arXiv research

Let $M$ be a finite volume oriented Riemannian manifold of dimension $n\geq 3$ and curvature in $[-b^2,-1]$, with thick-thin decomposition $M=M(thick)\cup M(thin)$. Denote by $λ_k(M(thick))$ the k-th eigenvalue for the Laplacian on $M(thick)$, with Neumann boundary conditdions. We show that $λ_k(M(thick))/3\leq λ_k(M)$…

We show that a complete hyperbolic n-manifold has a geodesic triangulation such that the tetrahedra contained in the thick part are L-bilipschitz diffeomorphic to the standard Euclidean n-simplex, for some constant L depending only on the dimension and the constant used to define the thick-thin decomposition of M.

We show that thick morphisms (or microformal morphisms) between smooth (super)manifolds, introduced by us before, are classical limits of `quantum thick morphisms' defined here as particular oscillatory integral operators on functions.

We show that every closed, virtually fibered hyperbolic 3-manifold contains immersed, quasi-Fuchsian surfaces with convex cores of arbitrarily large thickness.

We show that the diameter of the skinning map of an acylindrical hyperbolic 3-manifold M is bounded on thick Teichmueller geodesic rays by a constant depending only on the thickness of the ray and the topological type of the boundary of M.

The study proves a dichotomy for minimal hypersurfaces in certain thick manifolds.

We construct a geometric decomposition for the convex core of a thick hyperbolic 3-manifold M with bounded rank. Corollaries include upper bounds in terms of rank and injectivity radius on the Heegaard genus of M and on the radius of any embedded ball in the convex core of M.

Equivalence of norms on manifolds with curvature bounds established.

The paper defines a condition for the finiteness of mapping class groups of Heegaard splittings.

We describe a natural decomposition of a normal complex surface singularity $(X,0)$ into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the…

Minimal hypersurfaces scarring along a fixed one in certain manifolds.

We prove that simple, thick hyperbolic P-manifolds of dimension >2 exhibit Mostow rigidity. We also prove a quasi-isometry rigidity result for the fundamental groups of simple, thick hyperbolic P-manifolds of dimension >2. The key tool in the proofs of these rigidity results is a strong form of the Jordan separation th…

We provide an alternative, simpler proof of the existence of thick triangulations for noncompact $\mathcal{C}^1$ manifolds. Moreover, this proof is simpler than the original one given in \cite{pe}, since it mainly uses tools of elementary differential topology. The role played by curvatures in this construction is also…

There are several types of equation of motion of elastic wires. In this paper, we treat an equation taking account of the thickness of wire. The equation was introduced by Caflisch and Maddocks on plane curves, and they proved the existence of solutions. Koiso and Sugimoto generalized the result to any dimensional Eucl…

New theory of distributions on spaces with singular submanifolds.

Path-connectivity of thick laminations on high-genus surfaces.

In this paper, we give an algorithm to build all compact orientable atoroidal Haken 3-manifolds with tori boundary or closed orientable Haken 3-manifolds, so that in both cases, there are embedded closed orientable separating incompressible surfaces which are not tori. Next, such incompressible surfaces are related to …

Let X be a building of uniform thickness q+1. L^2-Betti numbers of X are reinterpreted as von-Neumann dimensions of weighted L^2-cohomology of the underlying Coxeter group. The dimension is measured with the help of the Hecke algebra. The weight depends on the thickness q. The weighted cohomology makes sense for all re…

We prove that every Teichmuller geodesic of a finite type surface contains a string of intersecting long, thick and dominant segments, such that the distance between consecutive segments is bounded. This is key to obtaining some results about Teichmuller geodesics which mimic those for hyperbolic geodesics. These resul…

What length of rope (of given diameter) is required to tie a particular knot? To answer this question, we define some new notions of thickness for a space curve, one based on Gromov's distortion, and another generalizing the thickness of Litherland, Simon et al. We prove a basic inequality between these thickness measu…

New examples show limits of physical link isotopies.

A knot's thickness is measured by its β invariant, a new numerical invariant.

We give the first part of a proof of Thurston's Ending Lamination conjecture. In this part we show how to construct from the end invariants of a Kleinian surface group a ``Lipschitz model'' for the thick part of the corresponding hyperbolic manifold. This enables us to describe the topological structure of the thick pa…

The paper provides explicit bilipschitz bounds on Dehn fillings of hyperbolic 3-manifolds.

This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is pro…

The first algorithm for sampling the space of thick equilateral knots, as a function of thickness, will be described. This algorithm is based on previous algorithms of applying random reflections. To prove the existence of the algorithm, we describe a method for turning any knot into the regular planar polygon using on…

Method estimates section thickness and XY anisotropy in ssEM images.

Algorithm computes knot Floer complex for knots of thickness one.

In this paper we explore relationships between divergence and thick groups, and with the same techniques we estimate lengths of shortest conjugators. We produce examples, for every positive integer n, of CAT(0) groups which are thick of order n and with polynomial divergence of order n+1, both these phenomena are new. …

Tubes in manifolds require wide spaces.

For right-angled Coxeter groups $W_Γ$, we obtain a condition on $Γ$ that is necessary and sufficient to ensure that $W_Γ$ is thick and thus not relatively hyperbolic. We show that Coxeter groups which are not thick all admit canonical minimal relatively hyperbolic structures; further, we show that in such a structure, …

We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds and use this to prove existence of universal bounds on the principal curvatures of surfaces embedded in hyperbolic 3-manifolds.

We describe some problems, observations, and conjectures concerning thickness and packing density of knots and links in $\sp^3$ and $\R^3$. We prove the thickness of a nontrivial knot or link in $\sp^3$ is no more than $\fracπ{4}$, the thickness of a Hopf link. We also give arguments and evidence supporting the conject…

New examples of gordian unlinks show different rope geometries.

Study Betti numbers of manifolds converging to covers, deriving new convergence results.

Thick morphisms link quantum mechanics and supermanifolds.

The paper explores new algebraic structures and morphisms in graded settings.

The paper introduces boundary thickness as a measure for improving model robustness.

New theorem shows certain curved surfaces are uniquely identified by their geodesic lengths.

We prove a formula for the normal injectivity radius(thickness)i(K,M)for C^{1,1} compact submanifolds K^k of complete Riemannian manifolds M^n in terms of geometric focal distance and double critical points. We also prove the C^1 compactness of the set of all compact submanifolds K contained in a compact subset D of a …

We introduce a new quasi-isometry invariant of 2-dimensional right-angled Coxeter groups, the hypergraph index, that partitions these groups into infinitely many quasi-isometry classes, each containing infinitely many groups. Furthermore, the hypergraph index of any right-angled Coxeter group can be directly computed f…

Let $T$ be a graph in a compact, orientable 3--manifold $M$ and let $Γ$ be a subgraph. $T$ can be placed in bridge position with respect to a Heegaard surface $H$. We show that if $H$ is what we call $(T,Γ)$-c-weakly reducible in the complement of $T$ then either a "degenerate" situation occurs or $H$ can be untelescop…

New proof shows knot Floer thickness limits bad domains in diagrams.

Study on reducing surgeries on knots, developing thickness and genus bounds.

We show the existence of a thick thin decomposition of the domain of a pseudo holomorphic curve with boundary. The geometry of the thick part is bounded uniformly in the energy. Furthermore, in the thick part, there is a uniform bound on the differential which is exponential in the energy. The thin part consists of ann…

We prove that thick groups (and more generally thick graphs) have trivial Floyd boundary. This shows a wide class of finitely generated groups that are non-relatively hyperbolic have trivial Floyd boundary. In addition to giving new examples, our result provides a common proof and framework for many of the known result…

We introduce the notion of connection thickness of spheres in a Cayley graph, related to dead-ends and their retreat depth. It was well-known that connection thickness is bounded for finitely presented one-ended groups. We compute that for natural generating sets of lamplighter groups on a line or on a tree, connection…

A graphical calculus for microformal morphisms simplifies complex operations in classical and quantum physics.