Moduli of spherical tori with one conical point

Eremenko, Alexandre; Mondello, Gabriele; Panov, Dmitri

doi:10.2140/gt.2023.27.3619

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Moduli of spherical tori with one conical point

Alexandre Eremenko, Gabriele Mondello and Dmitri Panov

Geometry & Topology 27 (2023) 3619–3698

DOI: 10.2140/gt.2023.27.3619

Abstract

We determine the topology of the moduli space ${ℳ 𝒮}_{1, 1} (𝜗)$ of surfaces of genus one with a Riemannian metric of constant curvature $1$ and one conical point of angle $2 π 𝜗$ . In particular, for $𝜗 \in (2 m - 1, 2 m + 1)$ nonodd, ${ℳ 𝒮}_{1, 1} (𝜗)$ is connected, has orbifold Euler characteristic $- \frac{1}{12} m^{2}$ , and its topology depends on the integer $m > 0$ only. For $𝜗 = 2 m + 1$ odd, ${ℳ 𝒮}_{1, 1} (𝜗)$ has $⌈ \frac{1}{6} m (m + 1) ⌉$ connected components. For $𝜗 = 2 m$ even, ${ℳ 𝒮}_{1, 1} (𝜗)$ has a natural complex structure and it is biholomorphic to $ℍ^{2} ∕ G_{m}$ for a certain subgroup $G_{m}$ of $SL (2, ℤ)$ of index $m^{2}$ , which is nonnormal for $m > 1$ .

Keywords

spherical surfaces, moduli spaces, conical points, Belyi curves

Mathematical Subject Classification

Primary: 58D27

Secondary: 32Q20, 52B70

References

Publication

Received: 4 May 2021

Revised: 11 February 2022

Accepted: 9 April 2022

Published: 5 December 2023

Proposed: Benson Farb

Seconded: David M Fisher, Leonid Polterovich

Authors

Alexandre Eremenko
Department of Mathematics Purdue University West Lafayette, IN United States

Gabriele Mondello
Dipartimento di Matematica Guido Castelnuovo Sapienza Università of Roma Roma Italy

Dmitri Panov
Department of Mathematics King’s College London London United Kingdom

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Volume 27, issue 9 (2023)