Related papers: Orderings of mapping class groups after Thurston

We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single…

For every $n\ge 2$, the {\em surface Houghton group} $\mathcal B_n$ is defined as the asymptotically rigid mapping class group of a surface with exactly $n$ ends, all of them non-planar. The groups $\mathcal B_n$ are analogous to, and in…

We prove that the pure braid groups on closed, orientable surfaces are bi-orderable, and that the pure braid groups on closed, non-orientable surfaces have generalized torsion, thus they are not bi-orderable.

Using the canonical JSJ splitting, we describe the outer automorphism group $\Out(G)$ of a one-ended word hyperbolic group $G$. In particular, we discuss to what extent $\Out(G)$ is virtually a direct product of mapping class groups and a…

We describe the development of the theory of automatic groups. We begin with a historical introduction, define the concepts of automatic, biautomatic and combable groups, derive basic properties, then explain how hyperbolic groups and the…

Clean markings on surfaces were a key component in Masur and Minsky's hierarchy machinery, which proved to be a powerful tool in the study of mapping class groups. We construct a marking graph for irreducible finite-type Artin groups which…

Consider a one-ended word-hyperbolic group. If it is the fundamental group of a graph of free groups with cyclic edge groups then either it is the fundamental group of a surface or it contains a finitely generated one-ended subgroup of…

In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian punctured surface groups without accidental parabolics. In this note we prove that pre-images of points are precisely end-points of leaves of the ending lamination…

We show that mapping class groups of surfaces of genus at least two contain elements of infinite order that are not conjugate to their inverses, but whose powers have bounded torsion lengths. In particular every homogeneous…

Hyperelliptic mapping class groups are defined either as the centralizers of hyperelliptic involutions inside mapping class groups of oriented surfaces of finite type or as the inverse images of these centralizers by the natural…

The study of geometric group theory has suggested several theorems related to subdivision tilings that have a natural hyperbolic structure. However, few examples exist. We construct subdivision tilings for the complement of every…

For an infinite type surface $\Sigma$, we consider the space of (marked) convex hyperbolic structures on $\Sigma$, denoted $H(\Sigma)$, with the Fenchel-Nielsen topology. The (big) mapping class group acts faithfully on this space allowing…

We give the asymptotic growth of the number of (multi-)arcs of bounded length between boundary components on complete finite-area hyperbolic surfaces with boundary. Specifically, if $S$ has genus $g$, $n$ boundary components and $p$…

For a group $G$, we construct a quasi morphism from its left orderings and the map from the space of left orderings to the second bounded cohomology. We show that these maps reflect various properties of the group orderings.

We show that continuous epimorphisms between a class of subgroups of mapping class groups of orientable infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms. This class of subgroups includes the pure mapping…

We consider an oriented surface S and a cellular complex X of curves on S, defined by Hatcher and Thurston in 1980. We prove by elementary means, without Cerf theory, that the complex X is connected and simply connected. From this we derive…

A normal subgroup of the (extended) mapping class group of a surface is said to be geometric if its automorphism group is the mapping class group. We prove that in the case of the Cantor tree surface, every normal subgroup is geometric. We…

By analyzing known presentations of the pure mapping groups of orientable surfaces of genus $g$ with $b$ boundary components and $n$ punctures, we show that these groups are isomorphic to some groups related to the braid groups and the…

In this note we prove that the mapping class group of a compact topological manifold $M$ with boundary is of finite type, under assumptions on its dimension and connectivity.

A topology is defined on the mapping class group of a compact connected orientable surface. It is shown that a notion of "genericity" on subsets of the mapping class group arises from this definition. Many plausible results follow from this…