Hurwitz ball quotients

Abstract

We consider the analogue of Hurwitz curves, smooth projective curves $C$ of genus $g\ge 2$ that realize equality in the Hurwitz bound $|\mathrm{Aut}(C)|\le 84(g-1)$, to smooth compact quotients $S$ of the unit ball in ${\mathbb{C}}^2$. When $S$ is arithmetic, we show that $|\mathrm{Aut}(S)|\le 288e(S)$, where $e(S)$ is the (topological) Euler characteristic, and in the case of equality show that $S$ is a regular cover of a particular Deligne--Mostow orbifold. We conjecture that this inequality holds independent of arithmeticity, and note that work of Xiao makes progress on this conjecture and implies the best-known lower bound for the volume of a complex hyperbolic $2$-orbifold.

Several improvements incorporating referee's comments. To appear in Math. Z