151 research outputs found

    Hurwitz ball quotients

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    We consider the analogue of Hurwitz curves, smooth projective curves CC of genus gβ‰₯2g \ge 2 that realize equality in the Hurwitz bound ∣Aut(C)βˆ£β‰€84(gβˆ’1)|\mathrm{Aut}(C)| \le 84 (g - 1), to smooth compact quotients SS of the unit ball in C2\mathbb{C}^2. When SS is arithmetic, we show that ∣Aut(S)βˆ£β‰€288e(S)|\mathrm{Aut}(S)| \le 288 e(S), where e(S)e(S) is the (topological) Euler characteristic, and in the case of equality show that SS 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 22-orbifold.Comment: Several improvements incorporating referee's comments. To appear in Math.

    A Cantor set with hyperbolic complement

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    We construct a Cantor set in S^3 whose complement admits a complete hyperbolic metric
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