Moduli spaces of complex affine and dilation surfaces

Paul Apisa; Matt Bainbridge; Jane Wang

doi:10.1515/crelle-2023-0005

Article

Moduli spaces of complex affine and dilation surfaces

Paul Apisa , Matt Bainbridge and Jane Wang

Published/Copyright: February 23, 2023

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2023 Issue 796

Abstract

We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech [W. A. Veech, Flat surfaces, Amer. J. Math. 115 1993, 3, 589–689], we show that the moduli space 𝒜 g , n ⁢ ( 𝒎 ) of genus g affine surfaces with cone points of complex order 𝒎 = ( m 1 ⁢ … , m n ) is a holomorphic affine bundle over ℳ g , n , and the moduli space 𝒟 g , n ⁢ ( 𝒎 ) of dilation surfaces is a covering space of ℳ g , n . We then classify the connected components of 𝒟 g , n ⁢ ( 𝒎 ) and show that it is an orbifold- K ⁢ ( G , 1 ) , where G is the framed mapping class group of [A. Calderon and N. Salter, Framed mapping class groups and the monodromy of strata of Abelian differentials, preprint 2020].

Funding source: National Science Foundation

Award Identifier / Grant number: DMS 1803625

Funding source: Simons Foundation

Award Identifier / Grant number: 713192

Funding statement: During the preparation of this paper, the first author was partially supported by NSF Postdoctoral Fellowship DMS-1803625. Research of the second author is supported in part by the Simons Foundation, Grant No. 713192.

Acknowledgements

The second author is grateful to Eduard Duryev for inspiring conversations on dilation surfaces some years ago. In particular, we learned from him the main ideas of the proof of Veech’s Theorem 2.1. We are also grateful to Christopher Zhang for useful comments on a previous draft.

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Received: 2022-05-20

Revised: 2023-01-13

Published Online: 2023-02-23

Published in Print: 2023-03-01

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https://doi.org/10.1515/crelle-2023-0005