Related papers: Train tracks and the Gromov boundary of the comple…

This work is the extension of the results by the author in [7] and [6] for low-genus surfaces. Let $S$ be an orientable, connected surface of finite topological type, with genus $g \leq 2$, empty boundary, and complexity at least $2$; as a…

Let $S$ be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures…

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 $\lam$…

Suppose that S is a surface of genus two or more, with exactly one boundary component. Then the curve complex of S has one end.

In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the $n$-torus admits a fibre whose homological size is bounded below by some…

A taming symplectic structure provides an upper bound on the area of an approximately pseudoholomorphic curve in terms of its homology class. We prove that, conversely, an almost complex manifold with such an area bound admits a taming…

We prove that a sequence of holomorphic discs with totally real boundary conditions has a subsequence that Gromov converges to a stable holomorphic map of genus zero with connected boundary provided that the sequence is bounded and has…

The local Lipschitz property is shown for the graph avoiding multiple point intersection with lines directed in a given cone. The assumption is much stronger than those of Marstrand's well-known theorem, but the conclusion is much stronger…

Coning off a collection of uniformly quasiconvex subsets of a Gromov hyperbolic space leaves a new space, called the cone-off. Kapovich and Rafi generalized work of Bowditch to show this space is still Gromov hyperbolic. We show that the…

We prove that the Gromov boundary of every hyperbolic group is homeomorphic to some Markov compactum. Our reasoning is based on constructing a sequence of covers of $\partial G$, which is quasi-$G$-invariant wrt. the ball $N$-type (defined…

We define an electrification of the curve graph of a surface S of finite type and use it to identify the Poisson boundary of a random walk on the mapping class group of S with some logarithmic moment condition as a stationary measure on the…

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…

The ray graph is a Gromov hyperbolic graph on which the mapping class group of the plane minus a Cantor set acts by isometries. We give a description of the Gromov boundary of the ray graph in terms of cliques of long rays on the plane…

Suppose $\tau$ is a train track on a surface $S$. Let $C(\tau)$ be the set of isotopy classes of simple closed curves carried by $\tau$. Masur and Minsky [2004] prove $C(\tau)$ is quasi-convex inside the curve complex $C(S)$. We prove the…

Consider a closed marked flat surface $S$ of genus $g\geq 2$ and area 1 and its universal covering $\tilde{S}$. We show that the measure class of the Hausdorff measure of the Gromov boundary of $\tilde{S}$ uniquely determines $S$.

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…

Let $N$ be a connected finite type nonorientable surface with or without boundary components and punctures. We prove that the graph of nonseparating curves of $N$ is connected and Gromov hyperbolic with a constant which does not depend on…

We develop a formalism that allows us to describe Markov compacta with finite sets of diagrams that are building blocks of the entire sequence. This encodes complex, continuous spaces with discrete collections of combinatorial objects. We…

Let $S$ be a connected orientable surface of finite topological type. We prove that there is an exhaustion of the curve complex $\mathcal{C}(S)$ by a sequence of finite rigid sets.

An outer automorphism of a free group is geometric if it can be represented by a homeomorphism of a compact surface. Bestvina and Handel gave an algorithmic characterization of geometric irreducible outer automorphisms using relative train…