Big Mapping Class Groups: An Overview

Aramayona, Javier; Vlamis, Nicholas G.

doi:10.1007/978-3-030-55928-1_12

Cited by 32 publications

(31 citation statements)

References 72 publications

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“…For example, the Blooming Cantor tree surface (genus(S) = ∞ and Ends(S) = Ends np (S) = C) and the Cantor tree surface (≃ S 2 \ C and genus(S) = 0) satisfy the hypothesis of the previous theorem. See [AV20], for details.…”

Section: Birman Exact Sequencementioning

confidence: 99%

“…In the Appendix of that paper, Dahmani et al proved that the mapping class group of the compact surfaces have the R ∞ -property except for a few low complexity cases, using geometric group theoretic methods. Recently, there has been a surge of activities on infinite-type surfaces and corresponding groups, namely the big mapping class groups (see [AV20]).…”

Section: Introductionmentioning

confidence: 99%

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Twisted Conjugacy in Big Mapping Class Groups

Bhunia¹,

Krishna²

2021

Preprint

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Let G be a group and ϕ be an automorphism of G. Two elements x, y of G are said to be ϕ-twisted conjugate if y = gxϕ(g) −1 for some g ∈ G. A group G has the R ∞ -property if the number of ϕ-twisted conjugacy classes is infinite for every automorphism ϕ of G. In this paper we prove that the big mapping class group MCG(S) possesses the R ∞ -property under some suitable conditions on the infinite-type surface S. As an application we also prove that the big mapping class group possesses the R ∞ -property if and only if it satisfies the S ∞ -property.

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Section: Birman Exact Sequencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Twisted Conjugacy in Big Mapping Class Groups

Bhunia¹,

Krishna²

2021

Preprint

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“…We first show M aps(X) is a topological group. That composition is continuous follows from the fact that the map P H(X) × P H(X) → P H(X) defined by (( f , ĝ), ( f , ĝ )) → ( f f , ĝ ĝ) is continuous, being the restriction of the analogous map ( X → X) 4 → ( X → X) 2 .…”

Section: Topologymentioning

confidence: 99%

“…See Section 4. We thus propose M aps(X) as the "big Out(F n )" equivalent of mapping class groups of surfaces of infinite type (or "big mapping class groups"), for a survey of the subject see [2]. Comparison with mapping class groups has shown to be very useful in the study of Out(F n ), and we expect that comparison between M aps(X) and big mapping class groups will likewise prove fruitful.…”

Section: Introductionmentioning

confidence: 99%

Groups of proper homotopy equivalences of graphs and Nielsen Realization

Algom-Kfir¹,

Bestvina²

2021

Preprint

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For a locally finite connected graph X we consider the group M aps(X) of proper homotopy equivalences of X. We show that it has a natural Polish group topology, and we propose these groups as an analog of big mapping class groups. We prove the Nielsen Realization theorem: if H is a compact subgroup of M aps(X) then X is proper homotopy equivalent to a graph Y so that H is realized by simplicial isomorphisms of Y .

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“…Mapping class groups of finite-type surfaces have been a classical field of study for several decades, and within the past decade there has been newfound interest in the study of mapping class groups of infinite-type, also known as big, surfaces. See [3] for a recent survey on the topic of mapping class groups of infinite-type surfaces.…”

Section: Introductionmentioning

confidence: 99%

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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Big Mapping Class Groups: An Overview

Cited by 32 publications

References 72 publications

Twisted Conjugacy in Big Mapping Class Groups

Twisted Conjugacy in Big Mapping Class Groups

Groups of proper homotopy equivalences of graphs and Nielsen Realization

Coarse Geometry of Pure Mapping Class Groups of Infinite Graphs

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