On minimal actions of Polish groups
type: Item
instance of: scholarly article
title: On minimal actions of Polish groups
zbMATH Open document ID: 0923.54030
DOI: 10.1016/S0166-8641(97)00143-0
published in: Topology and its Applications
publication date: 1999-10-25T00:00:00Z
review text: This is an important paper, establishing the existence of a monothetic (that is, topologically singly generated) topological group \(G\) such that every continuous action of \(G\) on a compact space has a fixed point. This is done by using the technique of concentration of measure on high-dimensional structures (see the book [\textit{M. Gromov}, Metric structures for Riemannian and non-Riemannian spaces, Prog. Math. 152, Birkhäuser Verlag (1999)] for the most recently written and very fresh introduction to the topic). According to an important result by \textit{W. A. Veech} [Bull. Am. Math. Soc. 83, 775-830 (1977; Zbl 0384.28018)], every locally compact group \(G\) acts freely on a suitable compact space. An opposite sort of behaviour is captured by saying that a topological group \(G\) is extremely amenable [\textit{E. Granirer}, Math. Scand. 20, 93-113 (1967; Zbl 0204.03501)] or, as suggested in the paper under review, has the fixed point on compacta property (f.p.c.), if every continuous action of \(G\) on a compact space has a fixed point. Since such groups cannot be locally compact in view of Veech's theorem, first examples were difficult to come by, and they indeed looked more like elaborately arranged pathologies, see [\textit{W. Herer} and \textit{J. P. R. Christensen}, Math. Ann. 213, 203-210 (1975; Zbl 0311.28002)] and also [\textit{W. Banaszczyk}, Math. Ann. 264, 485-493 (1983; Zbl 0502.22010)]. However, recent developments have revealed that among the `massive', infinite-dimensional, groups, the fixed point on compacta property is rather common. The technique of concentration of measure on high-dimensional structures had led \textit{M. Gromov} and \textit{V. D. Milman} [Am. J. Math. 105, 843-854 (1983; Zbl 0522.53039)] to establish the f.p.c. property of the unitary group \(U(l_2)_s\) of an infinite-dimensional Hilbert space with the strong operator topology, though here one needs to mention that, strictly speaking, these two authors had assumed an unnec…
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 43A05
zbMATH DE Number: 1307998
zbMATH Keywords: concentration of measure
zbMATH Keywords: Lévy groups
zbMATH Keywords: fixed point on compacta property
zbMATH Keywords: Bohr neighbourhoods
author: Eli Glasner
reviewed by: Vladimir G. Pestov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1307998