A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general
construction of many actions of groups on quasi-trees. The groups we can handle include …

In this work, we study the asymptotic geometry of the map** class group and Teichmuller
space. We introduce tools for analyzing the geometry of “projection” maps from these spaces …

We study the geometry of non-relatively hyperbolic groups. Generalizing a result of
Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively …

Let g,n be an orientable surface of genus g with n punctures. We study actions of the
map** class group Mod_g,n of g,n via Hodge-theoretic and arithmetic techniques. We …

We develop a theory of convex cocompact subgroups of the map** class group MCG of a
closed, oriented surface S of genus at least 2, in terms of the action on Teichmüller space …

Hierarchically hyperbolic spaces provide a common framework for studying map** class
groups of finite-type surfaces, Teichmüller space, right-angled Artin groups, and many other …

We study the large scale geometry of map** class groups ℳ CG (S), using hyperbolicity
properties of curve complexes. We show that any self quasi-isometry of ℳ CG (S)(outside a …

We introduce the notion of a coarse median on a metric space. This satisfies the axioms of a
median algebra up to bounded distance. The existence of such a median on a geodesic …

Divergence functions of a metric space estimate the length of a path connecting two points $
A $, $ B $ at distance $\le n $ avoiding a large enough ball around a third point $ C $. We …

In the present paper we study two sequences of real numbers associated to a symplectic
diffeomorphism:¶• The uniform norm of the differential of its n-th iteration;¶• The word length …