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On Følner sets in topological groups

Part of: Topological and differentiable algebraic systems Connections with other structures, applications Abstract harmonic analysis

Published online by Cambridge University Press:  16 May 2018

Friedrich Martin Schneider  and
Andreas Thom
Friedrich Martin Schneider
Affiliation:
Institute of Algebra, TU Dresden, 01062 Dresden, Germany email [email protected]
Andreas Thom
Affiliation:
Institute of Geometry, TU Dresden, 01062 Dresden, Germany email [email protected]
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Abstract

We extend Følner’s amenability criterion to the realm of general topological groups. Building on this, we show that a topological group $G$ is amenable if and only if its left-translation action can be approximated in a uniform manner by amenable actions on the set  $G$ . As applications we obtain a topological version of Whyte’s geometric solution to the von Neumann problem and give an affirmative answer to a question posed by Rosendal.

Keywords

topological groupamenabilityalmost invarianceparadoxicalitycoarse geometry

MSC classification

Primary: 54H11: Topological groups 54H20: Topological dynamics 22A10: Analysis on general topological groups 43A07: Means on groups, semigroups, etc.; amenable groups
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 7 , July 2018 , pp. 1333 - 1361
Copyright
© The Authors 2018 

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References

Babenko, I. K. and Bogatyĭ, S. A., Amenability of the substitution group of formal power series , Izv. Ross. Akad. Nauk Ser. Mat. 75 (2011), 19–34.Google Scholar
Barroso, C. S., Mbombo, B. R. and Pestov, V. G., On topological groups with an approximate fixed point property , An. Acad. Brasil. Ciênc. 89 (2017), 19–30.Google Scholar
Carderi, A. and Thom, A. B., An exotic group as limit of finite special linear groups , Ann. Inst. Fourier 68 (2018), 257–273.Google Scholar
Ceccherini-Silberstein, T., Grigorchuk, R. I. and de la Harpe, P., Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces , Proc. Steklov Inst. Math. 224 (1999), 57–97.Google Scholar
Day, M. M., Amenable semigroups , Illinois J. Math. 1 (1957), 509–544.CrossRefGoogle Scholar
Følner, E., On groups with full Banach mean value , Math. Scand. 3 (1955), 243–254.Google Scholar
Gaboriau, D. and Lyons, R., A measurable-group-theoretic solution to von Neumann’s problem , Invent. Math. 177 (2009), 533–540.CrossRefGoogle Scholar
Gromov, M. and Milman, V. D., A topological application of the isoperimetric inequality , Amer. J. Math. 105 (1983), 843–854.CrossRefGoogle Scholar
Hall, P., On representatives of subsets , J. Lond. Math. Soc. (2) 10 (1935), 26–30.Google Scholar
Kechris, A. S., Global aspects of ergodic group actions, Mathematical Surveys and Monographs, vol. 160 (American Mathematical Society, Providence, RI, 2010).Google Scholar
Lubotzky, A. and Shalom, Y., Finite representations in the unitary dual and Ramanujan groups , in Discrete geometric analysis, Contemporary Mathematics, vol. 347 (American Mathematical Society, Providence, RI, 2004), 173–189.Google Scholar
Marks, A. and Unger, S., Baire measurable paradoxical decompositions via matchings , Adv. Math. 289 (2016), 397–410.Google Scholar
Namioka, I., Følner’s conditions for amenable semi-groups , Math. Scand. 15 (1964), 18–28.Google Scholar
Naor, A. and Peres, Y., L p compression, traveling salesmen, and stable walks , Duke Math. J. 157 (2011), 53–108.Google Scholar
Neufang, M., Pachl, J. and Salmi, P., Uniform equicontinuity, multiplier topology and continuity of convolution , Arch. Math. (Basel) 104 (2015), 367–376.Google Scholar
von Neumann, J., Über die analytischen Eigenschaften von Gruppen linearer Transformationen und ihrer Darstellungen , Math. Z. 30 (1929), 3–42.Google Scholar
Ol’šanskiĭ, A. J., On the question of the existence of an invariant mean on a group , Uspekhi Mat. Nauk 35 (1980), 199–200.Google Scholar
Ore, O., Graphs and matching theorems , Duke Math. J. 22 (1955), 625–639.Google Scholar
Pachl, J., Uniform spaces and measures, Fields Institute Monographs, vol. 30 (Springer, New York, 2013).Google Scholar
Pachl, J. and Steprāns, J., Continuity of convolution on SIN groups , Canad. Math. Bull. 60 (2017), 845–854.Google Scholar
Pestov, V. G., On free actions, minimal flows, and a problem by Ellis , Trans. Amer. Math. Soc. 350 (1998), 4149–4165.Google Scholar
Pestov, V. G., Dynamics of infinite-dimensional groups: the Ramsey–Dvoretzky–Milman phenomenon, University Lecture Series, vol. 40 (American Mathematical Society, Providence, RI, 2006).Google Scholar
Pestov, V. G., A theorem of Hrushovski–Solecki–Vershik applied to uniform and coarse embeddings of the Urysohn metric space , Topology Appl. 155 (2008), 1561–1575.CrossRefGoogle Scholar
Pestov, V. G., Review of [ BB11 ], MathSciNet; MR 2830241 (2012e:43002), URL: http://www.ams.org/mathscinet-getitem?mr=2830241.Google Scholar
Rickert, N. W., Amenable groups and groups with the fixed point property , Trans. Amer. Math. Soc. 127 (1967), 221–232.Google Scholar
Rosenblatt, J. M., A generalization of Følner’s condition , Math. Scand. 33 (1973), 153–170.CrossRefGoogle Scholar
Rosendal, C., Coarse geometry of topological groups, book manuscript (2017),http://homepages.math.uic.edu/∼rosendal/PapersWebsite/Coarse-Geometry-Book17.pdf.Google Scholar
Rosendal, C., Equivariant geometry of Banach spaces and topological groups , Forum Math. Sigma 5 (2017), e22; 62 pages.Google Scholar
Schneider, F. M., About von Neumann’s problem for locally compact groups, J. Noncommut. Geom., to appear. Preprint (2017), arXiv:1702.07955 [math.GR].Google Scholar
Schneider, F. M. and Thom, A. B., Topological matchings and amenability , Fund. Math. 238 (2017), 167–200.CrossRefGoogle Scholar
Stern, J., Some applications of model theory in Banach space theory , Ann. Math. Logic 9 (1976), 49–121.Google Scholar
Stern, J., Ultrapowers and local properties of Banach spaces , Trans. Amer. Math. Soc. 240 (1978), 231–252.Google Scholar
Weil, A., Sur les espaces à structure uniforme et sur la topologie générale, Actualités Scientifiques et Industrielles, vol. 551 (Hermann, Paris, 1937).Google Scholar
Whyte, K., Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture , Duke Math. J. 99 (1999), 93–112.Google Scholar