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Berenstein, Alexander; Zamora, Rafael

Isometry groups of Borel randomizations. (English) Zbl 1484.03090

Notre Dame J. Formal Logic 61, No. 2, 297-316 (2020).
Summary: We study global dynamical properties of the isometry group of the Borel randomization of a separable complete structure. We show that if properties such as the Rokhlin property, topometric generics, and extreme amenability hold for the isometry group of the structure, then they also hold in the isometry group of the randomization.

MSC:

03E15 Descriptive set theory
54H11 Topological groups (topological aspects)
22F50 Groups as automorphisms of other structures
22A05 Structure of general topological groups

Keywords:

topological groups; isometry groups; Borel randomizations; rokhlin property; topometric spaces; topometric groups
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\textit{A. Berenstein} and \textit{R. Zamora}, Notre Dame J. Formal Logic 61, No. 2, 297--316 (2020; Zbl 1484.03090)
Full Text: DOI arXiv Euclid
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References:

[1] Ben Yaacov, I., “Continuous and random Vapnik-Chervonenkis classes,” Israel Journal of Mathematics, vol. 173 (2009), pp. 309-33. · Zbl 1184.03020
[2] Ben Yaacov, I., A. Berenstein, and J. Melleray, “Polish topometric groups,” Transactions of the American Mathematical Society, vol. 365 (2013), pp. 3877-97. · Zbl 1295.03026
[3] Ben Yaacov, I., and H. J. Keisler, “Randomizations of models as metric structures,” Confluentes Mathematici, vol. 1 (2009), pp. 197-223. · Zbl 1185.03068
[4] Giordano, T., and V. Pestov, “Some extremely amenable groups,” Comptes Rendus Mathématique, Académie des Sciences, Paris, vol. 334 (2002), pp 273-78. · Zbl 0995.43001
[5] Halmos, P. R., Lectures on Ergodic Theory, Chelsea, New York, 1960. · Zbl 0096.09004
[6] Hodges, W., I. Hodkinson, D. Lascar, and S. Shelah, “The small index property for \(\omega \)-stable \(\omega \)-categorical structures and for the random graph,” Journal of the London Mathematical Society (2), vol. 48 (1993), pp. 204-18. · Zbl 0788.03039
[7] Hrushovski, E., “Extending partial isomorphisms of graphs,” Combinatorica, vol. 12 (1992), pp. 411-16. · Zbl 0767.05053
[8] Ibarlucía, T., “Automorphism groups of randomized structures,” Journal of Symbolic Logic, vol. 82 (2017), pp. 1150-79. · Zbl 1404.22051
[9] Kaïchouh, A., “Variations on automatic continuity,” preprint, http://math.univ-lyon1.fr/homes-www/kaichouh/Variations
[10] Kaïchouh, A., and F. Le Maître, “Connected Polish groups with ample generics,” Bulletin of the London Mathematical Society, vol. 47 (2015), pp. 996-1009. · Zbl 1373.03081
[11] Kechris, A. S., and C. Rosendal, “Turbulence, amalgamation, and generic automorphisms of homogeneous structures,” Proceedings of the London Mathematical Society (3), vol. 94 (2007), pp. 302-50. · Zbl 1118.03042
[12] Kwiatkowska, A., and M. Malicki, “Automorphism groups of countable structures and groups of measurable functions,” Israel Journal of Mathematics, vol. 230 (2019), pp. 335-60. · Zbl 1429.22002
[13] Malicki, M., “Consequences of the existence of ample generics and automorphism groups of homogeneous metric structures,” Journal of Symbolic Logic, vol. 81 (2016), pp. 876-86. · Zbl 1403.03087
[14] Moore, C. C., “Group extensions and cohomology for locally compact groups, IV,” Transactions of the American Mathematical Society, vol. 221 (1976), pp. 35-58. · Zbl 0366.22006
[15] Pestov, V., Dynamics of Infinite-Dimensional Groups: The Ramsey-Dvoretzky-Milman Phenomenon, vol. 40 of University Lecture Series, American Mathematical Society, Providence, 2006. · Zbl 1123.37003
[16] Pestov, V., and F. M. Schneider, “On amenability and groups of measurable maps,” Journal of Functional Analysis, vol. 273 (2017), pp. 3859-74. · Zbl 1412.43002
[17] Sabok, M., “Automatic continuity for isometry groups,” Journal of the Institute of Mathematics of Jussieu, vol. 18 (2019), pp. 561-90. · Zbl 07051730
[18] Solecki, S., “Extending partial isometries,” Israel Journal of Mathematics, vol. 150 (2005), pp. 315-31. · Zbl 1124.54012
[19] Truss, J. K., “Generic automorphisms of homogeneous structures,” Proceedings of the London Mathematical Society (3), vol. 65 (1992), pp. 121-41. · Zbl 0723.20001
[20] Vershik, A. M., “Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of Urysohn space,” Topology and Its Applications, vol. 155 (2008), 1618-26. · Zbl 1160.54020
[21] Wheeden, R. · Zbl 1326.26007
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.