Dynamics and the classification of geometries on surfaces

Abstract: Many interesting dynamical systems arise from the classification of locally homogeneous geometric structures and flat connections on manifolds. Their classification mimics that of Riemann surfaces by the Riemann moduli space, which identifies as the quotient of Teichmueller space of marked Riemann surfaces by the action of the mapping class group. However, unlike Riemann surfaces, these actions are generally chaotic. A striking elementary example is Baues's theorem that the deformation space of complete affine structures on the 2-torus is the plane with the usual linear action of GL(2,Z) (the mapping class group of the torus). We discuss specific examples of these dynamics for some simple surfaces, where the relative character varieties appear as cubic surfaces in affine 3-space. Complicated dynamics seems to accompany complicated topologies, which we interpret as (possibly singular) hyperbolic structures on surfaces.

algebraic geometrydifferential geometrygroup theorygeometric topologynumber theory

Audience: general audience

International Conference on Discrete groups, Geometry and Arithmetic

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