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(A) We study in this paper some connections between the Fra¨ıssé theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. A prime concern of topological dynamics is the study of continuous actions of (Hausdorff) topological groups G on (Hausdorff) compact spaces… (More)

- VLADIMIR PESTOV, V. PESTOV
- 1999

This is an account of one man's view of the current perspective of theory of topological groups. We survey some recent developments which are, from our viewpoint, indicative of the future directions, concentrating on actions of topologi-cal groups on compacta, embeddings of topological groups, free topological groups, and 'massive' groups (such as groups of… (More)

(A) We study in this paper some connections between the Fra¨ıssé theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. A prime concern of topological dynamics is the study of continuous actions of (Haus-dorff) topological groups G on (Hausdorff) compact spaces… (More)

- V. PESTOV
- 1997

This small survey of basic universal constructions related to the actions of topological groups on compacta is centred around a new result — an intrinsic description of extremely amenable topological groups (i.e., those having a fixed point in each compactum they act upon), solving a 1967 problem by Granirer. Another old problem whose solution (in the… (More)

- Vladimir Pestov
- Bulletin of Symbolic Logic
- 2008

This is an introductory survey of the emerging theory of two new classes of (discrete, countable) groups, called hyperlinear and sofic groups. They can be characterized as subgroups of metric ultraproducts of families of, respectively, uni-tary groups U (n) and symmetric groups S n , n ∈ N. Hyperlinear groups come from theory of operator algebras (Connes'… (More)

- Vladimir Pestov
- 2005

We prove that the isometry group Iso (U) of the universal Urysohn metric space U equipped with the natural Polish topology is a Lévy group in the sense of Gromov and Milman, that is, admits an approximating chain of compact (in fact, finite) subgroups , exhibiting the phenomenon of concentration of measure. This strengthens an earlier result by Vershik… (More)

- Ilya Volnyansky, Vladimir Pestov
- 2009 Second International Workshop on Similarity…
- 2009

We offer a theoretical validation of the curse of dimensionality in the pivot-based indexing of datasets for similarity search, by proving, in the framework of statistical learning, that in high dimensions no pivot-based indexing scheme can essentially outperform the linear scan. A study of the asymptotic performance of pivot-based indexing schemes is… (More)

- VLADIMIR PESTOV, V. PESTOV
- 2008

In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group Iso (U) of isome-tries of Urysohn's universal complete separable metric space U, equipped with the compact-open topology, acts upon an arbitrary… (More)

- Su Gao, Vladimir Pestov
- 2008

We show that every abelian Polish group is the topo-logical factor-group of a closed subgroup of the full unitary group of a separable Hilbert space with the strong operator topology. It follows that all orbit equivalence relations induced by abelian Polish group actions are Borel reducible to some orbit equivalence relations induced by actions of the… (More)

- Vladimir Pestov
- 1997

We exhibit abelian topological groups admitting no nontrivial strongly continuous irreducible representations in Banach spaces. Among them are some abelian Banach–Lie groups and some monothetic subgroups of the uni-tary group of a separable Hilbert space.