Summary: AN INTRODUCTION TO HYPERLINEAR AND SOFIC GROUPS
VLADIMIR G. PESTOV AND ALEKSANDRA KWIATKOWSKA
Abstract. This is an edited write-up of lecture notes of the 7-th Appalachian set
theory workshop of the same title led by the first named author at the Cornell
University on November 22, 2008. A draft version of the notes was prepared by
the second named author. This presentation is largely complementary to the earlier
survey by the first-named author (Hyperlinear and sofic groups: a brief guide, Bull.
Symb. Logic 14 (2008), pp. 449-480).
1. Motivation: group matrix models in the sense of classical
In these lectures, we will deal with a class of groups called hyperlinear groups, as
well as its (possibly proper) subclass, that of sofic groups. One natural way to get into
this line of research is through the theory of operator algebras. Here, the hyperlinear
groups are sometimes referred to as "groups admitting matrix models". This can be
indeed interpreted as a genuine model-theoretic statement, within a suitable version of
logic. Namely, a group G is said to admit matrix models if every existential sentence
of the first-order theory of G is satisfied in matrix groups.
What makes the concept interesting -- and difficult to work with -- is that at the
matrix group end it is not the classical first-order logic that one has in mind, but
rather a version of continuous logic with truth values in the unit interval [0, 1]. By