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...

We prove that the image of the mapping class group by the representations arising in the SU (2)-TQFT is infinite, provided that the genus g ≥ 2 and the level of the theory r = 2, 3, 4, 6 (and r = 10 for g = 2). In particular it follows that the quotient groups M g /N (t r ) by the normalizer of the r-th power of a Dehn twist t are infinite if g ≥ 3 and r = 2, 3, 4, 6, 8, 12.

The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman and Kuperberg's G 2 quantum invariants. Our method consists of the study of Markov traces on a suitable tower of quotients of cubic Hecke algebras extending Jones approach.

The minimum number of critical points of a small codimension smooth map between two manifolds is computed. Some partial results for the case of higher codimension when the manifolds are spheres are also given.

Between the rank 3 quotients of cubic Hecke algebras there is essentially one of maximal dimension. We prove it has a unique Markov trace having values in a torsion module. Therefore the description of a Markov trace on the cubic Hecke algebra corresponding to x 3 -j-1 and having the parameters (1,1) is derived. Thus we obtain a numerical link invariant of finite degree, and define a whole sequence of 3 rd order Vassiliev invariants. Contents1. Introduction 513 2. The quotients of H(Q,3) 517 3. Markov traces on K^y) 524 4. Link groups and invariants 528 5. Graphical reduction of obstructions 533 A.

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

The aim of this paper (inspired from a problem of Habegger) is to describe the set of cubical decompositions of compact manifolds mod out by a set of combinatorial moves analogous to the bistellar moves considered by Pachner, which we call bubble moves. One constructs a surjection from this set onto the the bordism group of codimension one immersions in the manifold. The connected sums of manifolds and immersions induce multiplicative structures which are respected by this surjection. We prove that those cubulations which map combinatorially into the standard decomposition of R n for large enough n (called mappable), are equivalent. Finally we classify the cubulations of the 2-sphere.