Extensions of Veech groups I: A hyperbolic action

Spencer Dowdall, Matthew G. Durham, Christopher J. Leininger, Alessandro Sisto

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Given a lattice Veech group in the mapping class group of a closed surface $S$, this paper investigates the geometry of $\Gamma$, the associated $\pi_1S$--extension group. We prove that $\Gamma$ is the fundamental group of a bundle with a singular Euclidean-by-hyperbolic geometry. Our main result is that collapsing ``obvious'' product regions of the universal cover produces an action of $\Gamma$ on a hyperbolic space, retaining most of the geometry of $\Gamma$. This action is a key ingredient in the sequel where we show that $\Gamma$ is hierarchically hyperbolic and quasi-isometrically rigidity.
Original languageEnglish
Pages (from-to)757-805
JournalJournal of Topology
Issue number2
Early online date31 May 2023
Publication statusPublished - Jun 2023


  • math.GT
  • math.GR
  • 20F67, 20F65, 30F60, 57M60, 57M07


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