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ROOTS OF DEHN TWISTS ABOUT MULTICURVES

  • KASHYAP RAJEEVSARATHY (a1) and PRAHLAD VAIDYANATHAN (a1)

Abstract

A multicurve ${\mathcal{C}}$ in a closed orientable surface Sg of genus g is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. A left-handed Dehn twist $t_{\mathcal{C}}$ about ${\mathcal{C}}$ is the product of left-handed Dehn twists about the individual curves in ${\mathcal{C}}$ . In this paper, we derive necessary and sufficient conditions for the existence of a root of $t_{\mathcal{C}}$ in the mapping class group Mod(Sg). Using these conditions, we obtain combinatorial data that correspond to roots, and use it to determine upper bounds on the degree of a root. As an application of our theory, we classify all such roots up to conjugacy in Mod(S4). Finally, we establish that no such root can lie in the level m congruence subgroup of Mod(Sg), for m ≥ 3.

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References

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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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