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.

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...

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We study some aspects of the geometric representation theory of the Thompson and Neretin groups, suggested by their analogies with the diffeomorphism groups of the circle. We prove that the Burau representation of the Artin braid groups extends to a mapping class group A T related to Thompson's group T by a short exact sequence B ∞ ֒→ A T → T , where B ∞ is the infinite braid group. This non-commutative extension abelianises to a central extension 0A morphism from the above non-commutative extension to a reduced Pressley-Segal extension is then constructed, and the class gv is realised as a pull-back of the reduced Pressley-Segal class. A similar program is carried out for an extension of the Neretin group related to the combinatorial version of the Bott-Virasoro-Godbillon-Vey class.

We exhibit a finitely generated group M whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface S∞ of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus g with n boundary components, for any g ≥ 0 and n > 0. We construct a representation of M into the restricted symplectic group Spres(Hr) of the real Hilbert space generated by the homology classes of non-separating circles on S∞, which generalizes the classical symplectic representation of the mapping class groups. Moreover, we show that the first universal Chern class in H 2