An introduction to hyperlinear and sofic groups

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 Sn 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 AG, 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:

However, there will be some additional material not found in the survey, and some topics may be presented differently.

Suggested preliminary background reading: