Related papers: The hyperbolicity of the sphere complex via surger…
We prove that the curve graph $\calC^{(1)}(S)$ is Gromov-hyperbolic with a constant of hyperbolicity independent of the surface $S$. The proof is based on the proof of hyperbolicity of the free splitting complex by Handel and Mosher, as…
We show how to derive hyperbolicity of the free factor complex of $F_N$ from the Handel-Mosher proof of hyperbolicity of the free splitting complex of $F_N$, thus obtaining an alternative proof of a theorem of Bestvina-Feighn. We also show…
We prove that the free splitting complex of a finite rank free group, also known as Hatcher's sphere complex, is hyperbolic.
When two free factors A and B of a free group F_n are in "general position" we define the projection of B to the splitting complex (alternatively, the complex of free factors) of A. We show that the projections satisfy properties analogous…
We study the large scale geometry of the relative free splitting complex and the relative free factor complex of the rank $n$ free group $F_n$, relative to the choice of a free factor system of $F_n$, proving that these complexes are…
We develop the geometry of folding paths in Outer space and, as an application, prove that the complex of free factors of a free group of finite rank is hyperbolic.
The group $\Out$ of outer automorphisms of the free group has been an object of active study for many years, yet its geometry is not well understood. Recently, effort has been focused on finding a hyperbolic complex on which $\Out$ acts, in…
We define analogues of the graphs of free splittings, of cyclic splittings, and of maximally-cyclic splittings of $F_N$ for free products of groups, and show their hyperbolicity. Given a countable group $G$ which splits as…
We introduce the co-surface graph $\mathcal{CS}$ of a finitely generated free group $\mathbb{F}$ and use it to study the geometry of hyperbolic group extensions of $\mathbb{F}$. Among other things, we show that the Gromov boundary of the…
J Hempel [Topology, 2001] showed that the set of distances of the Heegaard splittings (S,V, h^n(V)) is unbounded, as long as the stable and unstable laminations of h avoid the closure of V in PML(S). Here h is a pseudo-Anosov homeomorphism…
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 give a distance estimate for the metric on the disk complex and show that it is Gromov hyperbolic. As another application of our techniques, we find an algorithm which computes the Hempel distance of a Heegaard splitting, up to an error…
A proof that the separating curve complex of the closed genus two surface has a quasi-distance formula and is delta hyperbolic using tools of Masur and Schleimer. This answers in the affirmative a Conjecture of Schleimer.
Given isometric actions by a group G on finitely many \delta-hyperbolic metric spaces, we provide a sufficient condition that guarantees the existence of a single element in G that is hyperbolic for each action. As an application we prove a…
We prove that an automorphism $\phi:F\to F$ of a finitely generated free group $F$ is hyperbolic in the sense of Gromov if it has no nontrivial periodic conjugacy classes.
This paper continues a geometric study of Harvey's Complex of Curves, whose ultimate goal is to apply the theory of hyperbolic spaces and groups to algorithmic questions for the Mapping Class Group and geometric properties of Kleinian…
The free splitting graph of a free group $F_n$ with $n\geq 2$ generators is a hyperbolic ${\rm Out}(F_n)$-graph which has a geometric realization as a sphere graph in the connected sum of $n$ copies of $S^1\times S^2$. We use this…
We show that many graphs naturally associated to a connected, compact, orientable surface are hierarchically hyperbolic spaces in the sense of Behrstock, Hagen and Sisto. They also automatically have the coarse median property defined by…
In this paper we provide a procedure to obtain a non-trivial HHS structure on a hyperbolic space. In particular, we prove that given a finite collection $\mathcal{F}$ of quasi-convex subgroups of a hyperbolic group $G$, there is an HHG…
We prove a structure theorem for pseudo-Anosov flows restricted to Seifert fibered pieces of three manifolds. The piece is called periodic if there is a Seifert fibration so that a regular fiber is freely homotopic, up to powers, to a…