A theorem of Hrushovski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space

A theorem proved by Hrushovski for graphs and extended by Solecki and Vershik (independently from each other) to metric spaces leads to a stronger version of ultrahomogeneity of the infinite random graph $R$, the universal Urysohn metric space \Ur, and other related objects. We show how the result can be used to average out uniform and coarse embeddings of \Ur (and its various counterparts) into normed spaces. Sometimes this leads to new embeddings of the same kind that are metric transforms and besides extend to affine representations of various isometry groups. As an application of this technique, we show that \Ur admits neither a uniform nor a coarse embedding into a uniformly convex Banach space.Comment: 23 pages, LaTeX 2e with Elsevier macros, a significant revision taking into account anonymous referee's comments, with the proof of the main result simplified and another long proof moved to the appendi

On sequences of finitely generated discrete groups

We consider sequences of finitely generated discrete subgroups Gamma_i=rho_i(Gamma) of a rank 1 Lie group G, where the representations rho_i are not necessarily faithful. We show that, for algebraically convergent sequences (Gamma_i), unless Gamma_i's are (eventually) elementary or contain normal finite subgroups of arbitrarily high order, their algebraic limit is a discrete nonelementary subgroup of G. In the case of divergent sequences (Gamma_i) we show that the limiting action on a real tree T satisfies certain semistability condition, which generalizes the notion of stability introduced by Rips. We then verify that the group Gamma splits as an amalgam or HNN extension of finitely generated groups, so that the edge group has an amenable image in the isometry group of T.Comment: 21 pages, 1 figur

Model theory of operator algebras III: Elementary equivalence and II_1 factors

We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.Comment: 16 page

Phase Operator Problem and Macroscopic Extension of Quantum Mechanics

To find the Hermitian phase operatorof a single-mode electromagnetic field in quantum mechanics, the Schroedinger representation is extended to a larger Hilbert space augmented by states with infinite excitation by nonstandard analysis. The Hermitian phase operator is shown to exist on the extended Hilbert space. This operator is naturally considered as the controversial limit of the approximate phase operators on finite dimensional spaces proposed by Pegg and Barnett. The spectral measure of this operator is a Naimark extension of the optimal probability operator-valued measure for the phase parameter found by Helstrom. Eventually, the two promising approaches to the statistics of the phase in quantum mechanics is unified by means of the Hermitian phase operator in the macroscopic extension of the Schroedinger representation.Comment: 26 pages, LaTeX, no figures, to appear in Ann. Phys. (N.Y.

Neocompact Sets and the Fixed Point Property

AbstractWe use a newly introduced concept of neocompactness to study problems from metric fixed point theory. In particular, we give a sufficient condition for a superreflexive Banach space X to have the fixed point property and obtain shorter proofs of some well-known results in that theory

Applications of Model Theory to Complex Analysis

We use a nonstandard model of analysis to study two main topics in complex analysis. UNIFORM CONTINUITY AND RATES OF GROWTH OF MEROMORPHIC FUNCTIONS is a unified nonstandard approach to several theories; the Julia-Milloux theorem and Julia exceptional functions, Yosida's class (A), normal meromorphic functions, and Gavrilov's Wp classes. All of these theories are reduced to the study of uniform continuity in an appropriate metric by means of S-continuity in the nonstandard model (which was introduced by A. Robinson). The connection with the classical Picard theorem is made through a generalization of a result of A. Robinson on S-continuous *-holomorphic functions. S-continuity offers considerable simplifications over the standard sequential approach and permits a new characterization of these growth requirements. BOUNDED ANALYTIC FUNCTIONS AS THE DUAL OF A BANACH SPACE is a nonstandard approach to the pre-dual Banach space for H∞(D) which was introduced by Rubel and Shields. A new characterization of the pre-dual by means of the nonstandard hull of a space of contour integrals infinitesimally near the boundary of an arbitrary region is given. A new characterization of the strict topology is given in terms of the infinitesimal relation: "h b k provided ||h-k|| is finite and h(z) ≈ k(z) for z∈(*D)". A new proof of the noncoincidence of the strict and Mackey topologies is given in the case of a smooth finitely connected region. The idea of the proof is that the infinitesimal relation: "h γ k provided ||h-k|| is finite and h(z) ≈ k(z) on nearly all of the boundary", gives rise to a compatible topology finer than the strict topology.</p

Deformations and stability in complex hyperbolic geometry

This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into $PU(n,1)$ and the problem of (quasiconformal) stability of deformations of such groups and manifolds in the sense of L.Bers and D.Sullivan. Despite Goldman-Millson-Yue rigidity results for such complex manifolds of infinite volume, we present different classes of such manifolds that allow non-trivial (quasi-Fuchsian) deformations and point out that such flexible manifolds have a common feature being Stein spaces. While deformations of complex surfaces from our first class are induced by quasiconformal homeomorphisms, non-rigid complex surfaces (homotopy equivalent to their complex analytic submanifolds) from another class are quasiconformally unstable, but nevertheless their deformations are induced by homeomorphisms