Vladimir Pestov : "An introduction to hyperlinear and sofic groups"

List of participants in this workshop

Lecture notes by Vladimir Pestov and Aleksandra Kwiatkowska (PDF; Final version 11/09)

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.

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

For those interested in 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. I suggest an ε-neighbourhood of pages 22-25, for any value of ε≥0.
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 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. Here is a list of notions whose knowledge will help: action of a group on a compact space, morphism of G-spaces, a shift (symbolic dynamical system), topological entropy of a subshift.
Some infinite group theory, including basic constructions of group theory, examples (e.g. finite groups of permutations), and such concepts as an amenable (discrete) group and residually finite group. Many other classes of groups will be mentioned in passing in connection with open problems (hyperbolic groups, one-relator groups, topologically amenable groups...) 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.
Basic topological group theory notions: a topological group, left-invariant and bi-invariant metrics, the unitary group U(n) equipped with the uniform and Hilbert-Schmidt distances. Abstract Harmonic Analysis, Vol. I by Hewitt and Ross is an excellent source. The unitary group, as well as a number of other topics on the list, is treated in Essential Results in Functional Analysis by Robert Zimmer.
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. However, the main concepts, results and open questions pertaining to hyperlinear groups can be stated and understood without referring to operator algebras: they are important for understanding the origin and motivation.

Travel and lodging

We had blocks of fifteen rooms set aside at each of the following two hotels until October 21. In both cases, the group name is "Appalachian set theory", the arrival date is Friday, November 21, and the departure date is Sunday, November 23.
- Best Western
  $99 per night plus 13% tax
  1020 Ellis Hollow Rd.
  Ithaca, NY 14850
  (607) 272-6100
  http://www.bestwesternnewyork.com/hotels/best-western-university-inn-ithaca/
- Hilton Garden Inn
  $119 per night plus 13% tax
  130 E. Seneca St.
  Ithaca, NY 14850
  (607) 277-8900
  http://www.hiltongardeninn.com/
Other useful websites include:
Getting to Malott Hall: Malott is at the corner of Tower Road and Garden Ave. You should enter Malott via the doors which are closest to the Tower Rd/Garden Ave intersection. Room 251 is one of the first lecture halls you will encounter. This entrance should be unlocked. The other entrance (near 406) likely will be locked.
The Best Western offers a shuttle service to Cornell (you should let them know in advance).
The walk to the department is a little over 1.5 miles.
You can also find a map of campus here: http://www.cornell.edu/maps/

Parking: Generally you do not need a parking permit to park on Campus on the weekend. There are, however, a large number of exceptions. Often they are marked with "reserved" or with a "moon" symbol and additional, self explanatory text. There is a parking lot adjacent to Malott (accessible via Garden Ave), but parking is limited there and it fills up quickly on the weekend. Additional parking can be found either East of Malott on Tower road or in the parking garage off of Hoy road (soon after you enter Campus if you enter where Dryden turns South and Hoy starts).

Coffee shop: There will be coffee, juice, bagels, etc. at the department starting at 8:45am. There is also a coffee shop (the "Queen of Tarts") on Maple (on the walking route) near Veterans Pl. There is also an excellent bagel shop at College and Oak (it also has a branch in the shopping plaza where the Best Western is located).

Participant travel support

Funds provided by the National Science Foundation will be used to reimburse some participant transportation and lodging expenses. Priority will be given to students and faculty who do not hold federal research grants. Please request such funds as far in advance of the meeting as possible by sending the following information to James Cummings and Ernest Schimmerling by email.

Your name, university affiliation, mailing address, phone number and email address
Your professional status and
- undergraduate students: please describe your background in set theory
- graduate students: please tell us your year and the name of your thesis advisor if you have one
- faculty: please tell us whether you hold a federal research grant
A brief statement about your interest in the workshop