Representations of surface groups with universally finite mapping class group orbit

Let $\Sigma_{g,n}$ be the orientable genus $g$ surface with $n$ punctures, where $2 - 2_g - n \lt 0$. Let\[\rho : \pi_1 (\Sigma_{g,n}) \to GL_m (\mathbb{C})\]be a representation. Suppose that for each finite covering map $f : \Sigma_{g^\prime, n^\prime} \to \Sigma_{g,n}$ 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.

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Litt is supported by NSF Grant DMS-2001196; Lawrence is supported by NSF Grant DMS-1705140.

Received 2 February 2021

Accepted 1 June 2021

Published 4 May 2023