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Coherent extension of partial automorphisms, free amalgamation and automorphism groups. (English) Zbl 1477.03124

Summary: We give strengthened versions of the Herwig-Lascar and Hodkinson-Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraïssé limit of any Fraïssé class that is the class of all \(\mathcal F\)-free structures (in the Herwig-Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.

MSC:

03C13 Model theory of finite structures
03C15 Model theory of denumerable and separable structures
20B27 Infinite automorphism groups
03C35 Categoricity and completeness of theories
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[1] Bhattacharjee, M. and Macpherson, D., A locally finite dense group acting on the random graph. Forum Mathematicum, vol. 17 (2005), no. 3, pp. 513-517. · Zbl 1093.20002
[2] Cameron, P. J., The random graph, The mathematics of Paul Erdös, II (Graham, R., Nesetril, J., and Butler, S., editors), Algorithms Combin, vol. 14, Springer, Berlin, 1997, pp. 333-351. · Zbl 0855.00014
[3] Henson, C. W., Countable homogeneous relational structures and \({\aleph_0}\)-categorical theories, this Journal, vol. 37 (1972), pp. 494-500. · Zbl 0259.02040
[4] Herwig, B., Extending partial isomorphisms for the small index property of many ω-categorical structures. Israel Journal of Mathematics, vol. 107 (1998), no. 1, pp. 93-123. · Zbl 0922.03044
[5] Herwig, B. and Lascar, D., Extending partial automorphisms and the profinite topology on free groups. American Mathematical Society, vol. 352 (1999), pp. 1985-2021. · Zbl 0947.20018
[6] Hodges, W., Hodkinson, I., Lascar, D., and Shelah, S., The small index property for ω-stable, ω-categorical structures and for the random graph. Journal of the London Mathematical Society, vol. 2 (1993), no. 2, pp. 204-218. · Zbl 0788.03039
[7] Hodkinson, I. and Otto, M., Finite conformal hypergraph covers and Gaifman cliques in finite structures. The Bulletin of Symbolic Logic, vol. 9 (2003), no. 3, pp. 387-405. · Zbl 1058.03031
[8] Ivanov, A., Automorphisms of homogeneous structures. Notre Dame Journal of Formal Logic, vol. 46 (2005), no. 4, pp. 419-424. · Zbl 1096.03039
[9] Ivanov, A., Strongly bounded automorphism groups. Colloquium Mathematicum, vol. 105 (2006), no. 1, pp. 57-67. · Zbl 1098.20003
[10] Kechris, A. S. and Rosendal, C., Turbulence, amalgamation, and generic automorphisms of homogeneous structures. Proceedings of the London Mathematical Society, vol. 94 (2007), no. 2, pp. 302-350. · Zbl 1118.03042
[11] Mackey, G. W., Ergodic theory and virtual groups. Mathematische Annalen, vol. 166 (1966), pp. 187-207. · Zbl 0178.38802
[12] Macpherson, D., A survey of homogeneous structures. Discrete Mathematics, vol. 311 (2011), no. 15, pp. 1599-1634. · Zbl 1238.03032
[13] Macpherson, D. and Tent, K., Simplicity of some automorphism groups. Journal of Algebra, vol. 342 (2011), no. 1, pp. 40-52. · Zbl 1244.20002
[14] Macpherson, D. and Thomas, S., Comeagre conjugacy classes and free products with amalgamation. Discrete Mathematics, vol. 291 (2005), no. 1, pp. 135-142. · Zbl 1058.03035
[15] Rosendal, C., Finitely approximable groups and actions part I: The Ribes-Zalesskiĭ property, this Journal, vol. 76 (2011), no. 4, pp. 1297-1306. · Zbl 1250.03085
[16] Siniora, D., Automorphism groups of homogeneous structures. Ph.D. thesis, University of Leeds, 2017.
[17] Solecki, S., Extending partial isometries. Israel Journal of Mathematics, vol. 150 (2005), no. 1, pp. 315-331. · Zbl 1124.54012
[18] Thomas, S.. Reducts of random hypergraphs. Annals of Pure and Applied Logic, vol. 80 (1996), no. 2, pp. 165-193. · Zbl 0865.03025
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