Brown University

Mathematics Department Colloquium

Fridays – 4:30PM – Barus-Holley 190 (note corrected Room)

January 27

Bill Goldman, U. Maryland

Geometry and Dynamics of Surface Group Representations

Abstract: The space of representations of the fundamental group of a surface into a Lie group is a rich geometric object. Examples include symplectic vector spaces, Jacobi varieties and Teichmueller spaces. The topological symmetries of the surface acts on this space preserving a natural Poisson geometry. When the Lie group is compact, the action is chaotic and for representations corresponding to uniformizations by geometric structures, the action is proper.

In general the dynamics falls between these two extremes. In the case of a one-holed torus, the dynamics reduces to an action of the modular group on cubic surfaces related to the Markoff equation, where both chaotic and proper dynamics coexist.