arXiv research

Counterexample disproves Masur's criterion in Thurston metric.

We construct Weil-Petersson (WP) geodesic rays with minimal filling non-uniquely ergodic ending lamination which are recurrent to a compact subset of the moduli space of Riemann surfaces. This construction shows that an analogue of the Masur's criterion for Teichmüller geodesics does not hold for WP geodesics.

Example shows unique ergodicity of foliations on surfaces.

The paper provides a criterion for creating Haken manifolds with specific properties.

Extends Masur domain for 3-manifold study.

We calculate the Masur-Veech volume of gothic Teichmüller curves in genus four.

Computes the contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes.

Paper compares Bergman kernel and Masur-Veech measure on Teichmüller space.

We build an augmentation of the Masur-Minsky marking complex by Groves-Manning combinatorial horoballs to obtain a graph we call the augmented marking complex, $\mathcal{AM}(S)$. Adapting work of Masur-Minsky, we prove that $\mathcal{AM}(S)$ is quasiisometric to Teichmüller space with the Teichmüller metric. A similar …

Graphs of multicurves are hierarchically hyperbolic spaces.

We find the asymptotic expansion of Masur-Veech volumes for large genus.

The study predicts large genus behavior of quadratic differential volumes and constants.

This article is based on the lectures given at the ``Ecole thematique de theorie ergodique'', at the C.I.R.M. in Marseille, in April 2006. We give a complete proof of a theorem of Kerckhoff, Masur and Smillie on the unique ergodicity of the directional flow on a translation surface in almost every direction. The proof …

In this paper, we investigate the structure of the Gardiner-Masur boundary of Teichmuller space. Indeed, we will give a geometric description of boundary comparing to the Duchin-Leininger-Rafi compactification of the space of singular flat structures. We will obtain the coincidence between the Gardiner-Masur boundary a…

Extends Masur's divergence theorem to complex tori and Kummer surfaces.

The paper explores generalizations of Mirzakhani's recursion and computes volumes for physical gravity models.

Study of $n$-cylinder surfaces to calculate Masur-Veech volumes.

We introduce a deformation of Riemann surfaces and we are interested in the convergence of this deformation to a point of the Gardiner-masur boundary of Teichmueller space. This deformation, which we call the horocyclic deformation, is directed by a projective measured foliation and belongs to a certain horocycle in a …

Formula for Masur-Veech volumes in quadratic differentials with odd zeros.

We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichmüller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination converges to the Gardiner-Masur boundary. Using the embedding of flat metrics into th…

Formula for volumes of odd strata of quadratic differentials using graph intersection numbers.

Classifies orbit closures in translation surface strata.

Study calculates volumes and constants from intersection theory on abelian differential strata.

Survey of methods for computing volumes of moduli spaces.

Optimal geodesics connect boundary points in Teichmüller space.

New boundary constructed for mapping class group.

We show how to construct, for each $r \geq 3$, an ageometric, fully irreducible $φ\in Out(F_r)$ whose ideal Whitehead graph is the complete graph on $2r-1$ vertices. This paper is the second in a series of three where we show that precisely eighteen of the twenty-one connected, simplicial, five-vertex graphs are ideal …

Formulae for Masur-Veech volumes derived from intersection numbers of curves.

Formulae for Masur-Veech volumes and frequencies of geodesics derived from intersection numbers.

We calculate volumes of quadratic differentials using topological recursion.

Let $S_{g,p}$ denote the genus $g$ orientable surface with $p$ punctures. We show that nested train track sequences constitute $O((g,p)^{2})$-quasiconvex subsets of the curve graph, effectivizing a theorem of Masur and Minsky. As a consequence, the genus $g$ disk set is $O(g^{2})$-quasiconvex. We also show that splitti…

We give explicit bounds on the intersection number between any curve on a tight multigeodesic and the two ending curves. We use this to construct all tight multigeodesics and so conclude that distances in the curve graph are computable. The algorithm applies to all surfaces. We recover the finiteness result of Masur-Mi…

We show that the horofunction compactification of Teichmüller space with the Teichmüller metric is homeomorphic to the Gardiner-Masur compactification.

A 2008 general overview on Weil-Petersson geometry is offered. A preliminary plan for the subsequent CBMS lectures at Central Connecticut State University is included. Mirzakhani's solution of Witten-Kontsevich is not included - this work essentially requires its own lectures. Lectures on Mirzakhani's Witten-Kontsevich…

Square-tiled surfaces with fixed combinatorics become equidistributed in moduli spaces.

By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to…

Configurations of rigid collections of saddle connections are connected component invariants for strata of the moduli space of quadratic differentials. They have been classified for strata of Abelian differentials by Eskin, Masur and Zorich. Similar work for strata of quadratic differentials has been done in Masur and …

Paper describes principal boundaries of moduli spaces for abelian and quadratic differentials.

The paper calculates large genus limits for two types of Siegel-Veech constants.

A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira. Recurrent train tracks with a single switch which we call non-classical interval exchanges, form a subclass of linear involutions without flips. They are analogs of classical interval exchanges, and are…

Completed volumes match with combinatorial classes of the double ramification cycle.

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.

Using existing technology, we prove a Masur-Minsky style distance formula for flip- graph distance between two triangulations, expressed as a sum of the distances of the projections of these triangulations into arc graphs of the suitable subsurfaces of S.

Researchers derive asymptotic formulas for meander counts and probabilities.

This short survey illustrates the ideas of Teichmuller dynamics. As a model application we consider the asymptotic topology of generic geodesics on a "flat" surface and count closed geodesics and saddle connections. This survey is based on the joint papers with A.Eskin and H.Masur and with M.Kontsevich.

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

Let $M=H_{+}\cup_{S} H_{-}$ be a genus $g$ Heegaard splitting with Heegaard distance $n\geq κ+2$: (1) Let $c_{1}$, $c_{2}$ be two slopes in the same component of $\partial_{-}H_{-}$, such that the natural Heegaard splitting $M^{i}=H_{+}\cup_{S} (H_{-}\cup_{c_{i}} 2-handle)$ has distance less than $n$, then the distance…

Quadratic differentials on punctured surfaces link foliations and metric graphs.