Math Graduate Student Seminar

For a countable group Gamma, what are the possible sizes of Out(Gamma)? From the point of view of cardinality, the answer is not so interesting: if it is uncountable, then it has size continuum. There is a more suitable notion of Borel cardinality, coming from the theory of countable Borel equivalence relations in descriptive set theory. We will see that although there are many Borel cardinalities of varying complexity, Out(Gamma) is always among the lower end of them; namely, Out(Gamma) is always hyperfinite. This is joint work with Joshua Frisch.