McCullough, Darryl; Rajeevsarathy, Kashyap Roots of Dehn twists. (English) Zbl 1219.57019 Geom. Dedicata 151, 397-409 (2011). 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. Reviewer: Mustafa Korkmaz (Ankara) Cited in 3 ReviewsCited in 13 Documents 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 Keywords:surface; mapping class group; Dehn twist; nonseparating curve; root Citations:Zbl 1203.57007 Software:GAP PDFBibTeX XMLCite \textit{D. McCullough} and \textit{K. Rajeevsarathy}, Geom. Dedicata 151, 397--409 (2011; Zbl 1219.57019) Full Text: DOI arXiv References: [1] Edmonds A.: Surface symmetry I. Mich. Math. J. 29, 171–183 (1982) · Zbl 0511.57025 · doi:10.1307/mmj/1029002670 [2] Fenchel W.: Estensioni di gruppi discontinui e transformazioni periodiche delle superficie. Atti Accad. Naz Lincei. Rend. Cl. Sci. Fis. Mat. Nat. 5(8), 326–329 (1948) · Zbl 0036.12902 [3] Fenchel W.: Remarks on finite groups of mapping classes. Mat. Tidsskr. B 1950, 90–95 (1950) [4] GAP: Groups, Algorithms, and Programming. Available at the St. Andrews GAP website http://turnbull.mcs.st-and.ac.uk/\(\sim\)gap/ [5] Harvey W.J.: Cyclic groups of automorphisms of a compact Riemann surface. Quart. J. Math. Oxford Ser. 17(2), 86–97 (1966) · Zbl 0156.08901 · doi:10.1093/qmath/17.1.86 [6] Kerckhoff S.: The Nielsen realization problem. Bull. Am. Math. Soc. (N.S.) 2, 452–454 (1980) · Zbl 0434.57007 · doi:10.1090/S0273-0979-1980-14764-3 [7] Kerckhoff S.: The Nielsen realization problem. Ann. Math. 117(2), 235–265 (1983) · Zbl 0528.57008 · doi:10.2307/2007076 [8] Margalit D., Schleimer S.: Dehn twists have roots. Geom. Topol. 13, 1495–1497 (2009) · Zbl 1203.57007 · doi:10.2140/gt.2009.13.1495 [9] Matsumoto Y., Montesinos-Amilibia J.M.: Pseudo-periodic homeomorphisms and degeneration of Riemann surfaces. Bull. Am. Math. Soc. (N.S.) 30, 70–75 (1994) · Zbl 0797.30036 · doi:10.1090/S0273-0979-1994-00437-9 [10] Monden, N.: On roots of Dehn twists. arXiv:0911.5079 · Zbl 1302.57056 [11] Monden, N.: Personal communication [12] Nielsen J.: Abbildungklassen Endliche Ordung. Acta Math. 75, 23–115 (1943) · Zbl 0027.26601 · doi:10.1007/BF02404101 [13] McCullough, D.: Software for ”Roots of Dehn twists”. Available at http://www.math.ou.edu/\(\sim\)dmccullough/research/software.html · Zbl 1219.57019 [14] Nielsen J.: Die Struktur periodischer Transformationen von Flächen. Danske Vid. Selsk. Mat.-Fys. Medd. 1, 1–77 (1937) · JFM 63.0553.03 [15] Scott P.: The geometries of 3-manifolds. Bull. London Math. Soc. 15, 401–487 (1983) · Zbl 0561.57001 · doi:10.1112/blms/15.5.401 [16] Thurston, W.: The geometry and topology of three-manifolds. Available at http://msri.org/publications/books/gt3m [17] Wiman, A.: Über die hyperelliptischen Curven und diejenigen vom Geschlechte p = 3, welche eindeutigen Transformationen in sich zulassen, Bihang Kongl. Svenska Vetenskaps-Akademiens Handlinger, Stockholm, 1895–1896. [18] Zieschang H.: Finite groups of mapping classes of surfaces, Lecture Notes in Mathematics, 875. Springer, Berlin (1981) · Zbl 0472.57006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.