Embedding and nonembedding results in asymptotic geometry
by
Sergei Buyalo
St. Petersburg Department of Steklov Institute of Mathematics

We present a survey on quasi-isometric embedding and nonembedding results in asymptotic geometry. A various constructions of quasi-isometric embeddings of (basically hyperbolic) metric spaces into other metric spaces will be discussed. In particular, we explain how any Gromov hyperbolic group G can be embedded into (n+1)-fold product of binary metric trees, where n is the topological dimension of the boundary at infinity of G (this result is due to S. Buyalo, V. Schroeder, A. Dranishnikov).

The results discussed in the first part of the talk are in a definite sense optimal. This is the topic of the second part of the talk, where we discuss some quasi-isometric invariants of metric spaces that serve as obstacles to quasi-isometric embeddings. Most important are the subexponential corank and the hyperbolic dimension. Open questions and some conjectures will be discussed.

Date received: April 27, 2008