Representations of surface groups with universally finite mapping class group orbit

Abstract

Let ${\Sigma }_{g,n}$ be the orientable genus $g$ surface with $n$ punctures, where $2-2g-n<0$. Let $$\rho :{\pi }_1({\Sigma }_{g,n})\to G{L}_m(\mathbb{C})$$ be a representation. Suppose that for each finite covering map $f:{\Sigma }_{g^{\prime },n^{\prime }}\to {\Sigma }_{g,n}$, the orbit of (the isomorphism class of) $f^{\ast }(\rho )$ under the mapping class group $MCG({\Sigma }_{g^{\prime },n^{\prime }})$ of ${\Sigma }_{g^{\prime },n^{\prime }}$ is finite. Then we show that $\rho$ has finite image. The result is motivated by the Grothendieck-Katz $p$-curvature conjecture, and gives a reformulation of the $p$-curvature conjecture in terms of isomonodromy.

Updated to final publication version; minor typos corrected and several new examples given