Triangle coordinates for integral laminations on non-orientable surfaces.
problem Describing integral laminations on non-orientable surfaces.
method Introducing triangle coordinates and providing an explicit bijection.
result Explicit bijection between integral laminations and a set of integer vectors.
Efficient algorithm counts components of lamination on a disk.
problem Counting components of an integral lamination on a disk.
method Efficient algorithm using Dynnikov coordinates with arithmetic operations.
result Calculates the number of components with O(n2M) operations. We give a recipe to compute the geometric intersection number of an integral lamination with a particular type of integral lamination on an n-times punctured disk. This provides a way to find the geometric intersection number of two arbitrary integral laminations when combined with an algorithm of Dynnikov and Wiest.
Researchers compute the ratio between two normalizations of Thurston measure on measured laminations.
problem Computing the ratio between two normalizations of Thurston measure.
method Using the integral and symplectic structures on the space of measured laminations.
result Computed the ratio between two normalizations of Thurston measure.
The Heights Theorem is extended to all Riemann surfaces with a first kind fundamental group.
problem Establishing the Heights Theorem for all Riemann surfaces.
method Extending the theorem to all surfaces with a first kind fundamental group, using measured laminations and straightening horizontal trajectories.
result The horizontal map is injective for arbitrary Riemann surfaces with a conformal hyperbolic metric.
Study geodesic rays in Teichmüller space and their limits in Thurston's boundary.
problem Characterize limits of geodesic rays in Thurston's boundary of Teichmüller space.
method Analyze geodesic rays in T(D) and their limits in PMLbdd(D). result Geodesic rays in T(D) can limit to projective measured laminations with non-homotopic leaves. Volume of unit balls defined by quadratic differentials is not proper and has integrable volume.
problem Characterizing the behavior of volume functions associated with quadratic differentials.
method Analyzing the Thurston volume of unit balls in measured lamination spaces.
result The volume function is not proper and is p-integrable for any 0<p<1. This paper investigates analytic properties of maps between hyperbolic surfaces, focusing on best Lipschitz maps and geodesic laminations.
problem Analyzing the properties of maps between hyperbolic surfaces, particularly best Lipschitz maps and their relationship to geodesic laminations.
method The authors produce best Lipschitz maps as limits of minimizers of p-Schatten integrals, addressing existence and regularity issues.
result The support of the measure dv, the derivative of a Lie algebra valued function v, lies on the canonical geodesic lamination constructed by Thurston.
Given a n-dimensional lamination endowed with a Riemannian metric, we introduce the notion of a multiplicative cocycle of rank d, where n and d are arbitrary positive integers. The holonomy cocycle of a foliation and its exterior powers as well as its tensor powers provide examples of multiplicative cocycles. Next, we …
Defines new lamination links in 3-manifolds and shows their properties.
problem Understanding lamination links in 3-manifolds.
method Introduces and defines oriented framed measured lamination links, showing their relationship to Seifert laminations.
result Taut Seifert laminations generalize minimal genus Seifert surfaces.
The paper studies Hamiltonian flows for pseudo-Anosov mapping classes on surfaces.
problem Understanding the dynamics of pseudo-Anosov mapping classes on Teichmüller spaces.
method Explicit formulae for Hamiltonian flows generated by invariant functions.
result Hamiltonian flows coincide with the action of pseudo-Anosov homeomorphisms at time one.
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…
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…
Characterizes geodesic laminations on surfaces.
problem Understanding geodesic laminations on surfaces.
method Topological characterization of geodesic laminations.
result Proved a topological characterization of geodesic laminations.
Proves a function's locally least gradient property if its level sets are minimal laminations.
problem Understanding the relationship between 1-harmonic functions and minimal laminations.
method Analyzes minimal laminations and their convergence properties, then applies to 1-harmonic functions.
result Proves a function is 1-harmonic if its level sets are minimal laminations.
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…
Extends Seifert's algorithm to lamination links.
problem Finding Seifert surfaces for lamination links.
method Generalizes Seifert's algorithm to framed oriented measured lamination links.
result Identifies the set of framed lamination links that bound Seifert laminations.
We construct a pair of transverse genuine laminations on an atoroidal 3-manifold admitting transversely orientable uniform 1-cochain. The laminations are induced by the uniform 1-cochain and they are indeed the "straightening" of the coarse laminations defined in [Ca], by using minimal surface techniques. Moreover, whe…
Generalizes existence of bending laminations for Kleinian groups.
problem Existence of bending laminations for Kleinian surface groups.
method Generalization of Bonahon and Otal's proof to include geometrically infinite groups.
result Compactness of Kleinian groups realizing specific laminations.
The aim of the paper is to investigate the relation between inverse limit of branched manifolds and codimension zero laminations. We give necessary and sufficient conditions for such an inverse limit to be a lamination. We also show that codimension zero laminations are inverse limits of branched manifolds. The inverse…
Path-connectivity of thick laminations on high-genus surfaces.
problem Path-connectivity of thick laminations on high-genus surfaces.
method Teichmüller ray analysis and subshift of finite type construction.
result Path-connectedness of the Morse boundary of the mapping class group.
Proves compactness for lamination spaces in complex manifolds.
problem Compactness of lamination spaces in complex manifolds.
method Introduced a Levy-Prokhorov metric topology on measured Riemann surface laminations and proved compactness using this topology.
result Compactness theorem for embedded measured Riemann surface laminations in almost complex manifolds.
Proves rigidity of homeomorphisms for lamination spaces.
problem Rigidity of homeomorphisms in lamination spaces.
method Analyzes homeomorphisms preserving geometric intersection.
result Two rigidity results for automorphism groups of lamination spaces.
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 L and measurable space X we define L-measures ν on X. If L is …
The paper stratifies projective measured laminations and identifies a group of transformations.
problem Stratifying the space of projective measured laminations.
method Introducing a natural stratification and proving rigidity results.
result The group of self-homeomorphisms preserving the stratification is identified with the extended mapping class group.
Study topological constraints for minimal hyperbolic surface laminations.
problem Understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations.
method Analyzes topological obstructions and embeddings of Cantor tree-like surfaces.
result All possible topological types of leaves can be simultaneously embedded in a lamination.
Let $(M,g,\si)$ be a compact spin manifold of dimension n≥2. Let λ1+(g~) be the smallest positive eigenvalue of the Dirac operator in the metric g~∈[g] conformal to g. We then define $\lamin(M,[g],\si) = \inf_{\tilde{g} \in [g]} λ_1^+(\tilde{g}) \Vol(M,\tilde{g})^{1/n} $. We show that $…
A persistent lamination for a knot K is an essential lamination in the complement of K, which remains essential after every non-trivial Dehn surgery along K. Having a persistent lamination implies, for example, that every manifold obtained by non-trivial Dehn surgery along K has universal cover R^3. In this paper we pr…
Study counts ergodic measures in surface lamination strata.
problem Counting ergodic measures in surface lamination strata.
method Determined through analysis of geodesic laminations.
result Number of ergodic measures identified in each stratum.
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."
We examine the dependence of the deformation obtained by bending quasi-Fuchsian structures on the bending lamination. We show that when we consider bending quasi-Fuchsian structures on a closed surface, the conditions obtained by Epstein and Marden to relate weak convergence of arbitrary laminations to the convergence …
We give a Thurston-like definition for laminations on higher Teichmuller spaces associated to a surface S and a semi-simple group G for G−SLm and PGLm. The case G=SL2 or PGL2 corresponds to the classical theory of laminations. Our construction involves positive configurations of points in the affine bui…
Study stretch laminations in hyperbolic 3-manifolds via circle-valued maps.
problem Characterize stretch laminations in hyperbolic 3-manifolds.
method Use Thurston norm and Dehn filling slope length to determine stretch laminations as unions of core curves.
result Show existence of infinitely many examples with fibration and only closed leaves.
Automorphisms of free groups yield invariant posets of lamination orbits.
problem Understanding the structure of free-by-cyclic groups through automorphisms.
method Analyzing the poset of attracting lamination orbits for free group automorphisms.
result The poset of lamination orbits is a commensurability invariant of free-by-cyclic groups.
We investigate the distribution of lengths obtained by intersecting a random geodesic with a geodesic lamination. We give an explicit formula for the distribution for the case of a maximal lamination and show that the distribution is independent of the surface and lamination. We also show how the moments of the distrib…
We give a general lower bound for the normal Gromov norm of genuine laminations in terms of the topology of the complementary regions. In the special case of 3-manifolds, this yields a generalization of Agol's inequality from incompressible surfaces to tight laminations. In particular, the inequality excludes the exist…
The study proves quasiconvexity for subgroups in hyperbolic group sequences.
problem Proving quasiconvexity for subgroups in hyperbolic group sequences.
method Using relationships between different types of algebraic laminations.
result Proves quasiconvexity for finitely generated infinite index subgroups of H. The study characterizes train tracks and measured laminations on infinite surfaces.
problem Characterizing geodesic laminations on infinite surfaces.
method Defining train tracks and parametrizing measured laminations by edge weight systems.
result A homeomorphism exists between bounded measured laminations and edge weight systems.
Study invariant measures on measured laminations for subgroups of mapping class group.
problem Classify invariant Radon measures on space of measured laminations for subgroups of mapping class group.
method Geometric approach, focusing on recurrent measured laminations, explicitly constructing ergodic measures.
result Show uniquely ergodic for divergence-type subgroups, generalize results for full mapping class group.
Regular normal form for braids derived from relaxation algorithm.
problem Representing and normalizing braid complexity.
method Relaxation algorithm on tight laminations of punctured disks.
result Normal form is regular and prefix-closed.
New proof for solenoidal laminations with symmetric leaves in dimensions other than 4.
problem Characterizing Riemannian solenoidal laminations with symmetric leaves.
method Proving homeomorphism to inverse limits of finite covers of compact locally-symmetric manifolds.
result Each lamination is homeomorphic to a specific inverse limit structure.
New distance function proves rigidity in geodesic lamination space.
problem Proving rigidity in geodesic lamination space.
method Introduced left Hausdorff distance function and proved rigidity result.
result Extended mapping class group is isomorphic to bijections preserving left Hausdorff convergence.
Study of laminations for pseudo-Anosov flows on three-manifolds.
problem Understanding laminations for pseudo-Anosov flows on three-manifolds.
method Analyzing laminations Λu± for pseudo-Anosov orbit space universal circles, using prelaminations and results from Barthelmé, Bonatti, and Mann. result Laminations Λu+ and Λu− are completely determined by prelaminations on the boundary of the orbit space. Generalizes Candel's theorem on curvature of laminated surfaces.
problem Finding curvature of laminated surfaces given certain conditions.
method Proves a generalized theorem using elliptic PDEs and Cheeger-Gromov topology.
result Unique laminated metric exists for given curvature function.
The paper proves the existence of non-trivial lamination in complex projective space.
problem Existence of non-trivial laminations in complex projective space.
method Using Donaldson's construction of asymptotically holomorphic submanifolds.
result The existence of a non-trivial Riemann surface lamination embedded in CP2. 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…
Since their introduction by Thurston, geodesic laminations on hyperbolic surfaces occur in many contexts. In this paper, we propose a generalization of geodesic laminations on locally CAT(0), complete, geodesic metric spaces, whose boundary at infinity of the universal cover is endowed with a invariant total cyclic ord…
Characterizes lamination spaces of graphs on a pair of pants.
problem Understanding lamination spaces of graphs on a pair of pants.
method Identifying lamination spaces as lattice polytopes and using graph exploration technique.
result Characterizes the polytopes that arise as lamination spaces of graphs on a pair of pants.