Compactifications of topological groups

Every topological group $G$ has some natural compactifications which can be a useful tool of studying $G$. We discuss the following constructions: (1) the greatest ambit $S(G)$ is the compactification corresponding to the algebra of all right uniformly continuous bounded functions on $G$; (2) the Roelcke compactification $R(G)$ corresponds to the algebra of functions which are both left and right uniformly continuous; (3) the weakly almost periodic compactification $W(G)$ is the envelopping compact semitopological semigroup of $G$ (`semitopological' means that the multiplication is separately continuous). The universal minimal compact $G$-space $X=M_G$ is characterized by the following properties: (1) $X$ has no proper closed $G$-invariant subsets; (2) for every compact $G$-space $Y$ there exists a $G$-map $X\to Y$. A group $G$ is extremely amenable, or has the fixed point on compacta property, if $M_G$ is a singleton. We discuss some results and questions by V. Pestov and E. Glasner on extremely amenable groups. The Roelcke compactifications were used by M. Megrelishvili to prove that $W(G)$ can be a singleton. They can be used to prove that certain groups are minimal. A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology.Comment: 17 page

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

A local characterization of Kazhdan projections and applications

We give a local characterization of the existence of Kazhdan projections for arbitary families of Banach space representations of a compactly generated locally compact group $G$. We also define and study a natural generalization of the Fell topology to arbitrary Banach space representations of a locally compact group. We give several applications in terms of stability of rigidity under perturbations. Among them, we show a Banach-space version of the Delorme--Guichardet theorem stating that property (T) and (FH) are equivalent for $\sigma$-compact locally compact groups. Another kind of applications is that many forms of Banach strong property (T) are open in the space of marked groups, and more generally every group with such a property is a quotient of a compactly presented group with the same property. We also investigate the notions of central and non central Kazhdan projections, and present examples of non central Kazhdan projections coming from hyperbolic groups.Comment: 31 pages. v2: small changes to the introduction. Added a discussion on the speed of convergence, and on a notion of positivity for Kazhdan constants (p 14). This version was submitted to a journal v3: small changes to the presentation, background details added on Banach space geometry. Accepted for publication to Commentarii Mathematici Helvetic

Adapting Search Theory to Networks

The CSE is interested in the general problem of locating objects in networks. Because of their exposure to search theory, the problem they brought to the workshop was phrased in terms of adapting search theory to networks. Thus, the first step was the introduction of an already existing healthy literature on searching graphs. T. D. Parsons, who was then at Pennsylvania State University, was approached in 1977 by some local spelunkers who asked his aid in optimizing a search for someone lost in a cave in Pennsylvania. Parsons quickly formulated the problem as a search problem in a graph. Subsequent papers led to two divergent problems. One problem dealt with searching under assumptions of fairly extensive information, while the other problem dealt with searching under assumptions of essentially zero information. These two topics are developed in the next two sections