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Automatic continuity for isometry groups. (English) Zbl 07051730

J. Inst. Math. Jussieu 18, No. 3, 561-590 (2019); erratum ibid. 21, No. 6, 2253-2255 (2022).
Summary: We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Ben Yaacov, Berenstein and Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group \(\mathrm{Aut}([0,1],\lambda)\), due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov.

MSC:

03E15 Descriptive set theory
54H11 Topological groups (topological aspects)
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
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[1] C. J.Ash, Inevitable sequences and a proof of the ‘type II conjecture’, in Monash Conference on Semigroup Theory (Melbourne, 1990), pp. 31-42 (World Sci. Publ., River Edge, NJ, 1991). · Zbl 1039.20505
[2] A. G.Atim and R. R.Kallman, The infinite unitary and related groups are algebraically determined Polish groups, Topology Appl.159(12) (2012), 2831-2840. · Zbl 1256.54060
[3] I.Ben Yaacov, A.Berenstein, C. W.Henson and A.Usvyatsov, Model theory for metric structures, in Model Theory with Applications to Algebra and Analysis, Vol. 2, London Mathematical Society Lecture Note Series, Volume 350, pp. 315-427 (Cambridge University Press, Cambridge, 2008). · Zbl 1233.03045
[4] I.Ben Yaacov, A.Berenstein and J.Melleray, Polish topometric groups, Trans. Amer. Math. Soc.365(7) (2013), 3877-3897. · Zbl 1295.03026
[5] I.Ben Yaacov and T.Tsankov, Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups, Trans. Amer. Math. Soc.368(11) (2016), 8267-8294. · Zbl 1350.22005
[6] J.Donald Monk and R.Bonnet (Eds.) Handbook of Boolean Algebras, Vol. 3 (North-Holland, Amsterdam, 1989).
[7] P.Dowerk and A.Thom, Bounded normal generation and invariant automatic continuity, Preprint, 2015, arXiv:1506.08549. · Zbl 1419.46038
[8] R. M.Dudley, Continuity of homomorphisms, Duke Math. J.28 (1961), 587-594. · Zbl 0103.01702
[9] D. H.Fremlin, Measure Theory, Vol. 3 (Torres Fremlin, Colchester, 2004). Measure algebras, Corrected second printing of the 2002 original. · Zbl 1165.28002
[10] S.Gao, Invariant Descriptive Set Theory, Pure and Applied Mathematics (Boca Raton), Volume 293 (CRC Press, Boca Raton, FL, 2009). · Zbl 1154.03025
[11] S.Gao and A. S.Kechris, On the classification of Polish metric spaces up to isometry, Mem. Amer. Math. Soc.161(766) (2003), viii+78. · Zbl 1012.54038
[12] E.Glasner, The group Aut(𝜇) is Roelcke precompact, Canad. Math. Bull.55(2) (2012), 297-302. · Zbl 1247.54045
[13] B.Herwig and D.Lascar, Extending partial automorphisms and the profinite topology on free groups, Trans. Amer. Math. Soc.352(5) (2000), 1985-2021. · Zbl 0947.20018
[14] G.Hjorth, Classification and Orbit Equivalence Relations, Mathematical Surveys and Monographs, Volume 75, (American Mathematical Society, Providence, RI, 2000). · Zbl 0942.03056
[15] W.Hodges, I.Hodkinson, D.Lascar and S.Shelah, The small index property for 𝜔-stable 𝜔-categorical structures and for the random graph, J. Lond. Math. Soc. (2)48(2) (1993), 204-218. · Zbl 0788.03039
[16] E.Hrushovski, Extending partial isomorphisms of graphs, Combinatorica12(4) (1992), 411-416. · Zbl 0767.05053
[17] G. E.Huhunaišvili, On a property of Uryson’s universal metric space, Dokl. Akad. Nauk SSSR (N.S.)101 (1955), 607-610.
[18] A.Kaïchouh, Variations on automatic continuity, Preprint, http://math.univ-lyon1.fr/homes-www/kaichouh/Variations/ on automatic continuity.pdf.
[19] R. R.Kallman, Uniqueness results for groups of measure preserving transformations, Proc. Amer. Math. Soc.95(1) (1985), 87-90. · Zbl 0577.28014
[20] R. R.Kallman, Uniqueness results for homeomorphism groups, Trans. Amer. Math. Soc.295(1) (1986), 389-396. · Zbl 0597.57019
[21] A. S.Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, Volume 156 (Springer, New York, 1995). · Zbl 0819.04002
[22] A. S.Kechris, Global Aspects of Ergodic Group Actions, Mathematical Surveys and Monographs, Volume 160 (American Mathematical Society, Providence, RI, 2010). · Zbl 1189.37001
[23] A. S.Kechris and B. D.Miller, Topics in Orbit Equivalence, Lecture Notes in Mathematics, Volume 1852 (Springer, Berlin, 2004). · Zbl 1058.37003
[24] A. S.Kechris and C.Rosendal, Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proc. Lond. Math. Soc. (3)94(2) (2007), 302-350. · Zbl 1118.03042
[25] J.Kittrell and T.Tsankov, Topological properties of full groups, Ergod. Th. & Dynam. Sys.30(2) (2010), 525-545. · Zbl 1185.37010
[26] M.Malicki, Consequences of the existence of ample generics and automorphism groups of homogeneous metric structures, J. Symbolic Logic81(3) (2016), 876-886. · Zbl 1403.03087
[27] M.Malicki, The automorphism group of the Lebesgue measure has no non-trivial subgroups of index <2^𝜔, Colloq. Math.133(2) (2013), 169-174. · Zbl 1307.22011
[28] K.Mann, Automatic continuity for homeomorphism groups and applications. With an appendix by Frédéric Le Roux and Mann, Geom. Topol.20(5) (2016), 3033-3056. · Zbl 1362.57044
[29] S.Mazur and S.Ulam, Sur les transformationes isométriques despaces vectoriels normés, C. R. Acad. Sci. Paris194 (1932), 946-948. · Zbl 0004.02103
[30] V.Pestov, Dynamics of Infinite-dimensional Groups, University Lecture Series, Volume 40, (American Mathematical Society, Providence, RI, 2006). The Ramsey-Dvoretzky-Milman phenomenon, Revised edition of Dynamics of Infinite-dimensional Groups and Ramsey-type Phenomena, Inst. Mat. Pura. Apl. (IMPA), Rio de Janeiro, 2005; MR2164572.
[31] V. G.Pestov, A theorem of Hrushovski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space, Topology Appl.155(14) (2008), 1561-1575. · Zbl 1152.46062
[32] J.-E.Pin and C.Reutenauer, A conjecture on the Hall topology for the free group, Bull. Lond. Math. Soc.23(4) (1991), 356-362. · Zbl 0754.20007
[33] L.Ribes and P. A.Zalesskii, On the profinite topology on a free group, Bull. Lond. Math. Soc.25(1) (1993), 37-43. · Zbl 0811.20026
[34] C.Rosendal, Automatic continuity in homeomorphism groups of compact 2-manifolds, Israel J. Math.166 (2008), 349-367. · Zbl 1155.54025
[35] C.Rosendal, Automatic continuity of group homomorphisms, Bull. Symbolic Logic15(2) (2009), 184-214. · Zbl 1173.03037
[36] C.Rosendal, Finitely approximable groups and actions. Part I: the Ribes-Zalesskiĭ property, J. Symbolic Logic76(4) (2011), 1297-1306. · Zbl 1250.03085
[37] C.Rosendal and S.Solecki, Automatic continuity of homomorphisms and fixed points on metric compacta, Israel J. Math.162 (2007), 349-371. · Zbl 1146.22003
[38] K.Slutsky, Automatic continuity for homomorphisms into free products, J. Symbolic Logic78(4) (2013), 1288-1306. · Zbl 1315.03084
[39] S.Solecki, Extending partial isometries, Israel J. Math.150 (2005), 315-331. · Zbl 1124.54012
[40] H.Steinhaus, Sur les distances des points dans les ensembles de mesure positive, Fund. Math.1 (1920), 93-104. · JFM 47.0179.02
[41] L.Stojanov, Total minimality of the unitary groups, Math. Z.187(2) (1984), 273-283. · Zbl 0528.22016
[42] K.Tent and M.Ziegler, The isometry group of the bounded Urysohn space is simple, Bull. Lond. Math. Soc.45(5) (2013), 1026-1030. · Zbl 1291.03075
[43] K.Tent and M.Ziegler, On the isometry group of the Urysohn space, J. Lond. Math. Soc. (2)87(1) (2013), 289-303. · Zbl 1273.03136
[44] T.Tsankov, Automatic continuity for the unitary group, Proc. Amer. Math. Soc.141(10) (2013), 3673-3680. · Zbl 1281.54010
[45] V. V.Uspenskij, On subgroups of minimal topological groups, Topology Appl.155(14) (2008), 1580-1606. · Zbl 1166.22002
[46] A. M.Vershik, Extensions of the partial isometries of the metric spaces and finite approximation of the group of isometries of urysohn space, Preprint, 2005.
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