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…

We introduce a natural stratification of the space of projective classes of measured laminations on a complete hyperbolic surface of finite area. We prove a rigidity result, namely, the group of self-homeomorphisms of the space of projective measured laminations that preserve such a stratification is in general identif…

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."

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…

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

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…

We prove two rigidity results for automorphism groups of the spaces ML(S) of measured laminations on a closed hyperbolic surface S and PML(S) of projective measured laminations on this surface. The results concern the homeomorphisms of ML(S) that preserve the geometric intersection between laminations and the homeomorp…

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

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 of measured laminations $\mathcal{ML}(Σ)$ associated to a topological surface $Σ$ of genus $g$ with $n$ punctures is an integral piecewise linear manifold of real dimension $6g-6+2n$. There is also a natural symplectic structure on $\mathcal{ML}(Σ)$ defined by Thurston. The integral and symplectic structures …

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 …

Bounds projective structure norms by bending lamination lengths.

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…

Geometrically boundary of surface moduli space defined.

Study of SL(2,R) representations on a once-punctured torus, showing Cantor set spectrum.

We describe a construction of ordered algebraic structures (ordered abelian semigroups, ordered commutative semirings, etc.) and describe applications to codimension-1 laminations. For a suitable ordered semi- algebraic structure $\mathbb L$ and measurable space $X$ we define $\mathbb L$-measures $ν$ on $X$. If $L$ is …

Study counts ergodic measures in surface lamination strata.

Study invariant measures on measured laminations for subgroups of mapping class group.

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 lamination, and (3) the sequence divides into a finite set of subsequences, each of wh…

Grafting a measured lamination on a hyperbolic surface defines a self-map of Teichmuller space, which is a homeomorphism by a result of Scannell and Wolf. In this paper we study the large-scale behavior of pruning, which is the inverse of grafting. Specifically, for each conformal structure $X \in \T(S)$, pruning $X$ g…

Geometric techniques reveal new insights into Gromov-Witten invariants.

We show that every topological surface lamination of a 3-manifold M is isotopic to one with smoothly immersed leaves. This carries out a project proposed by Gabai in [Problems in foliations and laminations, AMS/IP Stud. Adv. Math. 2.2 1--33]. Consequently any such lamination admits the structure of a Riemann surface la…

We introduce and define "oriented framed measured lamination links" in a 3-manifold $M$. These generalize oriented framed links in 3-manifolds, and are confined to 2-dimensional improperly embedded subsurfaces of the 3-manifold. Just as some framed links bound Seifert surfaces, so also some framed lamination links boun…

We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold $(X, J)$. To prove compactness result, we show that there is a suitable topology on the space of measured Riemann surface laminations induced by Levy-Prokhorov metric. As an application of th…

Study transverse measures on infinite type hyperbolic surfaces.

New combinatorial structures for Teichmüller spaces with Thurston's metric are explored.

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.

Let $X$ be an infinite Riemann surface equipped with its conformal hyperbolic metric such that the action of the covering group $π_1(X)$ on $\tilde{X}$ is of the first kind-i.e., the surface $X$ is equal to its convex core. We first prove that any geodesic lamination on $X$ is nowhere dense. Given a fixed geodesic pant…

We give a characterization of the action of the mapping class group on Thurston's space of measured laminations.

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 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…

The Masur domain is a subset of the space of projective measured geodesic laminations on the boundary of a 3-manifold M. This domain plays an important role in the study of the hyperbolic structures on the interior of M. In this paper, we define an extension of the Masur domain and explain that it shares a lot of prope…

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…

In a previous paper we introduced a notion of "genericity" for countable sets of curves in the curve complex of a surface S, based on the Lebesgue measure on the space of projective measured laminations in S. With this definition we prove that for each fixed g > 1 the set of irreducible genus g Heegaard splittings of h…

We consider a compact orientable hyperbolic 3-manifold with a compressible boundary. Suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination. We prove that if this limit projective lamination is doubly incompressi…

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

Fixed points found in Teichmüller space via anti-de Sitter geometry.

Generalizes existence of bending laminations for Kleinian groups.

Unique hyperbolic manifolds identified by boundary pleating.

In [4]: `The Riley slice of Schottky space', (Proc. London Math. Soc. 69 (1994), 72-90), Keen and Series analysed the theory of pleating coordinates in the context of the Riley slice of Schottky space R, the deformation space of a genus two handlebody generated by two parabolics. This theory aims to give a complete des…

We establish a Cauchy type inequality for the geometric intersection number between two 1-dimensional submanifolds in a surface. Some of the basic results in Thurston's theory of measured laminations on surfaces are derived from the Cauchy inequality.

Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are hyperbolic, ergodic harmonic measures are divided into two classes.

Extends earthquake and horocycle flows to new measures.

We prove that a transversely equicontinuous minimal lamination on a locally compact metric space $Z$ has a transversely invariant Radon measure. Moreover if the space $Z$ is compact, then the tranversely invariant Radon measure is shown to be unique up to a scaling.

New foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus are uniquely determined.

We generalize H. Seifert's algorithm for finding a Seifert surface for a knot or link. The generalization applies to "framed oriented measured lamination links." For knots, a Seifert surface determines a unique framing. In our setting, we analyze the set of framed lamination links which bound Seifert laminations and ar…

New geodesic rays found on hyperbolic surfaces with unique halo.