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


Authors: Itaï Ben Yaacov and Todor Tsankov
Journal: Trans. Amer. Math. Soc. 368 (2016), 8267-8294
MSC (2010): Primary 22F50, 22A15, 22A05; Secondary 03C45
Published electronically: June 29, 2016
MathSciNet review: 3546800
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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.


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Additional Information

Itaï Ben Yaacov
Affiliation: Université Claude Bernard – Lyon 1, Institut Camille Jordan, CNRS UMR 5208, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France

Todor Tsankov
Affiliation: Institut de Mathématiques de Jussieu–PRG, Université Paris 7, case 7012, 75205 Paris cedex 13, France

DOI: https://doi.org/10.1090/tran/6883
Keywords: WAP, stability, $\aleph_0$-categorical, group compactifications, Roelcke precompact, minimal groups, reflexively representable
Received by editor(s): May 12, 2015
Received by editor(s) in revised form: December 3, 2015
Published electronically: June 29, 2016
Additional Notes: This research was supported by the Institut Universitaire de France and ANR contract GrupoLoco (ANR-11-JS01-008).
Article copyright: © Copyright 2016 American Mathematical Society