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Roots of Dehn twists

Abstract

Margalit and Schleimer found examples of roots of the Dehn twist t C about a nonseparating curve C in a closed orientable surface, that is, homeomorphisms h such that h n  =  t C in the mapping class group. Our main theorem gives elementary number-theoretic conditions that describe the n for which an n th root of t C exists, given the genus of the surface. Among its applications, we show that n must be odd, that the Margalit-Schleimer roots achieve the maximum value of n among the roots for a given genus, and that for a given odd nn th roots exist for all genera greater than (n − 2)(n − 1)/2. We also describe all n th roots having n greater than or equal to the genus.

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Correspondence to Darryl McCullough.

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McCullough, D., Rajeevsarathy, K. Roots of Dehn twists. Geom Dedicata 151, 397–409 (2011). https://doi.org/10.1007/s10711-010-9541-4

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  • DOI: https://doi.org/10.1007/s10711-010-9541-4

Keywords

  • Surface
  • Mapping class
  • Dehn twist
  • Nonseparating
  • Curve
  • Root

Mathematics Subject Classification (2000)

  • Primary 57M99
  • Secondary 57M60