Finiteness results for flat surfaces: Large cusps and short geodesics

John Smillie, Barak Weiss

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

For fixed g and T we show the finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech group contains a cusp of hyperbolic co-area less than T. We obtain new restrictions on Veech groups: any non-elementary Fuchsian group can appear only finitely many times in a fixed stratum; any non-elementary Veech group is of finite index in its normalizer; and the quotient of ℍ by a non-lattice Veech group admits arbitrarily large embedded disks. A key ingredient of the proof is the finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech group contains a hyperbolic element with eigenvalue less than T.

Original languageEnglish
Pages (from-to)313-336
Number of pages24
JournalCommentarii Mathematici Helvetici
Volume85
Issue number2
DOIs
StatePublished - 9 Mar 2010

Keywords

  • Flat surface
  • Fuchsian group
  • Markov partition
  • Veech group

ASJC Scopus subject areas

  • Mathematics (all)

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