Hurwitz ball quotients

TL;DR

This paper explores bounds on automorphism groups of complex hyperbolic surfaces, establishing new inequalities for arithmetic cases and conjecturing their general validity, with connections to Deligne–Mostow orbifolds and volume bounds.

Contribution

It proves an automorphism bound for arithmetic complex hyperbolic surfaces and conjectures its general applicability, linking to orbifold covers and volume estimates.

Findings

Automorphism bound for arithmetic surfaces: | ext{Aut}(S)| extless= 288 e(S)

Equality case: S covers a Deligne–Mostow orbifold

Progress on volume bounds for complex hyperbolic 2-orbifolds

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 $|\mathrm{Aut}(S)| \le 288 e(S)$, 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.