Infinite-dimensional Polish groups and Property (T)

Ibarlucía, Tomás

doi:10.1007/s00222-020-00998-z

Cited by 4 publications

(2 citation statements)

References 26 publications

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“…It is natural to ask if there are examples of non-precompact topological groups which do have weak Dynamical Property (T). Given Corollary 7.5 and the recent result of Ibarlucía [25] that all Polish RPC groups have Property (T), perhaps it is true that every RPC group has weak Dynamical Property (T).…”

Section: Dynamical Property (T)mentioning

confidence: 99%

Big Ramsey degrees and topological dynamics

Zucker

2018

Groups Geom. Dyn.

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We consider Fraïssé structures whose objects have finite big Ramsey degree and ask what consequences this has for the dynamics of the automorphism group. Motivated by a theorem of D. Devlin about the partition properties of the rationals, we define the notion of a big Ramsey structure, a single structure which codes the big Ramsey degrees of a given Fraïssé structure. This in turn leads to the definition of a completion flow ; we show that if a Fraïssé structure admits a big Ramsey structure, then the automorphism group admits a unique universal completion flow. We also discuss the problem of when big Ramsey structures exist and explore connections to the notion of oscillation stability defined by Kechris, Pestov, and Todorčević [10]. Introduction, main definitions, and statements of theoremsConsider the statement of Ramsey's theorem: for every k, r < ω, we have ω → (ω) k r This "arrow notation" is shorthand for the following statement: for any coloring γ :[ω] k → r, there is an infinite S ⊆ ω so that |γ"([S] k )| = 1. We often say that S is monochromatic for γ. Ramsey's theorem can be generalized in many directions. Erdős and Rado [4] considered coloring finite tuples from larger cardinals while demanding larger monochromatic sets. Galvin and Prikry [7], and later Ellentuck [3], considered suitably definable colorings of the infinite subsets of ω. The generalization that we will consider in this paper is that of structural Ramsey theory. As a warmup, we will consider the structure Q, ≤ of rationals and their linear order. In what follows we will just write Q. Let us call a subset S ⊆ Q a dense linear order, or DLO for short, if the ordering on S inherited from Q is dense and contains no maximum or 2010 Mathematics Subject Classification. Primary: 22F50; Secondary: 03C15, 03E02, 03E75, 05D10, 37B20, 54D80, 54H20.

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Section: Dynamical Property (T)mentioning

confidence: 99%

Big Ramsey degrees and topological dynamics

Zucker

2018

Groups Geom. Dyn.

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“…Since then, the research on Property (T) for non-locally compact Polish groups has undergone a significant development, see e.g. [9], [59], [41], [27]. Curiously, the same cannot be said about Property (FH).…”

Section: Properties (T) and (Fh)mentioning

confidence: 99%

Large scale geometry of Banach-Lie groups

Ando¹,

Doucha²,

Matsuzawa³

2020

Preprint

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We initiate the large scale geometric study of Banach-Lie groups, especially of linear Banach-Lie groups. We show that the exponential length, originally introduced by Ringrose for unitary groups of C * -algebras, defines the quasi-isometry type of any connected Banach-Lie group. As an illustrative example, we consider unitary groups of separable abelian unital C *algebras with spectrum having finitely many components, which we classify up to topological isomorphism and up to quasi-isometry, in order to highlight the difference. The main results then concern the Haagerup property, and Properties (T) and (FH). We present the first non-trivial non-abelian and non-localy compact groups having the Haagerup property, most of them being non-amenable. These are the groups U2(M, τ ), where M is a semifinite von Neumann algebra with a normal faithful semifinite trace τ . Finally, we investigate the groups En(A), which are closed subgroups of GL(n, A) generated by elementary matrices, where A is a unital Banach algebra. We show that for n ≥ 3, all these groups have Property (T) and they are unbounded, so they have Property (FH) non-trivially. On the other hand, if A is an infinite-dimensional unital C * -algebra, then E2(A) does not have the Haagerup property. If A is moreover abelian and separable, then SL(2, A) does not have the Haagerup property.

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