arXiv research

Since their introduction by Thurston, measured geodesic laminations on hyperbolic surfaces occur in many contexts. In [Mor], we have introduced a notion of flat laminations on surfaces endowed with a half-translation structure (that is a singular flat surface with holonomy {+/-Id}, similar to geodesic laminations on hy…

In [Mor], we have introduced a notion of flat laminations on surfaces endowed with a flat structure, similar to geodesic laminations on hyperbolic surfaces. Here is a sequel to this article that aims at defining transversal measures on flat laminations similar to transversal measures on hyperbolic laminations, taking i…

Extends Masur domain for 3-manifold study.

Thurston's boundary to the universal Teichmüller space $T(\mathbb{D})$ is the space $PML_{bdd}(\mathbb{D})$ of projective bounded measured laminations of $\mathbb{D}$. A geodesic ray in $T(\mathbb{D})$ is of Teichmüller type if it shrinks vertical foliation of an integrable holomorphic quadratic differential. In a prio…

The space ML(F) of measured geodesic laminations on a given closed hyperbolic surface F has a canonical linear structure arising in fact from different sources in 2-dimensional hyperbolic (earthquake theory) or complex projective (grafting) geometry as well as in (2+1) Lorentzian one (globally hyperbolic spacetimes of …

New boundary for infinite-dimensional Teichmüller spaces.

The paper studies geodesics on surfaces with unique foliations, proving limit cycles.

Study counts ergodic measures in surface lamination strata.

A fundamental object in a hyperbolic 3-manifold M is its convex core C(M), defined as the smallest closed non-empty convex subset of M. We investigate the way the geometry of the boundary S of C(M) varies as we vary the hyperbolic metric of M. Thurston observed that the intrinsic metric of S is hyperbolic, and that its…

The paper stratifies projective measured laminations and identifies a group of transformations.

The paper proves the existence of non-trivial lamination in complex projective space.

Counting hyperbolic multi-geodesics with individual component lengths.

Study transverse measures on infinite type hyperbolic surfaces.

Proves the bending map is proper for hyperbolic 3-manifolds.

Geometric techniques reveal new insights into Gromov-Witten invariants.

New geodesic rays found on hyperbolic surfaces with unique halo.

Thurston's boundary to the universal Teichmüller space $T(\mathbb{H})$ is the set of asymptotic rays to the embedding of $T(\mathbb{H})$ in the space of geodesic currents; the boundary is identified with the projective bounded measured laminations $PML_{bdd}(\mathbb{H})$ of $\mathbb{H}$. We prove that each Teichmüller …

We calculate a projective space of essential measured laminations in a surface pair, which will be used in another paper to help describe spaces of "finite height laminations."

The study characterizes train tracks and measured laminations on infinite surfaces.

Let S be a non-exceptional oriented surface of finite type. We classify all Radon measures on the space of measured geodesic laminations for S which are invariant under the mapping class group.

The Bratteli diagram is an infinite graph which reflects the structure of projections in a C*-algebra. We prove that every strictly ergodic unimodular Bratteli diagram of rank 2g+m-1 gives rise to a minimal geodesic lamination with the m-component principal region on a surface of genus g greater or equal to 1. The proo…

Characterizes closures of mapping class group orbits on non-orientable surfaces.

Given a measured geodesic lamination on a hyperbolic surface, grafting the surface along multiples of the lamination defines a path in Teichmuller space, called the grafting ray. We show that every grafting ray, after reparametrization, is a Teichmuller quasi-geodesic and stays in a bounded neighborhood of a Teichmulle…

Given a measured lamination on a finite area hyperbolic surface we consider a natural measure Mon the real line obtained by taking the push-forward of the volume measure of the unit tangent bundle of the surface under an intersection function associated with the lamination. We show that the measure M gives summation id…

We define an ending lamination for a Weil-Petersson geodesic ray. Despite the lack of a natural visual boundary for the Weil-Petersson metric, these ending laminations provide an effective boundary theory that encodes much of its asymptotic CAT(0) geometry. In particular, we prove an ending lamination theorem (Theorem …

Random walks on mapping class groups identified with geodesic laminations.

Proves rigidity of homeomorphisms for lamination spaces.

Study automorphism groups of geodesic currents and measured laminations on surfaces.

Maps and measures on surfaces link best Lipschitz and least gradient functions.

Let $X_0$ be a complete borderless infinite area hyperbolic surface. We introduce Thurston's boundary to the Teichmüller space $T(X_0)$ of the surface $X_0$ using Liouville (geodesic) currents. Thurston's boundary to $T(X_0)$ is identified with the space $PML_{bdd}(X_0)$ of projective bounded measured laminations on $X…

Given two measured laminations mu and nu in a hyperbolic surface which fill up the surface, Kerckhoff [Lines of Minima in Teichmueller space, Duke Math J. 65 (1992) 187-213] defines an associated line of minima along which convex combinations of the length functions of mu and nu are minimised. This is a line in Teichmu…

We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmuller space, complementing a result of Scannell-Wolf on grafting by a fixed lamination. This result is used to study the relationship between the complex-analytic and geometric coordinate systems f…

The Heights Theorem is extended to all Riemann surfaces with a first kind fundamental group.

Let S be a closed surface of genus at least 2, and consider two measured geodesic laminations that fill S. Right earthquakes along these laminations are diffeomorphisms of the Teichmüller space of S. We prove that the composition of these earthquakes has a fixed point in the Teichmüller space. Another way to state this…

Extends classification of invariant measures on geodesic currents.

A measured laminations on the universal hyperbolic solenoid $§$ is, by our definition, a leafwise measured lamination with appropriate continuity for the transverse variations. An earthquakes on theuniversal hyperbolic solenoid $§$ is uniquely determined by a measured lamination on $§$; it is a leafwise earthquake with…

Study on the behavior of stretch lines in Teichmüller space.

Researchers compute the ratio between two normalizations of Thurston measure on measured laminations.

Nonorientable hyperbolic surfaces have fewer simple closed geodesics than expected.

We compare two relationships between quadratic differentials and measured geodesic laminations on hyperbolic Riemann surfaces (by foliations or complex projective structures). Each yields a homeomorphism $\ML(S) \to Q(X)$ for any conformal structure $X$ on a compact surface $S$. The main result is that these maps are n…

This paper investigates analytic properties of maps between hyperbolic surfaces, focusing on best Lipschitz maps and geodesic laminations.

Characterizes geodesic laminations on surfaces.

Counterexample disproves Masur's criterion in Thurston metric.

Study surface subgroups acting on projective space, finding bending laminations and spheres.

We show that the length function of a measured geodesic lamination is convex in Thurston's shear coordinates over Teichmüller space and strictly convex for generic laminations. We give some consequences of this result in the context of Thurston's asymmetric metric on Teichmüller space.

Analyzes convex structures in Teichmüller space unit tangent spheres.

Let $X$ be an infinite geodesically complete hyperbolic surface which can be decomposed into geodesic pairs of pants. We introduce Thurston's boundary to the Teichmüller space $T(X)$ of the surface $X$ using the length spectrum analogous to Thurston's construction for finite surfaces. Thurston's boundary using the leng…

Classifies geodesic planes outside convex core of geometrically finite 3-manifolds.