Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups

Ben Yaacov, Itaï; Tsankov, Todor

doi:10.1090/tran/6883

Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups
HTML articles powered by AMS MathViewer

by Itaï Ben Yaacov and Todor Tsankov PDF

Trans. Amer. Math. Soc. 368 (2016), 8267-8294 Request permission

Abstract:

We investigate the automorphism groups of $\aleph _0$-categorical structures and prove that they are exactly the Roelcke precompact Polish groups. We show that the theory of a structure is stable if and only if every Roelcke uniformly continuous function on the automorphism group is weakly almost periodic. Analysing the semigroup structure on the weakly almost periodic compactification, we show that continuous surjective homomorphisms from automorphism groups of stable $\aleph _0$-categorical structures to Hausdorff topological groups are open. We also produce some new WAP-trivial groups and calculate the WAP compactification in a number of examples.

References

Gisela Ahlbrandt and Martin Ziegler, Quasi-finitely axiomatizable totally categorical theories, Ann. Pure Appl. Logic 30 (1986), no. 1, 63–82. Stability in model theory (Trento, 1984). MR 831437, DOI 10.1016/0168-0072(86)90037-0
Dana Bartošová and Aleksandra Kwiatkowska, Lelek fan from a projective Fraïssé limit, Fund. Math. 231 (2015), no. 1, 57–79. MR 3361235, DOI 10.4064/fm231-1-4
Itaï Ben Yaacov, Alexander Berenstein, and C. Ward Henson, Model-theoretic independence in the Banach lattices $L_p(\mu )$, Israel J. Math. 183 (2011), 285–320. MR 2811161, DOI 10.1007/s11856-011-0050-4
Itaï Ben Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov, Model theory for metric structures, Model theory with applications to algebra and analysis. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 350, Cambridge Univ. Press, Cambridge, 2008, pp. 315–427. MR 2436146, DOI 10.1017/CBO9780511735219.011
Itaï Ben Yaacov, Alexander Berenstein, and Julien Melleray, Polish topometric groups, Trans. Amer. Math. Soc. 365 (2013), no. 7, 3877–3897. MR 3042607, DOI 10.1090/S0002-9947-2013-05773-X
Itaï Ben Yaacov and Alexander Usvyatsov, On $d$-finiteness in continuous structures, Fund. Math. 194 (2007), no. 1, 67–88. MR 2291717, DOI 10.4064/fm194-1-4
John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
Dikran Dikranjan and Michael Megrelishvili, Minimality conditions in topological groups, Recent progress in general topology. III, Atlantis Press, Paris, 2014, pp. 229–327. MR 3205486, DOI 10.2991/978-94-6239-024-9_{6}
D. Gamarnik, Minimality of the group $\textrm {Autohomeom}(C)$, Serdica 17 (1991), no. 4, 197–201. MR 1201115
Edward D. Gaughan, Topological group structures of infinite symmetric groups, Proc. Nat. Acad. Sci. U.S.A. 58 (1967), 907–910. MR 215940, DOI 10.1073/pnas.58.3.907
Eli Glasner, The group $\textrm {Aut}(\mu )$ is Roelcke precompact, Canad. Math. Bull. 55 (2012), no. 2, 297–302. MR 2957245, DOI 10.4153/CMB-2011-083-2
A. Grothendieck, Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), 168–186 (French). MR 47313, DOI 10.2307/2372076
E. Glasner and M. Megrelishvili, New algebras of functions on topological groups arising from $G$-spaces, Fund. Math. 201 (2008), no. 1, 1–51. MR 2439022, DOI 10.4064/fm201-1-1
Wilfrid Hodges, Ian Hodkinson, Daniel Lascar, and Saharon Shelah, The small index property for $\omega$-stable $\omega$-categorical structures and for the random graph, J. London Math. Soc. (2) 48 (1993), no. 2, 204–218. MR 1231710, DOI 10.1112/jlms/s2-48.2.204
Wilfrid Hodges, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993. MR 1221741, DOI 10.1017/CBO9780511551574
Trevor Irwin and Sławomir Solecki, Projective Fraïssé limits and the pseudo-arc, Trans. Amer. Math. Soc. 358 (2006), no. 7, 3077–3096. MR 2216259, DOI 10.1090/S0002-9947-06-03928-6
Alexander S. Kechris and Christian Rosendal, Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 302–350. MR 2308230, DOI 10.1112/plms/pdl007
Dugald Macpherson, A survey of homogeneous structures, Discrete Math. 311 (2011), no. 15, 1599–1634. MR 2800979, DOI 10.1016/j.disc.2011.01.024
Maciej Malicki, Generic elements in isometry groups of Polish ultrametric spaces, Israel J. Math. 206 (2015), no. 1, 127–153. MR 3319634, DOI 10.1007/s11856-014-1128-6
Michael G. Megrelishvili, Every semitopological semigroup compactification of the group $H_+[0,1]$ is trivial, Semigroup Forum 63 (2001), no. 3, 357–370. MR 1851816, DOI 10.1007/s002330010076
Michael G. Megrelishvili, Operator topologies and reflexive representability, Nuclear groups and Lie groups (Madrid, 1999) Res. Exp. Math., vol. 24, Heldermann, Lemgo, 2001, pp. 197–208. MR 1858149
Vladimir Pestov, The isometry group of the Urysohn space as a Lev́y group, Topology Appl. 154 (2007), no. 10, 2173–2184. MR 2324929, DOI 10.1016/j.topol.2006.02.010
Walter Roelcke and Susanne Dierolf, Uniform structures on topological groups and their quotients, Advanced Book Program, McGraw-Hill International Book Co., New York, 1981. MR 644485
Christian Rosendal, A topological version of the Bergman property, Forum Math. 21 (2009), no. 2, 299–332. MR 2503307, DOI 10.1515/FORUM.2009.014
Christian Rosendal, Global and local boundedness of Polish groups, Indiana Univ. Math. J. 62 (2013), no. 5, 1621–1678. MR 3188557, DOI 10.1512/iumj.2013.62.5133
Wolfgang Ruppert, Compact semitopological semigroups: an intrinsic theory, Lecture Notes in Mathematics, vol. 1079, Springer-Verlag, Berlin, 1984. MR 762985, DOI 10.1007/BFb0073675
Wolfgang A. F. Ruppert, On group topologies and idempotents in weak almost periodic compactifications, Semigroup Forum 40 (1990), no. 2, 227–237. MR 1029258, DOI 10.1007/BF02573269
Marcin Sabok, Automatic continuity for isometry groups, 2013. Preprint arXiv:1312.5141.
L. Stojanov, Total minimality of the unitary groups, Math. Z. 187 (1984), no. 2, 273–283. MR 753438, DOI 10.1007/BF01161710
Todor Tsankov, Unitary representations of oligomorphic groups, Geom. Funct. Anal. 22 (2012), no. 2, 528–555. MR 2929072, DOI 10.1007/s00039-012-0156-9
Todor Tsankov, Automatic continuity for the unitary group, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3673–3680. MR 3080189, DOI 10.1090/S0002-9939-2013-11666-7
V. V. Uspenskiĭ, The Roelcke compactification of groups of homeomorphisms, Topology Appl. 111 (2001), no. 1-2, 195–205.
Vladimir Uspenskiĭ, On the group of isometries of the Urysohn universal metric space, Comment. Math. Univ. Carolin. 31 (1990), no. 1, 181–182.
Vladimir Uspenskiĭ, The Roelcke compactification of unitary groups, Abelian groups, module theory, and topology (Padua, 1997), 1998, pp. 411–419.
Vladimir Uspenskiĭ, Compactifications of topological groups, Proceedings of the ninth prague topological symposium (2001), 2002, pp. 331–346 (electronic).
Vladimir Uspenskiĭ, On subgroups of minimal topological groups, Topology Appl. 155 (2008), no. 14, 1580–1606.
Frank O. Wagner, Relational structures and dimensions, Automorphisms of first-order structures, Oxford Sci. Publ., Oxford Univ. Press, New York, 1994, pp. 153–180. MR 1325473