Topology Atlas Document # taic-35 Topology Atlas Invited Contributions vol. 5 issue 1 (2000) 5-10

The concentration phenomenon and topological groups

Vladimir Pestov

Invited Contribution

Introduction

The phenomenon of concentration of measure on high-dimensional structures [21,11] (also known as the geometric law of large numbers) is an important development in modern analysis and geometry, manifesting itself across a wide range of mathematical sciences, particulaly geometric functional analysis [19,22,18,14], probability theory [33], graph theory [16], diverse fields of computer science [17, 1, 32, 27], and statistical physics [11].

Some of the most interesting recent developments related to concentration of measure occured in topological groups and compact G-spaces. The present author believes that those are only the tip of an iceberg and the potential for interaction of the phenomenon with topology is far greater than that, so that concentration deserves to be better known among topologists.

Overview

Informally, the main idea of the phenomenon of concentration of measure can be expressed by saying that on `high-dimensional' spaces equipped with distance and measure, `almost all' points are `very close' to every subset containing at least half of all points. Historically the first observed manifestation of the phenomenon is the Law of Large Numbers, where the space in question is the Hamming cube {0,1}n of all n-bit binary strings equipped with the Hamming distance and counting measure. Then in the 30's Paul Lévy has observed and stated explicitely the concentration phenomenon for the Euclidean spheres Sn. However, his results lay dormant until about 1970, when Vitali Milman [19] realized that the famous Dvoretzky's Theorem on almost spherical sections of convex bodies is a manifestation of the concentration phenomenon on spheres. The paper [19] marks the beginning of the modern era in geometry of high-dimensional structures. The paper [13] by Gromov and Milman, building on Milman's seminal paper [19], had initiated applications of the concentration phenomenon in topological transformation groups and related fields.

Concentration and fixed points

Let \Omega = (\Omega, \rho, \mu) be a metric space equipped with a probability measure (that is, \mu is a non-negative, countably additive function on Borel subsets of the metric space (\Omega, \rho), such that \mu(\emptyset) = 0 and \mu(\Omega>) = 1). Let A \subset \Omega be an arbitrary Borel subset with \mu(A) >= 1/2. Denote by O\epsilon(A) the open \epsilon-neighbourhood of A in \Omega.


Fig. 1. An illustration to the concentration of measure.

A heuristic way to formulate the phenomenon of concentration of measure is to say that

A rigorous way to capture the above is through invoking the asymptotic behaviour: say that a net of probability measures (\mu\alpha) on a metric space (X,\rho) concentrates if, whenever A\alpha \subset X satisfy liminf\alpha\mu\alpha(A\alpha) > 0, then for every \epsilon > 0 one has \mu\alphaO\epsilon(A\alpha) --> 1.

If now in addition a group G acts upon X in such a way that the measures (\mu\alpha) are in some suitable sense `almost invariant' under the action, then every equivariant compactification of X has a G-fixed point. This situation can be formalized either through the concept of a Lévy group (or, more generally, Lévy dynamical system), as in [13,20,21], or in a more general setting proposed by us in [30].

One significant trend of development stemming from the results of Gromov and Milman [13] and based on the above technique is as follows. Many `infinite-dimensional' topological groups G of importance have the fixed point on compacta property, or are extremely amenable, that is, every continuous action of G on a compact space has a fixed point.

Among such topological groups are: the infinite unitary groups U(H) with the strong operator topology [13,20,21], the group of measurable maps from the interval to the circle group with the L1-metric [8], the group of orientation-preserving homeomorphisms Homeo+(I) [24], the group of measure-preserving automorphisms of the standard sigma-finite Lebesgue space with the weak topology [7], and some others. This property is a drastically strengthened form of amenability, never possessed by locally compact groups [34].

It is worth noting that the concentration phenomenon is well-known, at the intuitive level, to be `similar' to Ramsey-type theorems (cf. [10]), and this parallel was made more explicit for topological transformation groups in [30].

Some open problems and research directions

Below is a sample of open problems (probably) related to the concentration phenomenon in topological (transformation) groups. These problems are located on the crossroads of topology with other disciplines and are therefore potentially capable of pumping new blood into the `general topological algebra.'

Bohr neighbourhoods of integers and dissipation of measure

Every extremely amenable abelian topological group G is minimally almost periodic, that is, admits no non-trivial continuous characters. It is unknown if the converse is true. In the case of monothetic groups, this question belongs to Glasner [8], who observed that a distinguishing example would answer in the negative the following old-standing question from combinatorial number theory and harmonic analysis, rooted in the work of Bologiuboff, Fø lner, Cotlar-Ricabarra, Ellis and Keynes, and Veech (cf. [5,4,35]). Recall that a subset S \subset Z is relatively dense if S+F = Z for some finite F. The Bohr topology on a topological abelian group is the topology induced by all continuous characters.

If S \subset Z is a relatively dense set, does S-S form a Bohr neighbourhood of identity?

To construct a counter-example to Glasner's question, one apparently needs to maintain a fine balance between minimal almost periodicity and the property that goes in the opposite direction from concentration: it is measure dissipation, cf. [11].

Ramsey-type theorems for metric spaces

It is worth exploring the link between the fixed point on compacta property and more sophisticated versions of Ramsey theorems for metric spaces, graphs, and other structures. (Cf. [30].) In particular, the central open question of Euclidean Ramsey theory (is every finite spherical metric space Ramsey? [9]) is linked to the existence of fixed points / concentration in the group of affine isometries of R\infty, equipped with the topology of pointwise convergence on the discrete topological space R\infty. (This is the tame topology known in representation theory.)

Concentration in injective von Neumann algebras

Examples of extremely amenable groups by Gromov and Milman (U(H)s, [13]) and by Glasner-Furstenberg-B. Weiss (L(X,U(1)), [8]) are of a common nature: both of them are unitary groups of injective von Neumann algebras equipped with the ultraweak topology [31]. Conversely, if a von Neumann algebra W is such that the unitary group with the ultraweak topology is amenable, then W is injective [15].

What are those injective von Neumann algebras (factors) whose unitary groups with the ultraweak topology are extremely amenable?

The Banach-Mazur problem

The Banach-Mazur problem asks whether a separable Banach space E with transitive norm is isometrically isomorphic to a Hilbert space. (Cf. [3].) The fixed point on compacta property for the transitive groups of isometries, if established, would answer the problem affirmatively at least in the particular case where E is isomorphic to a Hilbert space. Here the results from [14] may apply.

Conclusion

Since both the existence of fixed points and combinatorial properties of Ramsey type are very common and important throughout mathematics, the above trends lead to many interesting open questions. At the same time, we are still, quite obviously, at the very beginning of the road, with subtler aspects of concentration phenomenon not yet having made their way into the realm of topology.

In fact, metric spaces with measure (mm-spaces) form just one possible setting for studying the concentration phenomenon. For one thing, uniform spaces with measure provide the most obvious, and often necessary (for instance, in order to handle non-metrizable topological groups), generalization at no extra cost [28,29,30]. A radically new (`ergodic') setting for analyzing concentration has been proposed recently by Gromov [12]. The beginnings of yet another (`affine') setting are contained in the preprint by Giannopoulos and Milman [6]. Nothing is set in concrete yet!

It is sensible to expect the concentration phenomenon growing in importance in topology when (and if) the `probabilistic' treatment of topological spaces, fucntions, and other objects becomes as widespread as the similar approach already is in geometric functional analysis and graph theory. (One particularly interesting recent development here is Vershik's theorem [36] asserting that the completion of the set of integers equipped with a randomly chosen metric is almost surely isometrically isomorphic to the Urysohn metric space.) Here in particular the concept of phase transition, originated in statistical physics (cf. [23]) and already well-established in discrete mathematics [2], can move to the centre stage.

References

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School of Mathematical and Computing Sciences,
Victoria University of Wellington,
P.O. Box 600, Wellington,
New Zealand

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Date received: June 12, 2000

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