Topology Seminar (Introductory Talk): Group actions on the circle and surfaces

Seminar | January 18 | 2:10-3 p.m. | 736 Evans Hall

 Sebastian Hurtado, University of Chicago

 Department of Mathematics

A smooth group action on a manifold M is a group-morphism from a group G to the group of diffeomorphisms of M. If G = Z, the study of such actions is just the study of smooth dynamics and classification is impossible. However, if the group G is sufficiently rich or under some hypothesis in the type of action, classification is sometimes possible. A classical example is Holder's theorem (1901), which states that a group acting freely on the circle is abelian and conjugate to a group of rotations (a group acts freely if no non-trivial elements having fixed points in the circle).

I'll describe some of the motivating conjectures about group actions on surfaces and some recent techniques from dynamics (measure rigidity) and topology (mapping class groups of surfaces of infinite type) that can be used to study these type of questions.