Organizing Volumes of Right-Angled Hyperbolic Polyhedra

凸余紧直角Gromov-Thurston多面体

Convex cocompact right-angled Gromov-Thurston polyhedra

针对5维凸余紧双曲反射群的相关问题，本文采用与Lee和Marquis不同的方法，构造了直角情形的对应例子。

On volumes of hyperbolic right-angled polyhedra

In this paper we obtain new upper bounds on volumes of right-angled polyhedra in hyperbolic space $\mathbb{H}^3$ in three different cases: for ideal polyhedra with all vertices on the ideal hyperbolic boundary, for compact polytopes with only finite (or usual) vertices, and for finite volume polyhedra with vertices of both types.

右角双曲多面体的标准化体积谱

Spectra of normalized volumes of right-angled hyperbolic polyhedra

本文研究了双曲多面体的标准化体积谱，证明了理想右角双曲多面体的标准化体积谱在特定区间内具有离散和稠密的特性，同时给出了紧凑右角双曲多面体的标准化体积谱的范围和性质。

关于5到12维右角双曲多面体理想顶点和有限顶点数量下限的研究

On lower bounds for the number of ideal and finite vertices of right-angled hyperbolic polyhedra in dimensions from 5 to 12

研究通过几何方法确定了5维右角双曲多面体至少有3个理想顶点，7维至少有4个有限顶点，并通过递推关系扩展到12维。

Two-sided bounds for the volume of right-angled hyperbolic polyhedra

For a compact right-angled polyhedron $R$ in $\mathbb H^3$ denote by $\operatorname{vol} (R)$ the volume and by $\operatorname{vert} (R)$ the number of vertices. Upper and lower bounds for $\operatorname{vol} (R)$ in terms of $\operatorname{vert} (R)$ were obtained in \cite{A09}. Constructing a 2-parameter family of polyhedra, we show that the asymptotic upper bound $5 v_3 / 8$, where $v_3$ is the volume of the ideal regular tetrahedron in $\mathbb H^3$, is a double limit point for ratios $\operatorname{vol} (R) / \operatorname{vert} (R)$. Moreover, we improve the lower bound in the case $\ope…

Angled decompositions of arborescent link complements

This paper describes a way to subdivide a 3-manifold into angled blocks, namely polyhedral pieces that need not be simply connected. When the individual blocks carry dihedral angles that fit together in a consistent fashion, we prove that a manifold constructed from these blocks must be hyperbolic. The main application is a new proof of a classical, unpublished theorem of Bonahon and Siebenmann: that all arborescent links, except for three simple families of exceptions, have hyperbolic complements.

Two-sided combinatorial volume bounds for non-obtuse hyperbolic polyhedra

We give a method for computing upper and lower bounds for the volume of a non-obtuse hyperbolic polyhedron in terms of the combinatorics of the 1-skeleton. We introduce an algorithm that detects the geometric decomposition of good 3-orbifolds with planar singular locus and underlying manifold the 3-sphere. The volume bounds follow from techniques related to the proof of Thurston's Orbifold Theorem, Schläfli's formula, and previous results of the author giving volume bounds for right-angled hyperbolic polyhedra.

Creating subdivision rules from polyhedra with identifications

Cannon, Swenson, and others have proved numerous theorems about subdivision rules associated to hyperbolic groups with a 2-sphere at infinity. However, few explicit examples are known. We construct an explicit subdivision rule for many 3-manifolds from polyhedral gluings. The manifolds that satisfy the conditions include all closed manifolds created from right-angled hyperbolic polyhedra, as well as many 3-manifolds with toral or hyperbolic boundary.

随机无限理想角图与理想双曲多面体

Random infinite ideal angled graphs and ideal hyperbolic polyhedra

本文研究随机无限理想角图与理想双曲多面体的统一化和边界理论，揭示其ICP类型判定及边界一致性，扩展了双曲几何的理论边界。

Löbell多面体的几何与算术性质

Geometric and arithmetic properties of Löbell polyhedra

本文研究Löbell多面体的几何与算术性质，证明其短闭测地线可任意小且在可公度意义下仍为无限族，并精确刻画其中的拟算术成员。

广义双曲多面体和双曲链环体积的上界

Upper bounds for volumes of generalized hyperbolic polyhedra and hyperbolic links

本文研究了广义双曲多面体体积的上界，给出了线性依赖于边数的界限，并证明了在三角面和三价顶点条件下可改进该界，同时应用于获得具有超过8个扭转的双曲链环补集体积的新上界。

加厚曲面中链环的算术性与可公度性

Arithmeticity and commensurability of links in thickened surfaces

本文针对由常曲率曲面正则4价平铺构建的直角平铺链环，利用对称性论证等给出其算术性的完整刻画，结合典范分解与数论数据确定两两可公度条件。

Hyperbolic three-manifolds that embed geodesically

We prove that every complete finite-volume hyperbolic 3-manifold $M$ that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold $W$, which is also tessellated into right-angled regular polytopes (120-cells and ideal 24-cells). If $M$ is connected, then Vol($W$) < $2^{49}$Vol($M$). This applies for instance to the Borromean link complement. As a consequence, the Borromean link complement bounds geometrically a hyperbolic 4-manifold.

On the maximal volume of three-dimensional hyperbolic complete orthoschemes

A three-dimensional orthoscheme is defined as a tetrahedron whose base is a right-angled triangle and an edge joining the apex and a non-right-angled vertex is perpendicular to the base. A generalization, called complete orthoschemes, of orthoschemes is known in hyperbolic geometry. Roughly speaking, complete orthoschemes consist of three kinds of polyhedra; either compact, ideal or truncated. We consider a particular family of hyperbolic complete orthoschemes, which share the same base. They are parametrized by the "height", which represents how far the apex is from the base. We prove that th…

单模随机无限 trivalent 双曲多面体的类高斯-博内二分法

A Gauss-Bonnet-Type Dichotomy for Unimodular Random Infinite Trivalent Hyperbolic Polyhedra

本文针对三维双曲空间中的单模随机无限 trivalent 双曲多面体，建立共形类型的统一几何概率理论，通过局部几何量的期望给出其全局共形类型的二分判据，还证明了有限多面体的 Benjamini-Schramm 极限的抛物性并得到面随机游走的双曲速度结论。