Mathematics Colloquium: Dynamics on character varieties

Speaker: Sara Maloni, University of Virginia

Title: Dynamics on character varieties

Abstract: Given a surface S, the mapping class group Mod(S) is the group of isotopy classes of homeomorphisms of S. There is a natural action of this group on the set of conjugacy classes of representations from the fundamental group of S into the group PSL(2,R) of orientation preserving isometries of hyperbolic space. A famous theorem of Fricke proves that this action is properly discontinuously ("discrete/nice") on the representations which are discrete and faithful. Note that such representations correspond to marked isotopy classes of hyperbolic structures on S and constitute the famous Teichmueller space T(S) of S. Goldman conjectured that the action is ergodic ("chaotic") on the complement of T(S). One can generalize this discussion by considering the action of the outer automorphism group Out(H) on the set X(H, G) of conjugacy classes of representations from a group H into a Lie group G. In this talk, I will survey some of the results known, some of the many open conjectures in the field and some recent results in this direction. I will define the objects I will introduce and provide examples, so no advanced previous knowledge is required for the talk.

Time: Friday, January 31, 2020, 11:30-12:20 p.m. Note the unusual time!!

Place: Exploratory Hall, room 4106

Refreshments will be served at 11:00 a.m.