Automorphisms of Thurston's Space of Measured Laminations

We construct a geometric model for the mapping class group M of a non-exceptional oriented surface of finite type and use it to show that the action of M on the compact Hausdorff space of complete geodesic laminations is topologically amenable. As a consequence, the Novikov higher signature conjecture holds for every subgroup of M.

We give an infinite presentation for the mapping class group of a non-orientable surface with boundary components. The presentation is a generalization of the presentation given by the second author [15].

Let $M$ be a closed surface. By $\Homeo(M)$ we denote the group of orientation preserving homeomorphisms of $M$ and let $\MC(M)$ denote the Mapping class group. In this paper we complete the proof of the conjecture of Thurston that says that for any closed surface $M$ of genus $\g \ge 2$, there is no homomorphic section $\E:\MC(M) \to \Homeo(M)$ of the standard projection map $\Proj:\Homeo(M) \to \MC(M)$.

We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these `big' mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced...

Omori and the author have given an infinite presentation for the mapping class group of a compact non-orientable surface. In this paper, we give more simple infinite presentations for this group.

We survey recent developments on mapping class groups of surfaces of infinite topological type.

The mapping class group of a non-exceptional oriented surface of finite type admits a biautomatic structure.

This paper attempts to relate some ideas of Grothendieck in his Esquisse d'un programme and some of the recent results on 2-dimensional topology and geometry. Especially, we shall discuss the Teichm\"uller theory, the mapping class groups, $SL(2, \bold C)$ representation variety of surface groups, and Thurston's theory of measured laminations.

In this article we survey, and make a few new observations about, the surprising connection between sub-monoids of mapping class groups and interesting geometry and topology in low-dimensions.

We study the action of the elements of the mapping class group of a surface of finite type on the Teichm\"uller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurston's classification of ...