On roots of Dehn twists

Monden, Naoyuki

doi:10.1216/rmj-2014-44-3-987

Cited by 4 publications

(10 citation statements)

References 8 publications

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“…As was shown by McCullough, Rajeevsarathy [6] and independently by Monden [8], roots of Dehn twists constructed by Margalit and Schleimer [5] are of maximal possible degree. Since the geometric nature of our constructions of roots of crosscap slides and crosscap transpositions is quite similar to that of Margalit and Schleimer, it is natural to ask if the roots constructed in the proof of Main Theorem are also of maximal degree.…”

Section: Remark 33mentioning

confidence: 69%

Roots of crosscap slides and crosscap transpositions

Parlak

Stukow

2017

Period Math Hung

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Let N g denote a closed nonorientable surface of genus g. For g ≥ 2 the mapping class group M(N g ) is generated by Dehn twists and one crosscap slide (Y -homeomorphism) or by Dehn twists and a crosscap transposition. Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We give necessary and sufficient conditions for the existence of roots of crosscap slides and crosscap transpositions.

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Section: Remark 33mentioning

confidence: 69%

Roots of crosscap slides and crosscap transpositions

Parlak

Stukow

2017

Period Math Hung

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“…The following proposition and its proof is a nonorientable version of the result of Monden (Section 5 of [10]). Proposition 2.1.…”

Section: Preliminariesmentioning

confidence: 82%

“…Margalit and Schleimer [8] constructed a root of degree 2g + 1 of a Dehn twist t c about a nonseparating circle c in an orientable surface S g+1 , where g ≥ 1. Later McCullough, Rajeevsarathy and Monden proved that this is the maximal possible degree of a root of t c (Corollary 2.2 of [9] and Corollary C of [10]).…”

Section: Degrees Of Roots Of Dehn Twistsmentioning

confidence: 95%

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Roots of Dehn twists on nonorientable surfaces

Parlak

Stukow

2019

J. Knot Theory Ramifications

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Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We investigate the problem of roots of a Dehn twist tc about a nonseparating circle c in the mapping class group M(Ng) of a nonorientable surface Ng of genus g. We explore the existence of roots and, following the work of McCullough, Rajeevsarathy and Monden, give a simple arithmetic description of their conjugacy classes. We also study roots of maximal degree and prove that if we fix an odd integer n > 1, then for each sufficiently large g, tc has a root of degree n in M(Ng). Moreover, for any possible degree n we provide explicit expressions for a particular type of roots of Dehn twists about nonseparating circles in Ng.

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“…Fehrenbach and Los [8] have developed an algorithm that computes the roots of pseudo-Anosov mapping classes on oncepunctured surfaces. For the case of a single Dehn twist (or its power) and a product of Dehn twists, the answer to this question is well known [15,17,18,[22][23][24]. More recently, Dhanwani and Rajeevsarathy [6] have obtained necessary and sufficient conditions for the primitivity of periodic mapping classes.…”

Section: Introductionmentioning

confidence: 99%

General primitivity in the mapping class group

Kapari,

Rajeevsarathy

2021

Preprint

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For g ≥ 2, let Mod(Sg) be the mapping class group of the closed orientable surface Sg of genus g. In this paper, we obtain necessary and sufficient conditions under which a given pseudo-periodic mapping class can be a root of another up to conjugacy. Using this characterization, the canonical decomposition of (non-periodic) mapping classes, and some known algorithms, we give an efficient algorithm for computing roots of arbitrary mapping classes up to conjugacy. Furthermore, we derive realizable bounds on the degrees of roots of pseudo-periodic mapping classes in Mod(Sg), the Torelli group, the level-m subgroup of Mod(Sg), and the commutator subgroup of Mod(S2). In particular, we show that the highest possible (realizable) degree of a root of a pseudo-periodic mapping class F is 3q(F )(g +1)(g +2), where q(F ) is a unique positive integer associated with the conjugacy class of F . Moreover, this bound is realized by the roots of the powers of Dehn twist about a separating curve of genus [g/2] in Sg. Finally, for g ≥ 3, we show that any pseudo-periodic mapping class having a nontrivial periodic component that is not the hyperelliptic involution, normally generates Mod(Sg). Consequently, we establish that there exist roots of bounding pair maps and powers of Dehn twists that normally generate Mod(Sg).

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On roots of Dehn twists

Cited by 4 publications

References 8 publications

Roots of crosscap slides and crosscap transpositions

Roots of crosscap slides and crosscap transpositions

Roots of Dehn twists on nonorientable surfaces

General primitivity in the mapping class group

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