1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs

Hensel, Sebastian; Przytycki, Piotr; Webb, Richard

doi:10.4171/jems/517

Cited by 109 publications

(148 citation statements)

References 25 publications

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“…In this section, we discuss the regularity of "top-dimensional manifolds" in M. These play an important role in [Kleiner and Leeb 1997;Kapovich, Kleiner and Leeb 1998;. Our argument is analogous to those to be found there, though set in a somewhat different context.…”

Section: Cubulating Planesmentioning

confidence: 54%

“…For definiteness, we can interpret the dimension of a topological space to be its covering dimension. (Note that this differs from the notion of "topological rank" used in [Kleiner and Leeb 1997]. )…”

Section: Cubulating Planesmentioning

confidence: 82%

“…A more direct proof in a specific case is given in [Bowditch 2016b] (see Proposition 2.1 thereof). Analogous, but different, statements can be found in [Kleiner and Leeb 1997] and [Kapovich, Kleiner and Leeb 1998]. …”

Section: Introductionmentioning

confidence: 84%

“…The proof is based on an argument in Section 6.1 of [Kleiner and Leeb 1997]. A related, but slightly different statement can be found in [Behrstock, Kleiner, Minsky and Mosher 2012, Section 6].…”

Section: Cubulating Planesmentioning

confidence: 99%

“…We remark that if one assumes quasi-isometric rigidity as given in those papers, then one recovers (indirectly) that the quasi-isometry type of ‫(ލ‬ ) determines the topological type of (modulo a few low-dimensional exceptional cases) since it determines Map( ) up to isomorphism (see, for example, [Rafi and Schleimer 2011] for a proof that Map( ) determines ). Given this, Theorem 1.1 would be equivalent to asserting that any quasi-isometric embedding of ‫(ލ‬ ) into itself is necessarily a quasi-isometry (at least when ξ( ) ≥ 4).…”

Section: Introductionmentioning

confidence: 99%

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Large-scale rigidity properties of the mapping class groups

Bowditch

2018

Pacific J. Math.

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We study the coarse geometry of the mapping class group of a compact orientable surface. We show that, apart from a few low-complexity cases, any quasi-isometric embedding of a mapping class group into itself agrees up to bounded distance with a left multiplication. In particular, such a map is a quasi-isometry. This is a strengthening of the result of Hamenstädt and of Behrstock, Kleiner, Minsky and Mosher that the mapping class groups are quasi-isometrically rigid. In the course of proving this, we also develop the general theory of coarse median spaces and median metric spaces with a view to applications to Teichmüller space, and related spaces.

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Section: Cubulating Planesmentioning

confidence: 54%

Section: Cubulating Planesmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 84%

Section: Cubulating Planesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Large-scale rigidity properties of the mapping class groups

Bowditch

2018

Pacific J. Math.

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Big Mapping Class Groups: An Overview

Aramayona

Vlamis

2020

In the Tradition of Thurston

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Efficient geodesics and an effective algorithm for distance in the complex of curves

2016

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We give an algorithm for determining the distance between two vertices of the complex of curves. While there already exist such algorithms, for example by Leasure, Shackleton, and Webb, our approach is new, simple, and more effective for all distances accessible by computer. Our method gives a new preferred finite set of geodesics between any two vertices of the complex, called efficient geodesics, which are different from the tight geodesics introduced by Masur and Minsky. FIGURE 1. Vertices of C (S 2 ) with distance 4 and intersection number 12; this is the smallest possible intersection for vertices with distance 4 2 JOAN BIRMAN, DAN MARGALIT, AND WILLIAM MENASCO One goal of this paper is to give an algorithm for distance-the efficient geodesic algorithm-that actually can be implemented, at least for small distances. The third author and Glenn, Morrell, and Morse [9] have in fact already developed an implementation of our algorithm, called Metric in the Curve Complex [8]. Their program is assembling a data bank of examples as we write.Known examples. Let S g denote a closed, connected, orientable surface of genus g and let i min (g, d) denote the minimal intersection number for vertices of C (S g ) with distance d. The Metric in the Curve Complex program has been used to show that:(1) i min (2, 4) = 12 and (2) i min (3, 4) ≤ 21. The highly symmetric example in Figure 1-which realizes i min (2, 4)-was discovered using the program. See Section 2 for a discussion of this example and a proof using the methods of this paper that the distance is actually 4.We are only aware of one other explicit picture in the literature of a pair of vertices of C (S 2 ) that have distance four, namely, the example of Hempel that appears in the notes of Saul Schleimer [16, Figure 2] (see [9, Example 1.6] for a proof that the distance is 4). This example has geometric intersection number 25.Using the bounded geodesic image theorem [12, Theorem 3.1] of Masur and Minsky (as quantified by Webb [21]) it is possible to explicitly construct examples of vertices with any given distance; see [17, Section 6]. We do not know how to keep the intersection numbers close to the minimum with this method, but Aougab and Taylor did in fact use this method to give examples of vertices of arbitrary distance whose intersection numbers are close to the minimum in an asymptotic sense; see their paper [3] for the precise statement.Local infinitude. One reason why computations with the complex of curves are so difficult is that it is locally infinite and moreover there are infinitely many geodesics (i.e. shortest paths) between most pairs of vertices. Masur and Minsky [12] addressed this issue by finding a preferred set of geodesics, called tight geodesics, and proving that between any two vertices there are finitely many tight geodesics; see Section 2.2 for the definition. Our first goal is to give a new class of geodesics that still has finitely many elements connecting any two vertices but is more amenable to certain computations.Efficient geodesics. Ou...

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1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs

Cited by 109 publications

References 25 publications

Large-scale rigidity properties of the mapping class groups

Large-scale rigidity properties of the mapping class groups

Big Mapping Class Groups: An Overview

Efficient geodesics and an effective algorithm for distance in the complex of curves

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