Space Of Laminations Research Articles

Laminations and 2-filling rays on infinite type surfaces

The loop graph of an infinite type surface is an infinite diameter hyperbolic graph first studied in detail by Juliette Bavard. An important open problem in the study of infinite type surfaces is to describe the boundary of the loop graph as a space of geodesic laminations. We approach this problem by constructing the first examples of 2-filling rays on infinite type surfaces. Such rays have strong filling properties while failing to correspond to points on the boundary of the loop graph. As such they may be thought of as “fake boundary points.” We give multiple constructions using both a hands-on combinatorial approach and an approach using train tracks and automorphisms of flat surfaces. In addition, our approaches are sufficiently robust to describe all 2-filling rays with certain other basic properties as well as to produce uncountably many distinct mapping class group orbits.

A universal Cannon-Thurston map and the surviving curve complex

Using the Birman exact sequence for pure mapping class groups, we construct a universal Cannon-Thurston map onto the boundary of a curve complex for a surface with punctures we call surviving curve complex. Along the way we prove hyperbolicity of this complex and identify its boundary as a space of laminations. As a corollary we obtain a universal Cannon-Thurston map to the boundary of the ordinary curve complex, extending earlier work of the second author with Mj and Schleimer [Comment. Math. Helv. 86 (2011), pp. 769–816].

Harmonizing Graphene Laminate Spacing and Zinc‐Ion Solvated Structure toward Efficient Compact Capacitive Charge Storage

AbstractAqueous Zn‐ion hybrid capacitors (ZIHCs) present prominent potentials in flexible wearable electronics application scenarios due to their inherent high safety and low cost. Simultaneously, volumetric energy density is one of the crucial parameters to determine the lifespan of the wearable electronics, in which lightweight and miniaturization is a cardinal prerequisite for realistic application. In this work, an aqueous ZIHC is constructed by harmonizing interlayer spacing of the laminate graphene film and Zn‐ion solvation structure to improve the electrode space utilization. Laminate graphene film interspacing has been customized in the range of 0.72–0.81 nm via regulating the ratio of crumple graphene mediator, thereby optimizing the transport kinetics of large size hydrated Zn ions. Zn‐ion solvation structure is further tailored by introducing ZnCl2 electrolyte salt to accouple such regulated laminar ionic transport channel. In a result, the thus‐derived ZIHC demonstrates an ultralong cycling lifespan of 100 000 cycles (93.9% capacitance retention), a preeminent volumetric capacitance (235.4 F cm−3), and a remarkable specific area capacitance contribution (Cssa ≈ 72 µF cm−2). Quasi‐solid‐state ZIHC is assembled with ZnCl2 solution‐filled polyacrylamide gel electrolyte to concurrently achieve a superior areal capacitance of 1227 mF cm−2 and great mechanical flexibility toward practical wearable application.

Laurent phenomenon algebras arising from surfaces II: Laminated surfaces

It was shown by Fock and Goncharov (Dual Teichmuller and lamination spaces. Handbook of Teichmuller Theory, 2007), and Fomin et al. (Acta Math 201(1):83–146, 2008) that some cluster algebras arise from triangulated orientable surfaces. Subsequently, Dupont and Palesi (J Algebraic Combinatorics 42(2):429–472, 2015) generalised this construction to include unpunctured non-orientable surfaces, giving birth to quasi-cluster algebras. In Wilson (Int Math Res Notices 341, 2017) we linked this framework to Lam and Pylyavskyy’s Laurent phenomenon algebras (J Math 4(1):121–162, 2016), showing that unpunctured surfaces admit an LP structure. In this paper we extend quasi-cluster algebras to include punctured surfaces. Moreover, by adding laminations to the surface we demonstrate that all punctured and unpunctured surfaces admit LP structures. In short, we link two constructions which arose as seemingly unrelated generalisations of cluster algebras—one of the generalisations (quasi-cluster algebras) being based on triangulated surfaces, and the other (Laurent phenomenon algebras) based on the Laurent phenomenon. We thus provide a rich class of geometric examples in which to help study Laurent phenomenon algebras.

Laminations of a graph on a pair of pants

A lamination of a graph embedded on a surface is a collection of pairwise disjoint non-contractible simple closed curves drawn on the graph. In the case when the surface is a sphere with three punctures (a.k.a. a pair of pants), we first identify the lamination space of a graph embedded on that surface as a lattice polytope, then we characterize the polytopes that arise as the lamination space of some graph on a pair of pants. This characterizes the image of a simplified variant of the spectral map for the vector bundle Laplacian for a flat connection on a pair of pants. The proof uses a graph exploration technique akin to the peeling of planar maps.

On the asymptotic dimension of the curve complex

We give a bound, linear in the complexity of the surface, on the asymptotic dimension of the curve complex as well as the capacity dimension of the ending lamination space.

Evolution of nano-laminated structure formed by the thermally-assisted plastic deformation in dry sliding wear

Abstract The evolution of nano-laminated structure generated in dry sliding wear of a high strength martensite steel was investigated by using Scanning Electronic Microscopy (SEM) and Transmission Electron Microscopy (TEM). Gradient microstructure along the depth was observed with the top layer consisting of nano-laminated structure. During the sliding wear-induced plastic deformation, grains were refined and lamellar structures were formed as a result of generation of dislocation cell blocks bounded by geometrically necessary boundaries (GNBs) and incidental dislocation boundaries (IDBs). With increasing plastic shear strain and strain gradient, there is a continuous reduction in the spacing and length in dislocation cell blocks, leading to nano-laminated structures. The ultrafine nano-laminated structure with laminate spacing of 20 nm was observed at the sliding time of 120 min. However, the grain refinement is not a merely mechanical deformation process. The formation of Fe3O4 and FeO gave direct evidence that the highest temperature due to frictional heat could reach up to 600 °C. The result suggested that the formation of nano-laminated structure was governed by thermally-assisted plastic deformation.

Bijections of geodesic lamination space preserving left Hausdorff convergence

We introduce an asymmetric distance function, which we call the “left Hausdorff distance function”, on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity result for the action of the extended mapping class group of the surface on the space of geodesic laminations equipped with the topology induced from this distance. More specifically, we prove that there is a natural homomorphism from the extended mapping class group into the group of bijections of the space of geodesic laminations that preserve left Hausdorff convergence and that this homomorphism is an isomorphism.

Laminations from the symplectic double

Let S be a compact oriented surface with boundary together with finitely many marked points on the boundary, and let S^circ be the same surface equipped with the opposite orientation. We consider the doubleS_mathcal {D} obtained by gluing the surfaces S and S^circ along corresponding boundary components. We define a notion of lamination on the double and construct coordinates on the space of all such laminations. We show that this space of laminations is a tropical version of the symplectic double introduced by Fock and Goncharov. There is a canonical pairing between our laminations and the positive real points of the symplectic double. We derive an explicit formula for this pairing using the F-polynomials of Fomin and Zelevinsky.

A duality map for quantum cluster varieties from surfaces

We define a canonical map from a certain space of laminations on a punctured surface into the quantized algebra of functions on a cluster variety. We show that this map satisfies a number of special properties conjectured by Fock and Goncharov. Our construction is based on the “quantum trace” map introduced by Bonahon and Wong.

Cannon–Thurston maps for hyperbolic free group extensions

This paper gives a detailed analysis of the Cannon{Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free groups of nite rank at least 3 and consider a convex cocompact subgroup Out(F), i.e. one for which the orbit map from into the free factor complex of F is a quasi-isometric embedding. The subgroup determines an extension E of F, and the main theorem of Dowdall{Taylor (DT1) states that in this situation E is hyperbolic if and only if is purely atoroidal. Here, we give an explicit geometric description of the Cannon{Thurston maps @F! @E for these hyperbolic free group extensions, the existence of which follows from a general result of Mitra. In particular, we obtain a uniform bound on the multiplicity of the Cannon{Thurston map, showing that this map has multiplicity at most 2 rank(F). This theorem generalizes the main result of Kapovich and Lustig (KL5) which treats the special case where is innite cyclic. We also answer a question of Mahan Mitra by producing an explicit example of a hyperbolic free group extension for which the natural map from the boundary of to the space of laminations of the free group (with the Chabauty topology) is not continuous.

Subgroups of mapping class groups related to Heegaard splittings and bridge decompositions

Let \(M=H_1\cup _S H_2\) be a Heegaard splitting of a closed orientable 3-manifold M (or a bridge decomposition of a link exterior). Consider the subgroup \({\text {MCG}}^0(H_j)\) of the mapping class group of \(H_j\) consisting of mapping classes represented by orientation-preserving auto-homeomorphisms of \(H_j\) homotopic to the identity, and let \(G_j\) be the subgroup of the automorphism group of the curve complex \(\mathcal {CC}(S)\) obtained as the image of \({\text {MCG}}^0(H_j)\). Then the group \(G=\langle G_1, G_2\rangle \) generated by \(G_1\) and \(G_2\) acts on \(\mathcal {CC}(S)\) with each orbit being contained in a homotopy class in M. In this paper, we study the structure of the group G and examine whether a homotopy class can contain more than one orbit. We also show that the action of G on the projective lamination space of S has a non-empty domain of discontinuity when the Heegaard splitting satisfies R-bounded combinatorics and has high Hempel distance.

On the topology of ending lamination space

We show that if S is a finite type orientable surface of genus g and p punctures where 3g+p > 4, then EL(S) is (n-1)-connected and (n-1)-locally connected where dim(PML(S))=2n+1=6g+2p-7. Furthermore, if g=0, then EL(S) is homeomorphic to the p-4 dimensional Nobeling space.

On the homeomorphisms of the space of geodesic laminations on a hyperbolic surface

We prove that for any orientable connected surfaceSSof finite type which is not a sphere with at most four punctures or a torus with at most two punctures, the natural homomorphism from the extended mapping class group ofSSto the group of homeomorphisms of the space of geodesic laminations onSS, equipped with the Thurston topology, is an isomorphism.

A note on the rigidity of unmeasured lamination spaces

We show that every auto-homeomorphism of the unmeasured lamination space of an orientable surface of finite type is induced by a unique extended mapping class unless the surface is a sphere with at most four punctures or a torus with at most two punctures or a closed surface of genus 2 2 .

Explosion Resistance Investigation into Sandwich Structure of Steel Fiber Reinforced Concrete Support Plate

Abstract. In order to avoid the damage of precision guided weapons on target, a steel fiber reinforced concrete sandwich structure with different types of support was proposed. The failure characteristics of penetration plate, steel fiber reinforced concrete plate and support beam were studied by contact explosion test. The results showed that the using of steel fiber sheet sandwich structure could increase a resistance level for the project with the same thickness of the protective layer. The failure mode of steel fiber reinforced concrete sandwich structure under partial blast loading was still the local damage mode, the strength and thickness of steel fiber laminate material, height and spacing of support beam were the main factors to control the effect of wave attenuation in steel fiber reinforced concrete sandwich structure, if some measures to prevent vibration on steel fiber sheet or under-surface of vault would enhance the resistance of structure.

Geometry-Driven Charge Accumulation in Electrokinetic Flows between Thin, Closely Spaced Laminates

Fluid flows through anisotropic media are found in a wide variety of geophysical and biological systems. The macroscale behavior of these systems depends on the microstructure, which in turn may depend on local and global physical processes. Classically, geometric restrictions are needed to model these systems on the largest length scale, and we are interested in developing effective models which relax these restrictions. To explore the development of these multiscale models, we consider an array of closely spaced, purely dielectric rigid laminates with nonuniform thickness. The laminate thickness and spacing varies on a length scale much longer than the characteristic thickness of the laminates. In the spacing between the laminates, an electrically conducting fluid is driven by an applied electric field through electroosmosis and electrophoresis along with an applied pressure gradient. Debye layers occur at the laminate-fluid interface, which are assumed to be much smaller than the laminate thickness. From a modification of the classical homogenization approach that relies on a physical microscale constraint in place of a geometric constraint, we derive an effective set of equations that describe the fluid pressure, the anion and cation concentrations in the fluid, and the electric potential. Anisotropic dispersion effects in the electric field are included, and electroneutrality in the fluid is not imposed. We find that gradients in the laminate spacing can lead to charge accumulation when electroosmosis and the electrophoresis induced from the anisotropic dispersion effects balance.

Geosphere laminations in free groups

We construct geosphere laminations for free groups, which are codimension one analogues of geodesic laminations on surfaces. Other analogues that have been constructed by several authors are dimension-one instead of codimension-one. Our main result is that the space of such laminations is compact. This in turn is based on the result that crossing, in the sense of Scott-Swarup, is an open condition. Our construction is based on Hatcher’s normal form for spheres in the model manifold.

The ending lamination space of the five-punctured sphere is the Nöbeling curve

We prove that the ending lamination space of the five-punctured sphere is homeomorphic to the Nöbeling curve.

Almost filling laminations and the connectivity of ending lamination space

We show that if S is a finite type orientable surface of negative Euler characteristic which is not the 3-holed sphere, 4-holed sphere or 1-holed torus, then the ending lamination space of S is connected, locally path connected and cyclic.