# Representations of surface groups with universally finite mapping class group orbit

## TL;DR

This paper proves that certain surface group representations with finite mapping class group orbits across all finite covers must have finite image, linking to the Grothendieck-Katz p-curvature conjecture and isomonodromy.

## Contribution

It establishes a new criterion for finiteness of surface group representations based on their behavior under all finite covers and mapping class group actions.

## Key findings

- Representations with finite orbits under all finite covers have finite image.
- Provides a reformulation of the p-curvature conjecture via isomonodromy.
- Connects surface group representations to deep conjectures in algebraic geometry.

## Abstract

Let $\Sigma_{g,n}$ be the orientable genus $g$ surface with $n$ punctures, where $2-2g-n<0$. Let $$\rho: \pi_1(\Sigma_{g,n})\to GL_m(\mathbb{C})$$ be a representation. Suppose that for each finite covering map $f: \Sigma_{g', n'}\to \Sigma_{g, n}$, the orbit of (the isomorphism class of) $f^*(\rho)$ under the mapping class group $MCG(\Sigma_{g',n'})$ of $\Sigma_{g',n'}$ 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.