Colloquium: Tuesday March 31, 2:10 pm. Speaker: Jakub Gismatullin. Title: Metric approximations of groups

I will report some recent results on groups with good metric approximation properties, called weak sofic and weak hyperlinear groups. The class of weak sofic groups was introduced by L. Glebsky and L. M. Rivera, as a generalization of the notion of a sofic group, defined by B. Weiss and M. Gromov. Many open problems in group theory have been resolved for sofic and hyperlinear groups. These groups can be characterized as subgroups metric ultraproducts of families of certain metric groups: symmetric groups with the Hamming distance and unitary groups of finite rank with the Hilbert-Schmidt distance. In fact, there is another characterization of sofic groups: a group is weak sofic if an only if it can be embedded into an abstract quotient of a profinite group. Therefore one can define and study the class of weak hyperlinear groups, and give similar characterizations. In the second part of the talk i will concentrate on surjunctivity of (weak) sofic and hyperlinear groups.