College of Liberal Arts & Sciences
MATHEMATICS COLLOQUIUM - Kostyantyn Medynets (United States Naval Academy)
Abstract: A character of an infinite group G is a positive-definite class function. In view of the Gelfand-Naimark-Segal construction, having a complete description of characters is equivalent to the classification of the unitary representations of the group in question that generate II_1 factors.
Studying unitary representations of the infinite symmetric group S(N), Vershik noticed that many of its characters came from ergodic actions of S(N) on measure spaces $(X,\mu)$ by the formula $f(g) = \mu(FixedPoints(g))$. The question we are trying to address is for which infinite groups their characters can be given a similar dynamical interpretation.
We will reexamine some older results on the classification of characters for the Special Linear Group of Infinite Matrices over a finite field and Group of Rational Permutations of the unit interval. We will then classify characters of various inductive limits of symmetric groups (permutations of Bratteli diagrams) and the Higman-Thompson groups. In the case of Higman-Thompson groups, we show that these groups have no non-trivial characters. The absence of non-trivial characters have some implications in the theory of random subgroups.