Normal generators for mapping class groups are abundant

Lanier, Justin; Margalit, Dan

doi:10.48550/arxiv.1805.03666

Cited by 8 publications

(17 citation statements)

References 23 publications

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“…Further, any Type 2 cyclic action can be constructed from certain compatibilities on irreducible Type 1 actions [19,Theorem 2.24]. These results allow us to take an alternative approach which bypasses the usages of [14,Lemma 3.2]. However, we will use the well-suited curve criterion from [14].…”

Section: Normal Closure Of Pseudo-periodic Mapping Classesmentioning

confidence: 99%

“…These results allow us to take an alternative approach which bypasses the usages of [14,Lemma 3.2]. However, we will use the well-suited curve criterion from [14].…”

Section: Normal Closure Of Pseudo-periodic Mapping Classesmentioning

confidence: 99%

“…Recently, Margalit and Lanier [14] have proved that when g ≥ 3, any periodic mapping class, which is not a hyperelliptic involution, normally generates Mod(S g ). In Section 5, we generalize this result to pseudo-periodic mapping classes by applying the theory developed in [19] and the well-suited curve criterion from [14] (see Proposition 5.2). Keeping in mind that the nontrivial powers of Dehn twists and bounding pair maps do not normally generate Mod(S g ), we show the following as a final application.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Normal Closure Of Pseudo-periodic Mapping Classesmentioning

confidence: 99%

“…These results allow us to take an alternative approach which bypasses the usages of [14,Lemma 3.2]. However, we will use the well-suited curve criterion from [14].…”

Section: Normal Closure Of Pseudo-periodic Mapping Classesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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General primitivity in the mapping class group

Kapari,

Rajeevsarathy

2021

Preprint

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“…Further observation on reducible mapping classes gives an analogous criterion for them as in [LM18]. Let f ∈ Mod(S) be reducible with maximal invariant multicurve γ and ℓ T (f ) > 0.…”

Section: Mapping Classes With Small Asymptotic Translation Lengthsmentioning

confidence: 99%

“…Question 1.1 is motivated by a work of Lanier-Margalit [LM18] which showed that a pseudo-Anosov element with small minimal translation length normally generates the mapping class group where "small" means less than 1 2 log 2. The question is of course open-ended since it does not specify what we mean by "small" and "obvious counterexample".…”

Section: Introductionmentioning

confidence: 99%

Reducible normal generators for mapping class groups are abundant

Baik¹,

Kim²,

Wu³

2021

Preprint

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In this article, we study the normal generation of the mapping class group. We first show that a mapping class is normal generator if its restriction on the invariant subsurface normally generates the (pure) mapping class group of the subsurface. As an application, we provided a criterion for reducible mapping classes to normally generate the mapping class groups in terms of its asymptotic translation lengths on Teichmüller spaces. This is an analogue to the work of Lanier-Margalit dealing with pseudo-Anosov normal generators.

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Generating the mapping class group of a nonorientable surface by crosscap transpositions

Leśniak

Szepietowski

2017

Topology and its Applications

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A crosscap transposition is an element of the mapping class group of a nonorientable surface represented by a homeomorphism supported on a one-holed Klein bottle and swapping two crosscaps. We prove that the mapping class group of a compact nonorientable surface of genus g ≥ 7 is generated by conjugates of one crosscap transposition. In the case when the surface is either closed or has one boundary component, we give an explicit set of g + 2 crosscap transpositions generating the mapping class group.

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Normal generators for mapping class groups are abundant

Cited by 8 publications

References 23 publications

General primitivity in the mapping class group

General primitivity in the mapping class group

Reducible normal generators for mapping class groups are abundant

Generating the mapping class group of a nonorientable surface by crosscap transpositions

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