Georgia Institute of Technology College of Sciences

This is joint work with Aaron Landesman. There are a number of difficult open questions around representations of free and surface groups, which it turns out are accessible to methods from Hodge theory and arithmetic geometry. For example, I'll discuss applications of these methods to the following concrete theorem about surface groups, whose proof relies on non-abelian Hodge theory and the Langlands program:

Theorem. Let $\rho: \pi_1(\Sigma_{g,n})\to GL_r(\mathbb{C})$ be a representation of the fundamental group of a compact orientable surface of genus $g$ with $n$ punctures, with $r<\sqrt{g+1}$. If the conjugacy class of $\rho$ has finite orbit under the mapping class group of $\Sigma_{g,n}$, then $\rho$ has finite image.

This answers a question of Peter Whang. I'll also discuss closely related applications to the Putman-Wieland conjecture on homological representations of mapping class groups.