The First Integral Cohomology of Pure Mapping Class Groups

Aramayona, Javier; Patel, Priyam; Vlamis, Nicholas G.

doi:10.1093/imrn/rnaa229

Cited by 25 publications

(72 citation statements)

References 19 publications

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“…The following result is analogous to a result of Aramayona-Patel-Vlamis ( [2]) in the surface setting.…”

Section: Introductionsupporting

confidence: 74%

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Coarse Geometry of Pure Mapping Class Groups of Infinite Graphs

Domat¹,

Hoganson²,

Kwak³

2022

Preprint

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We discuss the large-scale geometry of pure mapping class groups of locally finite, infinite graphs, motivated from recent work by Algom-Kfir-Bestvina [1] and the work of Mann-Rafi [12] on the large-scale geometry of mapping class groups of infinitetype surfaces. Using the framework of Rosendal for coarse geometry of non-locally compact groups, we classify when the pure mapping class group of a locally finite, infinite graph is globally coarsely bounded (an analog of compact) and when it is locally coarsely bounded (an analog of locally compact).Our techniques also give lower bounds on the first integral cohomology of the pure mapping class group for some graphs and show that some of these groups have continuous actions on simplicial trees.

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“…The following result is analogous to a result of Aramayona-Patel-Vlamis ( [2]) in the surface setting.…”

Section: Introductionsupporting

confidence: 74%

“…Let Γ be an infinite, locally finite graph with at least two ends accumulated by loops. We show every such graph has non-CB pure mapping class groups following ideas from [2]. We first need some background on free factors.…”

Section: Flux Maps: Graphs With |E | ≥mentioning

confidence: 99%

Coarse Geometry of Pure Mapping Class Groups of Infinite Graphs

Domat¹,

Hoganson²,

Kwak³

2022

Preprint

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“…This coarse end behavior will also provide a convenient description of the surfaces S ± associated to an end-periodic homeomorphism as described above. To formalize this, we will first introduce a few more definitions and state a result of Aramayona-Patel-Vlamis [APV20].…”

Section: Preliminariesmentioning

confidence: 99%

“…If, for example, γ and f (γ) are disjoint, then |ϕ [γ] (f )| is the genus of the subsurface bounded by γ and f (γ) (with a negative sign if f (γ) is to the left of γ); see [APV20, Section 3] for a more precise description. Furthermore, they show that ϕ [γ] : PMap(S) → Z is a welldefined homomorphism that depends only on the homology class [γ]; see [APV20,Proposition 3.3].…”

Section: Preliminariesmentioning

confidence: 99%

End-periodic homeomorphisms and volumes of mapping tori

Field¹,

Kim²,

Leininger³

et al. 2021

Preprint

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Given an irreducible, end-periodic homeomorphism f : S → S of a surface with finitely many ends, all accumulated by genus, the mapping torus, M f , is the interior of a compact, irreducible, atoroidal 3-manifold M f with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of M f in terms of the translation length of f on the pants graph of S. This builds on work of Brock and Agol in the finite-type setting. We also construct a broad class of examples of irreducible, end-periodic homeomorphisms and use them to show that our bound is asymptotically sharp.

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“…Pure subgroups of big mapping class groups and their properties are studied e.g. in [3] and [19]. Note that PMod(S, Q) is a normal subgroup of Mod(S, Q).…”

Section: Introductionmentioning

confidence: 99%

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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The First Integral Cohomology of Pure Mapping Class Groups

Cited by 25 publications

References 19 publications

Coarse Geometry of Pure Mapping Class Groups of Infinite Graphs

Coarse Geometry of Pure Mapping Class Groups of Infinite Graphs

End-periodic homeomorphisms and volumes of mapping tori

Normal subgroups of big mapping class groups

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