Vladimir Pestov (University of Ottawa)

Workshop description:

This workshop will be dedicated to two new classes of (discrete, countable) groups, introduced fairly recently, the hyperlinear and sofic groups. Their interest for set theorists, model theorists and logicians stems from the fact that groups from both classes are most naturally defined as subgroups of ultraproducts of certain classical groups (even if alternative descriptions are also possible).
The ultraproducts in question are metric ultraproducts, formed very much like ultraproducts of Banach spaces. Hyperlinear groups can now be defined as those groups isomorphic to subgroups of ultraproducts of unitary groups of finite rank U(n) equipped with the Euclidean distance between matrices (normalized so as to make U(n) sit on a unit sphere), with regard to a non-principal ultrafilter on natural numbers. Similarly, sofic groups are those groups isomorphic to subgroups of metric ultraproducts of symmetric groups S_n of finite rank n, equipped with the normalized Hamming distance between permutations (counting the number of coordinates where two permutations differ).
Hyperlinear groups are motivated by Connes' Embedding Conjecture, which is one of the central unresolved problems in theory of operator algebras, while sofic groups were introduced by Gromov in connection with Gootschalk's Surjunctivity Conjecture (stating that no shift dynamical system A^G, where G is a countable group and A is a finite discrete space, contains a proper isomorphic copy of itself). It is still unknown, for example, if every group is hyperlinear and/or sofic, and generally open questions outnumber results.

The workshop lectures will roughly follow the recent survey article by the speaker:

• Hyperlinear and sofic groups: a brief guide. - arXiv:0804.3968v7 [math.GR], 28 pages, June 2008, accepted, to appear in The Bulletin of Symbolic Logic.

Suggested preliminary background reading:

• Basics of ultrafilters and ultraproducts as found in many sources on model theory, e.g., Bell and Slomson's Models and Ultraproducts which book is available on Amazon.com for an incredible $ 14.78 (in paperback). Maybe the soon-to-be published Model Theory for Metric Structures by Ben Yaacov, Berenstein, Henson, and Usvyatsov is even more relevant, especially since one can still download it for free.
• The very beginnings of topological and symbolic dynamics, which can be found in practically any book treating abstract topological dynamics. For instance, everything we'll ever need, including topological entropy, can be found in Glasner's Ergodic Theory via Joinings or Auslander's Minimal Flows and Their Extensions, but both books are an overkill. I need to think a bit of something more accessible here.
• Rudiments of C^*-algebras and von Neumann algebras, for instance, the first few chapters of Vaughan Jones' 2003 lectures available on the web for download.
• Some infintie group theory, including the concept of an amenable (discrete) group. For instance, Pierre de la Harpe's Topics in Geometric Group Theory is an excellent book both to read and to leaf through, and it is not expensive either. There are many other sources of course.