Minimal Asymptotic Translation Lengths of Torelli Groups and Pure Braid Groups on the Curve Graph

Baik, Hyungryul; Shin, Hyun Dong

doi:10.1093/imrn/rny273

Cited by 8 publications

(19 citation statements)

References 14 publications

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“…The construction involved in Theorem 4.1 can be modified to deal with the Torelli case. Such a modification gives an asymptote for L C (2g, g) which was already shown by [BS20] in a different way. See Remark 4.2.…”

Section: Introductionmentioning

confidence: 59%

“…We know that a way to construct the surface S n and map f n corresponding to 2 n α + β is as follows: Let S be the Z-fold cover corresponding to β restricted to S g 0 , f a lift of f 0 , h the deck transformation, then with a suitable choice of f we have S n = S/(h 2 n f ) and f n is lifted to h. Now consider a simple closed curve on a fundamental domain of S which is not homologous to the boundary, such that the homology class c represented by this curve γ is preserved by f . The existence of such a homology class is due to the construction in Baik-Shin [BS20]. Then…”

Section: Upper Boundsmentioning

confidence: 97%

“…In this section, we prove Theorem 1.2. The main idea is similar to the one used in the proof in [BS20] of that L C (2g, g) ≥ C/g for some constant. First note that for any f :…”

Section: Lower Boundmentioning

confidence: 98%

“…See Remark 4.2. Further, only the last assertion can also be deduced from Theorem 1.2 and [BS20]. See Section 4 for details.…”

Section: Introductionmentioning

confidence: 99%

“…We focus on specific dimensions of maximal invariant subspaces. In [BS20], Torelli pseudo-Anosovs are constructed in a concrete way based on Penner's or Thurston's construction. In a similar line of thought, we utilize finite cyclic covers of S 2 so that we get pseudo-Anosovs f ∈ Mod(S g ) with m(f ) = 2g − 1 and satisfying the upper bound in Theorem 3.3.…”

Section: Introductionmentioning

confidence: 99%

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Minimal asymptotic translation lengths on curve complexes and homology of mapping tori

Baik¹,

Kim²,

Wu³

2021

Preprint

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We consider pseudo-Anosov mapping classes on a closed orientable surface of genus g that fix a rank k subgroup of the first homology of the surface. We first show that there exists a uniform constant C > 0 so that the minimal asymptotic translation length on the curve complex among such pseudo-Anosovs is bounded below by C g(2g−k+1) . This interpolates between results of Gadre-Tsai and of the first author and Shin, who treated the cases of the entire mapping class group (k = 0) and the Torelli subgroup (k = 2g), respectively. We also discuss possible strategy to obtain an upper bound. Finally, we construct a pseudo-Anosov on a genus g surface whose maximal invariant subspace is of rank 2g − 1 and the asymptotic translation length is of ≍ 1/g for all g. Such pseudo-Anosovs are further shown to be unable to normally generate the whole mapping class groups. As Lanier-Margalit proved that pseudo-Anosovs with small translation lengths on the Teichmüller spaces normally generate mapping class groups, our observation provides a restriction on how small the asymptotic translation lengths on curve complexes should be if the similar phenomenon holds for curve complexes.

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Section: Introductionmentioning

confidence: 59%

Section: Upper Boundsmentioning

confidence: 97%

“…In this section, we prove Theorem 1.2. The main idea is similar to the one used in the proof in [BS20] of that L C (2g, g) ≥ C/g for some constant. First note that for any f :…”

Section: Lower Boundmentioning

confidence: 98%

“…See Remark 4.2. Further, only the last assertion can also be deduced from Theorem 1.2 and [BS20]. See Section 4 for details.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Minimal asymptotic translation lengths on curve complexes and homology of mapping tori

Baik¹,

Kim²,

Wu³

2021

Preprint

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On the finiteness property of hyperbolic simplicial actions: the right-angled Artin groups and their extension graphs

2022

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Topological and dynamical properties of Torelli groups of partitioned surfaces

Baik,

Shin,

Tranchida

2024

Geom Dedicata

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Putman introduced a notion of a partitioned surface which is a surface with boundary with decoration restricting how the surface can be embedded into larger surfaces, and defined the Torelli group of the partitioned surfaces. In this paper, we study some topological and dynamical aspects of the Torelli groups of partitioned surfaces. More precisely, first we obtain upper and lower bounds on the cohomological dimension of Torelli groups of partitioned surfaces and show that those two bounds coincide when at most three boundary components are grouped together in the partition of the boundary. Second, we study the asymptotic translation lengths of Torelli groups of partitioned surfaces on the corresponding curve complexes. We show that the minimal asymptotic translation length asymptotically behaves almost like the reciprocal of the Euler characteristic of the surface. This generalizes the previous result of the first and second authors on Torelli groups for closed surfaces.

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Minimal Asymptotic Translation Lengths of Torelli Groups and Pure Braid Groups on the Curve Graph

Cited by 8 publications

References 14 publications

Minimal asymptotic translation lengths on curve complexes and homology of mapping tori

Minimal asymptotic translation lengths on curve complexes and homology of mapping tori

On the finiteness property of hyperbolic simplicial actions: the right-angled Artin groups and their extension graphs

Topological and dynamical properties of Torelli groups of partitioned surfaces

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