Hurwitz ball quotients

Abstract

© 2014, Springer-Verlag Berlin Heidelberg. We consider the analogue of Hurwitz curves, smooth projective curves (Formula Presented) of genus (Formula Presented) that realize equality in the Hurwitz bound (Formula Presented), to smooth compact quotients (Formula Presented) of the unit ball in (Formula Presented). When (Formula Presented) is arithmetic, we show that (Formula Presented), where (Formula Presented) is the (topological) Euler characteristic, and in the case of equality show that (Formula Presented) 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 (Formula Presented)-orbifold.