arXiv research

Characterizes holonomies of convex projective cusps.

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

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

Minimal non-arithmetic hyperbolic 3-orbifold found with least volume.

Classifies worst approximable rational numbers using hyperbolic geometry.

Survey of analytic and geometric results on fibred cusp spaces.

Study bounds on cusp volumes of alternating knots on surfaces.

In this paper we explore coarse properties of cusp-decomposable manifolds first defined by Nguyên Phan. We describe the large scale geometry of the universal cover of a cusp-decomposable manifold and of quasi-isometries between two such universal covers. This description will provide us the tools to prove quasi-isometr…

We prove a Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds with asymptotic geometry at infinity. The asymptotic geometry at infinity is either a cusp bundle over a compact space (the fibered cusps) or a fiber bundle over a cone with a compact fiber (the fibered boundary). Many noncompact Einstein manifold…

We derive a sharp cusp count for finite volume complex hyperbolic surfaces which admit smooth toroidal compactifications. We use this result, and the techniques developed in [DiC12], to study the geometry of cusped complex hyperbolic surfaces and their compactifications.

Study of Eisenstein series linked to hyperbolic cusps.

Study of hyperbolic 3-manifolds via fractional Dehn twists and cusp geometry.

3D hyperbolic spaces have endless simple paths.

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

Researchers found the minimum volume of a 3-cusped hyperbolic 3-manifold.

Proves existence of circle patterns on surfaces with cusps.

New invariant measures knot geometry, improving volume-volume inequality.

This thesis investigates cusp cross-sections of arithmetic real, complex, and quaternionic hyperbolic $n$--orbifolds. We give a smooth classification of these submanifolds and analyze their induced geometry. One of the primary tools is a new subgroup separability result for general arithmetic lattices.

For a single cusped hyperbolic 3-manifold, Hodgson proved that there are only finitely many Dehn fillings of it whose trace fields have bounded degree. In this paper, we conjecture the same for manifolds with more cusps, and give the first positive results in this direction. For example, in the 2-cusped case, if a mani…

The cobordism invariance of the index on closed manifolds is reproved using the calculus of cusp pseudodifferential operators on a manifold with boundary. More generally, on a compact manifold with corners, the existence of a symmetric cusp differential operator of order 1 and of Dirac type near the boundary implies th…

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…

Napoleon's theorem in elementary geometry describes how certain linear operations on plane polygons of arbitrary shape always produce regular polygons. More generally, certain triangulations of a polygon that tiles R^2 admit deformations which keep fixed the symmetry group of the tiling. This gives rise to isolation ph…

Census of 10-tetrahedra hyperbolic 3-manifolds with 150,730 new examples.

We prove a bound relating the volume of a curve near a cusp in a hyperbolic manifold to its multiplicity at the cusp. The proof uses a hybrid technique employing both the geometry of the uniformizing group and the algebraic geometry of the toroidal compactification. There are a number of consequences: we show that for …

Study shows arithmetic properties of specific hyperbolic Dehn fillings.

Study shows flow convergence to smooth K-Ricci outside a divisor with cusp singularity.

Paper finds infinite family of minimal triangulations for complex 3D shapes.

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…

At a 3/2-cusp of a given plane curve $γ(t)$, both of the Euclidean curvature $κ_g$ and the affine curvature $κ_A$ diverge. In this paper, we show that each of $\sqrt{|s_g|}κ_g$ and $(s_A)^2 κ_A$ (called the Euclidean and affine normalized curvature, respectively) at a 3/2-cusp is a smooth function of the variable $t$, …

3-manifolds with similar completions have matching slopes and polynomials.

Study on random hyperbolic surfaces with many cusps, focusing on tight geodesics.

Study on geodesics and eigenvalues on random hyperbolic surfaces with cusps.

The paper shows caustics by reflection in projective Finsler metrics have at least four cusps.

Generalizes fully augmented links to doubled 3-manifolds with geometric bounds.

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

We study the geometry of fully augmented link complements in $S^3$ by looking at their link diagrams. We extend the method introduced by Thistlethwaite and Tsvietkova to fully augmented links and define a system of algebraic equations in terms of parameters coming from edges and crossings of the link diagrams. Combinin…

Complex hyperbolic triangle groups yield specific 3-manifolds at infinity.

A generalized cusp $C$ is diffeomorphic to $[0,\infty)$ times a closed Euclidean manifold. Geometrically $C$ is the quotient of a properly convex domain by a lattice, $Γ$, in one of a family of affine groups $G(ψ)$, parameterized by a point $ψ$ in the (dual closed) Weyl chamber for $SL(n+1,\mathbb{R})$, and $Γ$ determi…

Study projective deformations of hyperbolic 3-orbifolds with turnover ends.

3-manifolds are CR uniformized on spheres, proving a conjecture.

3-manifolds have covers with infinitely many ideal triangulations.

We study the geometry of the Margulis region associated with an irrational screw translation $g$ acting on the 4-dimensional real hyperbolic space. This is an invariant domain with the parabolic fixed point of $g$ on its boundary which plays the role of an invariant horoball for a translation in dimensions $\leq 3$. Th…

Spectro-Riemannian Graph Neural Networks integrate spectral and curvature signals for better graph representation learning.

The paper verifies no cosmetic surgeries on knots and 3-manifolds using hyperbolic geometry.

New noncompact Coxeter polytopes found in various dimensions.

Power series invariant of hyperbolic 3-manifolds matches knot invariants.

Sharp asymptotic behavior of Kähler-Einstein metrics on complex hyperbolic cusps.

The goal of this paper is to study the geometry of cusped complex hyperbolic manifolds through their compactifications. We characterize toroidal compactifications with non-nef canonical divisor. We derive effective very ampleness results for toroidal compactifications of finite volume complex hyperbolic manifolds. We e…