arXiv research

Classifies low-volume hyperbolic 3-manifolds with a maximal cusp.

Maximal cusps are not dense on Teichmüller space for infinite-type surfaces.

New tiling algorithm for hyperbolic 3-manifolds, characterizing cusp areas.

We compute the Cheeger constants of a collection of hyperbolic surfaces corresponding to maximal non-compact arithmetic Fuchsian groups, and to subgroups which are the rotation subgroup of maximal reflection groups. The Cheeger constants are geometric quantities, but relate to the smallest eigenvalues of Maass cusp for…

Characterizes holonomies of convex projective cusps.

Study of lambda lengths in figure eight knot complement using Eisenstein integers.

Combinatorial Ricci flow finds hyperbolic metrics on 3-manifolds.

The waist size of a cusp in an orientable hyperbolic 3-manifold is the length of the shortest nontrivial curve generated by a parabolic isometry in the maximal cusp boundary. Previously, it was shown that the smallest possible waist size, which is 1, is realized only by the cusp in the figure-eight knot complement. In …

New noncompact Coxeter polytopes found in various dimensions.

Any one-cusped hyperbolic manifold M with an unknotting tunnel tau is obtained by Dehn filling a cusp of a two-cusped hyperbolic manifold. In the case where M is obtained by "generic" Dehn filling, we prove that tau is isotopic to a geodesic, and characterize whether tau is isotopic to an edge in the canonical decompos…

We prove two conjectures of C. Gordon. We show that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10, and that the maximal intersection number between exceptional slopes is 8. The proof uses a combination of new geometric techniques and a rigorous computer-assisted calculation.

We introduce a new fundamental domain for the cusp stabilizer of a Hilbert modular group over a real quadratic field K=Q(sqrt n). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of the biplane. The region is the Cartesian product of the positive real…

The paper explores anomalous subvarieties in hyperbolic 3-manifolds and their geometric implications.

We consider hyperbolic 3-manifolds with either non-empty compact geodesic boundary, or some toric cusps, or both. For any such M we analyze what portion of the volume of M can be recovered by inserting in M boundary collars and cusp neighbourhoods with disjoint embedded interiors. Our main result is that this portion c…

We classify all the non-hyperbolic Dehn fillings of the complement of the chain-link with 3 components, conjectured to be the smallest hyperbolic 3-manifold with 3 cusps. We deduce the classification of all non-hyperbolic Dehn fillings of infinitely many 1-cusped and 2-cusped hyperbolic manifolds, including most of tho…

Let F be a surface and suppose that φ: F -> F is a pseudo-Anosov homeomorphism fixing a puncture p of F. The mapping torus M = M_φis hyperbolic and contains a maximal cusp C about the puncture p. We show that the area (and height) of the cusp torus bounding C is equal to the stable translation distance of φacting on th…

This study examines abnormal geodesics in 2D-Zermelo navigation problems, revealing their role in separating time minimal and maximal curves.

Proves cosets of certain subgroups in hyperbolic 3-manifold groups are conjugacy distinguished.

It is a theorem of Casson and Rivin that the complete hyperbolic metric on a cusp end ideal triangulated 3-manifold maximizes volume in the space of all positive angle structures. We show that the conclusion still holds if some of the tetrahedra in the complete metric are flat.

We consider a geometrically finite discrete group of conformal transformations of the sphere. Further we consider distributions which are supported on the limit set and are invariant with conformal weight. We estimate their regularity in terms of the conformal weight, the Hausdorff dimension of the limit set, and the m…

Study finds bounds for systole length on arithmetic punctured spheres.

We prove the meromorphic extension to C for the resolvent of the Laplacian on a class of geometrically finite hyperbolic manifolds with infinite volume and we give a polynomial bound on the number of resonances. This class notably contains the geometrically finite quotients with rational non-maximal rank cusps previous…

Study of rank 2 Kleinian groups with specific parabolic elements.

4-manifolds show every flat 3-manifold as cusp sections.

On geometrically finite hyperbolic manifolds $Γ\backslash H^{d}$, including those with non-maximal rank cusps, we give upper bounds on the number $N(R)$ of resonances of the Laplacian in disks of size $R$ as $R\to \infty$. In particular, if the parabolic subgroups of $Γ$ satisfy a certain Diophantine condition, the bou…

Let $S$ be a compact hyperbolic Riemann surface of genus $g \geq 2$. We call a systole a shortest simple closed geodesic in $S$ and denote by $\mathop{sys}(S)$ its length. Let $\mathop{msys(g)}$ be the maximal value that $\mathop{sys}(\cdot)$ can attain among the compact Riemann surfaces of genus $g$. We call a (global…

Study geodesics entering a fixed cusp neighborhood multiple times.

We find that cusp densities of hyperbolic knots in the 3-sphere are dense in [0,0.6826...] and those of links are dense in [0,0.853...]. We define a new invariant associated with cusp volume, the cusp crossing density, as the ratio between the cusp volume and the crossing number of a link, and show that cusp crossing d…

The paper classifies different types of cusps on plane curves.

The paper characterizes links in 3D from divides with cusps.

The paper explores cusp types in hyperbolic 4-manifolds and their commensurability classes.

The paper studies minimal surface entropy on hyperbolic 3-manifolds and compares it to the hyperbolic case.

Let $F$ be a closed essential surface in a hyperbolic 3-manifold $M$ with a toroidal cusp $N$. The depth of $F$ in $N$ is the maximal distance from points of $F$ in $N$ to the boundary of $N$. It will be shown that if $F$ is an essential pleated surface which is not coannular to the boundary torus of $N$ then the depth…

The paper classifies cusp types in hyperbolic knot complements.

New findings on cusped Borel Anosov representations and their properties.

We prove that non-compact finite volume hyperbolic 3-manifolds that satisfy a mild cohomological condition (infinitesimal rigidity) admit a family of properly convex deformations of their complete hyperbolic structure where the ends become generalized cusps of type 1 or type 2. We also discuss methods for controlling w…

Four hyperbolic 24-cell 4-manifolds with one cusp are identified.

Study of isometries on hyperbolic 3-manifold cusps.

We bound two global invariants of cusped hyperbolic manifolds: the length of the shortest closed geodesic (the systole), and the radius of the biggest embedded ball (the inradius). We give an upper bound for the systole, expressed in terms of the dimension and simplicial volume. We find a positive lower bound on the in…

Study of Eisenstein series linked to hyperbolic cusps.

We construct here two new examples of non-orientable, non-compact, hyperbolic 4-manifolds. The first has minimal volume $v_m = 4π^2/3$ and two cusps. This example has the lowest number of cusps among known minimal volume hyperbolic 4-manifolds. The second has volume $2\cdot v_m$ and one cusp. It has lowest volume among…

Study Dirac operator on cusped hyperbolic manifolds, finding spectrum properties.

Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.

Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.

Study Bergman kernels on complex hyperbolic cusps, generalizing previous results.

Recent work of Ballas, Cooper, and Leitner identifies $(n+1)$ types of $n$-dimensional convex projective cusps, one of which is the standard hyperbolic cusp. Work of Ballas-Marquis, and Ballas-Danciger-Lee give examples of these exotic (non-hyperbolic) type cusps in dimension $3$. Here an extension of the techniques of…

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that e…

The study shows how to embed cusp-decomposable manifolds quasi-isometrically.