Geometry & Topology

Equivariant concentration in topological groups

Friedrich Martin Schneider

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove that, if G is a second-countable topological group with a compatible right-invariant metric d and ( μ n ) n is a sequence of compactly supported Borel probability measures on G converging to invariance with respect to the mass transportation distance over d and such that ( spt μ n , d spt μ n , μ n spt μ n ) n concentrates to a fully supported, compact  mm –space ( X , d X , μ X ) , then X is homeomorphic to a G –invariant subspace of the Samuel compactification of G . In particular, this confirms a conjecture by Pestov and generalizes a well-known result by Gromov and Milman on the extreme amenability of topological groups. Furthermore, we exhibit a connection between the average orbit diameter of a metrizable flow of an arbitrary amenable topological group and the limit of Gromov’s observable diameters along any net of Borel probability measures UEB–converging to invariance over the group.

Article information

Source
Geom. Topol., Volume 23, Number 2 (2019), 925-956.

Dates
Received: 18 January 2018
Revised: 2 May 2018
Accepted: 14 July 2018
First available in Project Euclid: 17 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.gt/1555466433

Digital Object Identifier
doi:10.2140/gt.2019.23.925

Mathematical Reviews number (MathSciNet)
MR3939055

Zentralblatt MATH identifier
07056056

Subjects
Primary: 54H11: Topological groups [See also 22A05] 54H20: Topological dynamics [See also 28Dxx, 37Bxx] 22A10: Analysis on general topological groups 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Keywords
topological groups topological dynamics measure concentration observable distance observable diameter metric measure spaces

Citation

Schneider, Friedrich Martin. Equivariant concentration in topological groups. Geom. Topol. 23 (2019), no. 2, 925--956. doi:10.2140/gt.2019.23.925. https://projecteuclid.org/euclid.gt/1555466433


Export citation

References

  • A Arhangelskii, M Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics 1, Atlantis, Paris (2008)
  • J Auslander, Minimal flows and their extensions, North-Holland Mathematics Studies 153, North-Holland, Amsterdam (1988)
  • I Ben Yaacov, J Melleray, T Tsankov, Metrizable universal minimal flows of Polish groups have a comeagre orbit, Geom. Funct. Anal. 27 (2017) 67–77
  • A Bouziad, J-P Troallic, A precompactness test for topological groups in the manner of Grothendieck, Topology Proc. 31 (2007) 19–30
  • A Carderi, A Thom, An exotic group as limit of finite special linear groups, Ann. Inst. Fourier $($Grenoble$)$ 68 (2018) 257–273
  • R Ellis, Lectures on topological dynamics, W A Benjamin, New York (1969)
  • K Funano, Concentration of maps and group actions, Geom. Dedicata 149 (2010) 103–119
  • A L Gibbs, F E Su, On choosing and bounding probability metrics, Int. Stat. Rev. 70 (2002) 419–435
  • T Giordano, V Pestov, Some extremely amenable groups related to operator algebras and ergodic theory, J. Inst. Math. Jussieu 6 (2007) 279–315
  • E Glasner, B Tsirelson, B Weiss, The automorphism group of the Gaussian measure cannot act pointwise, Israel J. Math. 148 (2005) 305–329
  • E Glasner, B Weiss, Minimal actions of the group $\mathbb S(\mathbb{Z})$ of permutations of the integers, Geom. Funct. Anal. 12 (2002) 964–988
  • M Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics 152, Birkhäuser, Boston (1999)
  • M Gromov, V D Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983) 843–854
  • A S Kechris, V G Pestov, S Todorcevic, Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal. 15 (2005) 106–189
  • M Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs 89, Amer. Math. Soc., Providence, RI (2001)
  • J Melleray, L Nguyen Van Thé, T Tsankov, Polish groups with metrizable universal minimal flows, Int. Math. Res. Not. 2016 (2016) 1285–1307
  • V D Milman, Diameter of a minimal invariant subset of equivariant Lipschitz actions on compact subsets of ${\mathbb R}^k$, from “Geometrical aspects of functional analysis (1985/86)” (J Lindenstrauss, V D Milman, editors), Lecture Notes in Math. 1267, Springer (1987) 13–20
  • J Pachl, Uniform spaces and measures, Fields Institute Monographs 30, Springer (2013)
  • V G Pestov, On free actions, minimal flows, and a problem by Ellis, Trans. Amer. Math. Soc. 350 (1998) 4149–4165
  • V Pestov, Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups, Israel J. Math. 127 (2002) 317–357
  • V Pestov, Dynamics of infinite-dimensional groups: the Ramsey–Dvoretzky–Milman phenomenon, University Lecture Series 40, Amer. Math. Soc., Providence, RI (2006)
  • V Pestov, The isometry group of the Urysohn space as a Le\'vy group, Topology Appl. 154 (2007) 2173–2184
  • V Pestov, Concentration of measure and whirly actions of Polish groups, from “Probabilistic approach to geometry” (M Kotani, M Hino, T Kumagai, editors), Adv. Stud. Pure Math. 57, Math. Soc. Japan, Tokyo (2010) 383–403
  • V G Pestov, F M Schneider, On amenability and groups of measurable maps, J. Funct. Anal. 273 (2017) 3859–3874
  • S T Rachev, L Rüschendorf, Mass transportation problems, II: Applications, Springer (1998)
  • F M Schneider, Equivariant dissipation in non-archimedean groups, preprint (2018) To appear in Israel J. Math.
  • F M Schneider, A Thom, On Følner sets in topological groups, Compos. Math. 154 (2018) 1333–1361
  • T Shioya, Metric measure geometry: Gromov's theory of convergence and concentration of metrics and measures, IRMA Lectures in Mathematics and Theoretical Physics 25, Eur. Math. Soc., Zürich (2016)
  • S Solecki, Actions of non-compact and non-locally compact Polish groups, J. Symbolic Logic 65 (2000) 1881–1894
  • V Uspenskij, On universal minimal compact $G$–spaces, Topology Proc. 25 (2000) 301–308
  • V V Uspenskij, On subgroups of minimal topological groups, Topology Appl. 155 (2008) 1580–1606
  • C Villani, Optimal transport: old and new, Grundl. Math. Wissen. 338, Springer (2009)
  • J de Vries, Elements of topological dynamics, Mathematics and its Applications 257, Kluwer, Dordrecht (1993)
  • A Weil, Sur les espaces à structure uniforme et sur la topologie générale, Act. Sci. Ind. 551 (1937) 39 pages