arXiv subjects
Publications and source records attributed to Vladimir Pestov.
This is a selection of open problems dealing with ``large'' (non locally compact) topological groups and concerning extreme amenability (fixed point on compacta property), oscillation stability, universal minimal flows and other aspects of universality, and unitary representations.
A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a $C^\ast$-algebra $A$ is nuclear if and only if the unitary group $U(A)$ with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of $L^p(0,1)$, $1\leq p<\infty$, extending a classical result of Gromov and Milman ($p=2$). We show that a measure class preserving equivalence relation $\mathcal R$ on a standard Borel space is amenable if and only if the full group $[{\mathcal R}]$, equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a nontrivial space in the sense of Gromov occuring in the automorphism groups of injective factors of type $III$.
We prove that the isometry group $\Iso(\Ur)$ of the universal Urysohn metric space $\Ur$ 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 stating that $\Iso(\Ur)$ has a dense locally finite subgroup.
We suggest a variation of the Hellerstein--Koutsoupias--Papadimitriou indexability model for datasets equipped with a similarity measure, with the aim of better understanding the structure of indexing schemes for similarity-based search and the geometry of similarity workloads. This in particular provides a unified approach to a great variety of schemes used to index into metric spaces and facilitates their transfer to more general similarity measures such as quasi-metrics. We discuss links between performance of indexing schemes and high-dimensional geometry. The concepts and results are illustrated on a very large concrete dataset of peptide fragments equipped with a biologically significant similarity measure.
The paper has been withdrawn by the author as the main technical result (Th. 4.6) is false, as well as Corol. 4.3 on which it is based. The question of oscillation stability of the Urysohn space remains therefore open.
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 isometries of Urysohn's universal complete separable metric space $\mathbb U$, equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The same is true if $\U$ is replaced with any generalized Urysohn metric space $U$ that is sufficiently homogeneous. Modulo a recent theorem by Uspenskij that every topological group embeds into a topological group of the form $\Iso(U)$, our result implies that every topological group embeds into an extremely amenable group (one admitting an invariant multiplicative mean on bounded right uniformly continuous functions). By way of the proof, we show that every topological group is approximated by finite groups in a certain weak sense. Our technique also results in a new proof of the extreme amenability (fixed point on compacta property) for infinite orthogonal groups. Going in the opposite direction, we deduce some Ramsey-type theorems for metric subspaces of Hilbert spaces and for spherical metric spaces from existing results on extreme amenability of infinite unitary groups and groups of isometries of Hilbert spaces.
These are introductory notes on some aspects of concentration of measure in the presence of an acting group and its links to Ramsey theory.
Let $F\subseteq H\subseteq G$ be closed subgroups of a locally compact group. In response to a 1972 question by Eymard, we construct an example where the homogeneous factor-space $G/F$ is amenable in the sense of Eymard-Greenleaf, while $H/F$ is not. (In our example, $G$ is discrete.) As a corollary which answers a 1990 question by Bekka, the induced representation $\ind_H^G(ρ)$ can be amenable in the sense of Bekka even if $ρ$ is not amenable. The second example, answering another question by Bekka, shows that $\ind_H^G(ρ)$ need not be amenable even if both the representation $ρ$ and the coset space $G/H$ are amenable.
We show that every abelian Polish group is the topological 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 unitary group.
We discuss some aspects of approximating functions on high-dimensional data sets with additive functions or ANOVA decompositions, that is, sums of functions depending on fewer variables each. It is seen that under appropriate smoothness conditions, the errors of the ANOVA decompositions are of order $O(n^{m/2})$ for approximations using sums of functions of up to $m$ variables under some mild restrictions on the (possibly dependent) predictor variables. Several simulated examples illustrate this behaviour.
We discuss some techniques related to equivariant compactifications of uniform spaces and amenability of topological groups. In particular, we give a new proof of a recent result by Glasner and Weiss describing the universal minimal flow of the infinite symmetric group ${\mathfrak S}_\infty$ with the standard Polish topology, and extend Bekka's concept of an amenable representation, enabling one to deduce non-amenability of the Banach--Lie groups $\GL(L_p)$ and $\GL(\ell_p)$, $1\leq p <\infty$.
A topological group $G$ is extremely amenable if every continuous action of $G$ on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology but not with the uniform one. Strengthening a de la Harpe's result, we show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group.
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 topological groups on compacta, embeddings of topological groups, free topological groups, and `massive' groups (such as groups of homeomorphisms of compacta and groups of isometries of various metric spaces).
It is shown that every separable abelian topological group is isomorphic with a topological subgroup of a monothetic group (that is, a topological group with a single topological generator). In particular, every separable metrizable abelian group embeds into a metrizable monothetic group. More generally, we describe all topological groups that can be embedded into monothetic groups: they are exactly abelian topological groups of weight $\leq\frak c$ covered by countably many translations of every nonempty open subset.
The aim of this paper is to propose a geometric framework for modelling similarity search in large and multidimensional data spaces of general nature, which seems to be flexible enough to address such issues as analysis of complexity, indexability, and the `curse of dimensionality.' Such a framework is provided by the concept of the so-called similarity workload, which is a probability metric space $Ω$ (query domain) with a distinguished finite subspace $X$ (dataset), together with an assembly of concepts, techniques, and results from metric geometry. They include such notions as metric transform, $\e$-entropy, and the phenomenon of concentration of measure on high-dimensional structures. In particular, we discuss the relevance of the latter to understanding the curse of dimensionality. As some of those concepts and techniques are being currently reinvented by the database community, it seems desirable to try and bridge the gap between database research and the relevant work already done in geometry and analysis.
We suggest that the curse of dimensionality affecting the similarity-based search in large datasets is a manifestation of the phenomenon of concentration of measure on high-dimensional structures. We prove that, under certain geometric assumptions on the query domain $Ω$ and the dataset $X$, if $Ω$ satisfies the so-called concentration property, then for most query points $x^\ast$ the ball of radius $(1+\e)d_X(x^\ast)$ centred at $x^\ast$ contains either all points of $X$ or else at least $C_1\exp(-C_2\e^2n)$ of them. Here $d_X(x^\ast)$ is the distance from $x^\ast$ to the nearest neighbour in $X$ and $n$ is the dimension of $Ω$.
It is proved that a discrete group $G$ is amenable if and only if for every unitary representation of $G$ in an infinite-dimensional Hilbert space $\cal H$ the maximal uniform compactification of the unit sphere $\s_{\cal H}$ has a $G$-fixed point, that is, the pair $(\s_{\cal H},G)$ has the concentration property in the sense of Milman. Consequently, the maximal $U({\cal H})$-equivariant compactification of the sphere in a Hilbert space $\cal H$ has no fixed points, which answers a 1987 question by Milman. This is a version as of November 19, 1998, incorporating some revisions.
We discuss the problem of finding an analogue of the concept of a topological space in supergeometry, motivated by a search for a procedure to compactify a supermanifold along odd coordinates. In particular, we examine the topologies naturally arising on the sets of points of locally ringed superspaces, and show that in the presence of a nontrivial odd sector such topologies are never compact. The main outcome of our discussion is that not only the usual framework of supergeometry (the theory of locally ringed spaces), but the more general approach of the functor of points, need to be further enlarged.