Big mapping class groups and rigidity of the simple circle

Calegari, Danny; Chen, Lvzhou

doi:10.1017/etds.2020.43

Cited by 15 publications

(20 citation statements)

References 13 publications

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“…When S = R 2 , Lemmas 6.6 and 6.8 from [10] show that the normal closure of Γ (r) is equal to Γ. This proves N div = Γ = Γ 0 in the special case of S = R 2 since each mapping class in Γ (r) is supported in a dividing disk in this situation.…”

Section: An Immediate Corollary Ismentioning

confidence: 73%

“…First note that N div ⊂ Γ 0 since each mapping class supported in a disk becomes trivial under the forgetful map to Mod(S). To prove the converse, we shall use notation and terminology consistent with [10]. Recall that a short ray in S is an isotopy class of proper embedded ray from ∞ to some point in K, and a lasso is a homotopically essential properly embedded copy of R in S from ∞ to ∞.…”

Section: An Immediate Corollary Ismentioning

confidence: 99%

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Normal subgroups of big mapping class groups

Calegari¹,

Chen²

2021

Preprint

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Let S be a surface and let Mod(S, K) be the mapping class group of S permuting a Cantor subset K ⊂ S. We prove two structure theorems for normal subgroups of Mod(S, K).(Purity:) if S has finite type, every normal subgroup of Mod(S, K) either contains the kernel of the forgetful map to the mapping class group of S, or it is 'pure' -i.e. it fixes the Cantor set pointwise.(Inertia:) for any n element subset Q of the Cantor set, there is a forgetful map from the pure subgroup PMod(S, K) of Mod(S, K) to the mapping class group of S, Q fixing Q pointwise. If N is a normal subgroup of Mod(S, K) contained in PMod(S, K), its image NQ is likewise normal. We characterize exactly which finite-type normal subgroups NQ arise this way.Several applications and numerous examples are also given. Contents 1. Introduction 1 1.1. Statement of results 2 Acknowledgment 3 2. The Purity Theorem 3 2.1. Applications 4 2.2. Proof of the Purity Theorem when S has at most one puncture 5 2.3. Proof of the Purity Theorem in general 11 3. The Inertia Theorem 12 3.1. Inert subgroups 13 3.2. Proof of the Inertia Theorem 15 3.3. Operations on normal subgroups 19 References 20

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Section: An Immediate Corollary Ismentioning

confidence: 73%

Section: An Immediate Corollary Ismentioning

confidence: 99%

Normal subgroups of big mapping class groups

Calegari¹,

Chen²

2021

Preprint

Self Cite

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“…In this topology, the property of a ray having a (transverse) self-intersection is open. Thus the set of simple rays is closed on S 1 C , and is also nowhere dense [6,Lemma 3.3].…”

Section: Background On Rays and Loopsmentioning

confidence: 99%

Laminations and 2-filling rays on infinite type surfaces

Chen¹,

Rasmussen²

2020

Preprint

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The loop graph of an infinite type surface is an infinite diameter hyperbolic graph first studied in detail by Juliette Bavard. An important open problem in the study of infinite type surfaces is to describe the boundary of the loop graph as a space of geodesic laminations. We approach this problem by constructing the first examples of 2-filling rays on infinite type surfaces. Such rays accumulate onto geodesic laminations which are in some sense filling, but without strong enough properties to correspond to points in the boundary of the loop graph. We give multiple constructions using both a hands-on combinatorial approach and an approach using train tracks and automorphisms of flat surfaces. In addition, our approaches are sufficiently robust to describe all 2-filling rays with certain other basic properties as well as to produce uncountably many distinct mapping class group orbits.

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“…Relationship to the ray graph. The ray graph was introduced by Calegari in [5] and is defined as follows. The ray graph is the graph whose vertex set is the set of isotopy classes of proper rays, with interior in the complement of the Cantor set K ⊂ R 2 , from a point in K to infinity, and whose edges (of length 1) are the pairs of such rays that can be realized disjointly.…”

Section: Introductionmentioning

confidence: 99%

The grand arc graph

Bar-Natan¹,

Verberne²

2021

Preprint

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In this article, we construct a new simplicial complex for infinite-type surfaces, which we call the grand arc graph. We show that if the end space of a surface has at least three different self-similar equivalence classes of maximal ends, then the grand arc graph is infinite-diameter and δ-hyperbolic. We also show that the mapping class group acts on the grand arc graph by isometries and that the action is quasi-continuous, which is a coarse relaxation of a continuous action. When the surface has stable maximal ends, we also show that this action has finitely many orbits.

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Big mapping class groups and rigidity of the simple circle

Abstract: Let $\unicode[STIX]{x1D6E4}$ denote the mapping class group of the plane minus a Cantor set. We show that every action of $\unicode[STIX]{x1D6E4}$ on the circle is either trivial or semiconjugate to a unique minimal action on the so-called simple circle.

Cited by 15 publications

References 13 publications

Normal subgroups of big mapping class groups

Normal subgroups of big mapping class groups

Laminations and 2-filling rays on infinite type surfaces

The grand arc graph

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