An extension of the Burau representation to a mapping class group associated to Thompson’s group 𝑇

Kapoudjian

2008

Geom. Topol.

Self Cite

Pursuing our investigations on the relations between Thompson groups and mapping class groups, we introduce the group T ] (and its companion T ) which is an extension of the Ptolemy-Thompson group T by the braid group B 1 on infinitely many strands. We prove that T ] is a finitely presented group by constructing a complex on which it acts cocompactly with finitely presented stabilizers, and derive from it an explicit presentation. The groups T ] and T are in the same relation with respect to each other as the braid groups B nC1 and B n , for infinitely many strands n. We show that both groups embed as groups of homeomorphisms of the circle and their word problem is solvable. 20F36, 57M07; 20F38, 20F05, 57N05 IntroductionThe first relationships between Thompson's groups and braid groups were brought to light in the article [25] by P Greenberg and V Sergiescu, which is devoted to the construction and the homological study of extensions of Thompson's groups F and T by the stable braid group B Thompson's groups are not tree automorphisms, but are induced by piecewise tree automorphisms [28]. Therefore, a natural question is to find a way of lifting those elements to automorphisms of an appropriate structure. The answer proposed by [28] and [22] is to lift them to mapping classes of homeomorphisms of particular surfaces. Indeed, both groups A T and B are mapping class groups of infinite surfaces which are thickenings of suitable regular trees; the surfaces are endowed with an extra structure that must be, not globally, but only asymptotically preserved by the mapping classeshence the notion of asymptotic mapping class group. This extra structure may consist of a decomposition of the surface into pairs of pants, hexagons, hexagons with punctures, and so on.The surface D ] considered for the construction of the asymptotic mapping class group T ] is the planar thickened binary tree, which is punctured along an infinite discrete subset of points. The extra structure consists of a decomposition into suitably punctured hexagons. The asymptotic mapping class group that one obtains this way is an extension T ] of T by the group of braids B 1 on infinitely many strands (corresponding to the punctures). Therefore, T ] is quite similar to, but simpler than A T .This new group T ] seems interesting and worthy of deeper study. Compared with B , the definition of T ] presents new features, for instance, the dependence on the extra structure is now clearly manifest. We can choose two sets of punctures leading to homeomorphic surfaces for which the associated groups are not isomorphic. We obtain that way another group T , which is a sort of twin brother of T ] . Although T ] and T share the same properties, they are different. Our main result is the following: T (see Penner [35]). The terminology used here for T is expected to stress on its link with the Penner-Ptolemy groupoid. The group T ] is essentially different from BV (and B ), being an extension by the whole group of braids, and not only the pure braids. Moreover, it is know...

Section: Introductionmentioning

confidence: 99%

The braided Ptolemy–Thompson group is finitely presented

Kapoudjian

2008

Geom. Topol.

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“…0 ; E a 1 // of the Ptolemy group, where E a 1 is the oriented edge in the base triangle of the Farey triangulation 0 next to E a 0 . Let us explain now some details concerning the identification of the Ptolemy groupoid appearing in Lochak-Schneps' picture with that considered by the present authors (see also in [14], [28]). Lochak and Schneps defined two generators of P T , which are the two local moves below:…”

Section: The Thompson Group T Is Asynchronously Combablementioning

confidence: 58%

“…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%

“…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: 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

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.

On a Universal Mapping Class Group of Genus Zero

Kapoudjian²

2004

Geom. funct. anal.

106

The aim of this paper is to introduce a group containing the mapping class groups of all genus zero surfaces. Roughly speaking, such a group is intended to be a discrete analogue of the diffeomorphism group of the circle. One defines indeed a universal mapping class group of genus zero, denoted B. The latter is a nontrivial extension of the Thompson group V (acting on the Cantor set) by an inductive limit of pure mapping class groups of all genus zero surfaces. We prove that B is a finitely presented group, and give an explicit presentation of it.