncollapsed’ structures of infinite Morley rank obtained by the ab initio construction and the (unstable) \(\aleph_0\)-categorical pseudoplanes. The simplicity of the automorphism groups of these follows from results which generalize work of D. Lascar [J. Symb. Log. 57, No. 1, 238–251 (1992; Zbl 0785.03018)] and of K. Tent and M. Ziegler [J. Lond. Math. Soc., II. Ser. 87, No. 1, 289–303 (2013; Zbl 1273.03136)].

MSC:

03C15 Model theory of denumerable and separable structures
03C60 Model-theoretic algebra
20B07 General theory for infinite permutation groups
20B27 Infinite automorphism groups
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References:

[1] Evans, David M., \( \aleph_0\)-categorical structures with a predimension, Ann. Pure Appl. Logic, 116, 157-186 (2002) · Zbl 1002.03022
[2] Gardener, T., Infinite-dimensional classical groups, J. Lond. Math. Soc. (2), 51, 219-229 (1995) · Zbl 0833.20059
[3] Ghadernezhad, Zaniar, Automorphism groups of generic structures (June 2013), Universität Münster, PhD dissertation
[4] Ghadernezhad, Zaniar; Tent, Katrin, New simple groups with a BN-pair, J. Algebra, 414, 72-81 (2014) · Zbl 1311.20033
[6] Hrushovski, Ehud, A new strongly minimal set, Ann. Pure Appl. Logic, 62, 147-166 (1993) · Zbl 0804.03020
[8] Kechris, Alexander S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156 (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0819.04002
[9] Konnerth, Reinhold, Automorphism groups of differentially closed fields, Ann. Pure Appl. Logic, 118, 1-60 (2002) · Zbl 1052.03015
[10] Lascar, Daniel, Les automorphismes d’un ensemble fortement minimal, J. Symbolic Logic, 57, 238-251 (1992) · Zbl 0785.03018
[11] Macpherson, Dugald; Tent, Katrin, Simplicity of some automorphism groups, J. Algebra, 342, 40-52 (2011) · Zbl 1244.20002
[12] Rosenberg, Alex, The structure of the infinite general linear group, Ann. of Math., 68, 278-294 (1958) · Zbl 0128.25501
[13] Schreier, J.; Ulam, S., Über die Permutationsgruppe der natürlichen Zahlenfolge, Studia Math., 4, 134-141 (1933) · Zbl 0008.20003
[14] Tent, Katrin, A note on the model theory of generalized polygons, J. Symbolic Logic, 65, 692-702 (2000) · Zbl 0965.03051
[15] Tent, Katrin, Very homogeneous generalized \(n\)-gons of finite Morley rank, J. Lond. Math. Soc. (2), 62, 1-15 (2000) · Zbl 0960.51004
[16] Tent, Katrin; Ziegler, Martin, On the isometry group of the Urysohn space, J. Lond. Math. Soc. (2), 87, 289-303 (2013) · Zbl 1273.03136
[17] Wagner, Frank O., Relational structures and dimensions, (Kaye, R.; Macpherson, D., Automorphisms of First-Order Structures (1994), Oxford University Press: Oxford University Press Oxford), 153-180 · Zbl 0813.03020
[18] Wagner, Frank O., Simple Theories (2000), Kluwer: Kluwer Dordrecht · Zbl 0948.03032
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