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Roots of Dehn twists. (English) Zbl 1219.57019

In the mapping class group of a closed surface of genus \(g\), it is easy to find roots of the Dehn twist about a separating simple closed curve. For a nonseparating simple closed curve \(C\), D. Margalit and S. Schleimer [Geom. Topol. 13, No. 3, 1495–1497 (2009; Zbl 1203.57007)] found examples of homeomorphisms \(h\) such that \(h^{2g-1}=t_C\), where \(t_C\) is the Dehn twist about \(C\). The main result of this paper gives elementary number-theoretic conditions that describes those \(n\) for which there exists a homeomorphism \(h\) such that \(h^n=t_C\). Some of the interesting applications are the following: if an \(n^{th}\) root of \(t_C\) exists, then \(n\) must be odd; if \(n>2g-1\) then \(t_C\) has no \(n^{th}\) root; if \(n\) is odd and if \(g>(n-1)(n-2)/2\) then there exists an \(n^{th}\) root.

MSC:

57M99 General low-dimensional topology
57M60 Group actions on manifolds and cell complexes in low dimensions
20F38 Other groups related to topology or analysis

Citations:

Zbl 1203.57007

Software:

GAP
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References:

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