On a Universal Mapping Class Group of Genus Zero

Funar, Louis; Kapoudjian, Christophe

doi:10.1007/s00039-004-0480-9

Cited by 30 publications

(106 citation statements)

References 22 publications

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“…An algebraic relation between T and the braid groups has been discovered in an article due to P. Greenberg and V. Sergiescu ([21]). Since then, several works ( [6], [7], [10], [11], [14], [15], [16], [28]) have contributed to improve our understanding of the links between Thompson groups and mapping class groups of surfaces -including braid groups.…”

Section: Statements and Resultsmentioning

confidence: 99%

“…However, BV is not related to the group of Greenberg-Sergiescu constructed and studied in [21], but rather to our universal mapping class group in genus zero (cf. [14]). …”

Section: Statements and Resultsmentioning

confidence: 99%

“…In particular one can identify T with the group of asymptotically rigid homeomorphisms modulo isotopy of the ribbon tree D (cf. [28] and [14]). …”

Section: Remark 13mentioning

confidence: 98%

“…Recall from [14] that an almost automorphism (or piecewise tree automorphism) of some infinite binary tree T is given by a combinatorial isomorphism T nT 0 ! T nT 1 between the complements of two finite binary subtrees T 0 ; T 1 T .…”

Section: Corollary 12 the Thompson Group T Is Asynchronously Combablementioning

confidence: 99%

“…This picture suggests another approach to T , as a group of mapping classes of homeomorphisms of infinite surfaces (see [28] and [14]). The surfaces below will be oriented and all homeomorphisms considered in the sequel will be orientationpreserving, unless the opposite is explicitly stated.…”

mentioning

confidence: 99%

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The braided Ptolemy–Thompson group is asynchronously combable

Funar

Kapoudjian

2011

Comment. Math. Helv.

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Abstract. The braided Ptolemy-Thompson group T? is an extension of the Thompson group T by the full braid group B 1 on infinitely many strands and both of them can be viewed as mapping class groups of certain infinite planar surfaces. The main result of this article is that T ? (and in particular T ) is asynchronously combable. The result is new already for the group T . The method of proof is inspired by Lee Mosher's proof of automaticity of mapping class groups.Mathematics Subject Classification (2010). 57N05, 20F38, 57M07, 20F34.

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Section: Statements and Resultsmentioning

confidence: 99%

“…However, BV is not related to the group of Greenberg-Sergiescu constructed and studied in [21], but rather to our universal mapping class group in genus zero (cf. [14]). …”

Section: Statements and Resultsmentioning

confidence: 99%

“…In particular one can identify T with the group of asymptotically rigid homeomorphisms modulo isotopy of the ribbon tree D (cf. [28] and [14]). …”

Section: Remark 13mentioning

confidence: 98%

Section: Corollary 12 the Thompson Group T Is Asynchronously Combablementioning

confidence: 99%

mentioning

confidence: 99%

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The braided Ptolemy–Thompson group is asynchronously combable

Funar

Kapoudjian

2011

Comment. Math. Helv.

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On the automorphism group of the asymptotic pants complex of an infinite surface of genus zero

Funar

Nguyen

2016

Math. Nachr.

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The braided Thompson group B is an asymptotic mapping class group of a sphere punctured along the standard Cantor set, endowed with a rigid structure. Inspired from the case of finite type surfaces we consider a HatcherThurston cell complex whose vertices are asymptotically trivial pants decompositions. We prove that the automorphism group B 1 2 of this complex is also an asymptotic mapping class group in a weaker sense. Moreover B The study of such automorphisms groups in the pro-finite or pro-unipotent categories seems fundamental in Grothendieck's program. For instance, although the pro-finite pants complexes are still rigid the automorphism group of the corresponding pro-finite curve complexes is a version of the Grothendieck-Teichmüller group ([18] and references there).Simpler versions of this general question concern the solenoids, whose study was started in [2], and then infinite type surfaces corresponding to direct limits. The purpose of this article is to make progress in the second case using the formalism of asymptotically rigid homeomorphisms and braided Thompson groups developed in [9], [10]. A previous result in this direction is the rigidity theorem proved in [8] for an infinite type planar surface related to the Thompson group T (see [4]).In this note we will consider an infinite surface obtained from the sphere by deleting the standard Cantor set from the equator. Since its mapping class group is a topological group, the authors of [9] introduced a smaller subgroup B called asymptotically rigid mapping class group, which was proved to be finitely presented. As its name suggests, one restricts to mapping classes of those homeomorphisms which preserve an extra structure on the surface, but only outside of large enough compact sub-surfaces.There were different but closely related versions of such asymptotically rigid mapping class groups considered independently by Brin ([3]) and Dehornoy ([6], [7]). All of them are usually designed by the generic term of

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Big Mapping Class Groups: An Overview

Aramayona

Vlamis

2020

In the Tradition of Thurston

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On a Universal Mapping Class Group of Genus Zero

Cited by 30 publications

References 22 publications

The braided Ptolemy–Thompson group is asynchronously combable

The braided Ptolemy–Thompson group is asynchronously combable

On the automorphism group of the asymptotic pants complex of an infinite surface of genus zero

Big Mapping Class Groups: An Overview

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