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Cutting sequences on square-tiled surfaces

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Abstract

We characterize cutting sequences of infinite geodesics on square-tiled surfaces by considering interval exchanges on specially chosen intervals on the surface. These interval exchanges can be thought of as skew products over a rotation, and we convert cutting sequences to symbolic trajectories of these interval exchanges to show that special types of combinatorial lifts of Sturmian sequences completely describe all cutting sequences on a square-tiled surface. Our results extend the list of families of surfaces where cutting sequences are understood to a dense subset of the moduli space of all translation surfaces.

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Authors and Affiliations

  1. Wake Forest University, Winston-Salem, NC, USA

    Charles C. Johnson

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  1. Charles C. Johnson
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Correspondence to Charles C. Johnson.

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The author thanks the referee for their careful reading of the original manuscript and for their many helpful remarks for improving the exposition.

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Johnson, C.C. Cutting sequences on square-tiled surfaces. Geom Dedicata 190, 53–80 (2017). https://doi.org/10.1007/s10711-017-0227-z

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  • DOI: https://doi.org/10.1007/s10711-017-0227-z

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