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Hurwitz ball quotients

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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.

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Acknowledgments

I am indebted to Domingo Toledo for discussions about visualizing the orbifold structure on a Deligne–Mostow quotient. Any insight into the geometry of these spaces not in the standard literature should be considered his, not mine. I also want to thank the referee for suggestions that undoubtedly improved this paper.

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  1. Temple University, Philadelphia, PA, USA

    Matthew Stover

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  1. Matthew Stover
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Correspondence to Matthew Stover.

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This material is based upon work supported by the National Science Foundation under Grant Number NSF 0943832.

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Stover, M. Hurwitz ball quotients. Math. Z. 278, 75–91 (2014). https://doi.org/10.1007/s00209-014-1306-6

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