# Character Varieties, Dynamics and Arithmetic

## Dynamics on character varieties

In these two talks, I will describe how the classification of locally homogeneous geometric structures (closely related to flat connections) leads to interesting dynamical systems.

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 $GL(2,\mathbb Z)$ (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, as well as the symplectic leaves of invariant Poisson structures, Complicated dynamics seems to accompany complicated topology, which we can them as (possibly singular) hyperbolic structures on surfaces.