Related papers: The Boundary at Infinity of the Curve Complex and …
In this survey paper we give a proof of hyperbolicity of the complex of curves for a non-exceptional surface S of finite type combining ideas of Masur/Minsky and Bowditch. We also shortly discuss the relation between the geometry of the…
This paper is devoted to study of transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We…
The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. It is not hard to see that the…
We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the "hyperbolic directions" in that space. This boundary is a quasi-isometry invariant and thus produces…
In this paper, the Teichm{\"u}ller spaces of surfaces appear from two points of views: the conformal category and the hyperbolic category. In contrast to the case of surfaces of topologically finite type, the Teichm{\"u}ller spaces…
Let $S$ be an orientable surface with negative Euler characteristic. For $k \in \mathbb{N}$, let $\mathcal{C}_{k}(S)$ denote the $\textit{k-curve graph}$, whose vertices are isotopy classes of essential simple closed curves on $S$, and…
We prove, for any n, that there is a closed connected orientable surface S so that the hyperbolic space H^n almost-isometrically embeds into the Teichm\"uller space of S, with quasi-convex image lying in the thick part. As a consequence,…
We construct a metric simplicial complex which is an almost isometric model of the moduli space M(S) of Riemann surfaces. We then use this model to compute the "tangent cone at infinity" of M(S): it is the topological cone on the quotient…
Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichm\"uller space, right-angled Artin groups, and many other cubical groups. Given such a space $\mathcal X$, we build…
We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm of the quasicircle in the sense of…
Unlike the case of surfaces of topologically finite type, there are several different Teichm\"uller spaces that are associated to a surface of topological infinite type. These Teichm\"uller spaces first depend (set-theoretically) on whether…
We characterize which cobounded quasigeodesics in the Teichmueller space T of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path gamma in T, we show that gamma is a quasigeodesic with…
Let $S$ be a closed orientable surface with genus $g\geq 2$. For a sequence $\s_i$ in the Teichm\"uller space of $S$, which converges to a projective measured lamination $[\lam]$ in the Thurston boundary, we obtain a relation between $\lam$…
We study the Teichm\"uller metric on the Teichm\"uller space of a surface of finite type, in regions where the injectivity radius of the surface is small. The main result is that in such regions the Teichm\"uller metric is approximated up…
We prove that all hierarchically hyperbolic spaces have finite asymptotic dimension and obtain strong bounds on these dimensions. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the…
We consider a quotient space of the Bers boundary of Teichm\"{u}ller space, which we call the reduced Bers boundary, by collapsing each quasi-conformal deformation space into a point. This reduced Bers boundary turns out to be independent…
We study topological properties of random closed curves on an orientable surface $S$ of negative Euler characteristic. Letting $\gamma_{n}$ denote the conjugacy class of the $n^{th}$ step of a simple random walk on the Cayley graph driven…
We consider here the $3$-sphere $\mathbf S^3$ seen as the boundary at infinity of the complex hyperbolic plane $\mathbf{H}^2_{\mathbf C}$. It comes equipped with a contact structure and two classes of special curves. First $\mathbf…
Given a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending…
Masur and Minsky showed that the curve graph is quasi-isometric to the Teichm\"uller space electrified along its thin part, and hence the Teichm\"uller space is weakly relatively hyperbolic with respect to the thin part. In this paper, we…