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Geometriae Dedicata

, Volume 151, Issue 1, pp 397–409 | Cite as

Roots of Dehn twists

  • Darryl McCullough
  • Kashyap Rajeevsarathy
Original Paper

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.

Keywords

Surface Mapping class Dehn twist Nonseparating Curve Root 

Mathematics Subject Classification (2000)

Primary 57M99 Secondary 57M60 

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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OklahomaNormanUSA

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