[Paper Review] Hyperbolic orbifolds of small volume

Mikhail Belolipetsky|arXiv (Cornell University)|Feb 21, 2014

Geometric and Algebraic Topology4 citations

TL;DR

This paper surveys known results and conjectures on minimal-volume hyperbolic $n$-orbifolds and $n$-manifolds, focusing on arithmetic and non-arithmetic lattices in $\mathrm{PO}(n,1)$. It establishes super-exponentially decreasing lower bounds for volume via curvature and sphere-packing methods, highlighting a vast gap between known bounds and conjectured minimal volumes in high dimensions.

ABSTRACT

Volume is a natural measure of complexity of a Riemannian manifold. In this survey, we discuss the results and conjectures concerning n-dimensional hyperbolic manifolds and orbifolds of small volume.

Motivation & Objective

• To summarize current knowledge on the minimal volume problem for hyperbolic $n$-orbifolds and $n$-manifolds.
• To examine the role of arithmetic lattices and their volume bounds in low and high dimensions.
• To investigate the gap between known super-exponentially decreasing lower bounds and conjectured super-exponentially increasing minimal volumes in high dimensions.
• To explore the applicability of geometric and number-theoretic techniques, such as the Kazhdan–Margulis theorem and sphere-packing in cusps, to volume estimation.
• To identify open problems, particularly regarding non-arithmetic lattices and improved lower bounds for $\mu_n$, the minimal systole length in dimension $n$.

Proposed method

• Uses the Gauss–Bonnet theorem to relate volume to Euler characteristic in even dimensions, generalizing to orbifolds via orbifold Euler characteristic.
• Applies the Borel–Harish-Chandra theorem to construct arithmetic lattices from algebraic groups $\mathbf{G}$ over number fields $k$, with $\phi: \mathbf{G}(k\otimes_{\mathbb{Q}}\mathbb{R})^\circ \to \mathrm{PO}(n,1)^\circ$ having compact kernel.
• Employs the hyperbolic Pythagorean theorem and quasi-isometric embedding techniques to estimate distances in hyperbolic space via tree embeddings.
• Applies the disjoint bisectors test to show that if $\cosh(b_n) \geq \cosh(2.303)$, then the image of a tree under a map $\rho$ is a quasi-isometric embedding.
• Uses Wang’s bound on Zassenhaus neighborhoods and curvature estimates to derive a general lower bound for volume, decreasing super-exponentially with $n$.
• Applies Kellerhals’ method of cusp sphere packings to derive a lower bound for non-compact hyperbolic $n$-manifolds, proportional to $m \cdot \nu_n$, where $\nu_n \approx \frac{e\sqrt{n}}{n!}$.

Experimental results

Research questions

• RQ1 What are the known sharp lower bounds for the volume of arithmetic hyperbolic $n$-orbifolds, and how do they scale with $n$?
• RQ2 Can the minimal systole length $\mu_n$ in dimension $n$ be bounded below by a polynomial or inverse-polynomial function of $n$?
• RQ3 Why does the current best general lower bound for volume decrease super-exponentially with $n$, and how does this compare to the conjectured minimal volume?
• RQ4 To what extent can non-arithmetic lattices with small systole be used to improve volume lower bounds, and what obstacles prevent such constructions?
• RQ5 How do cusp geometry and sphere-packing density contribute to volume estimates in non-compact hyperbolic $n$-manifolds?

Key findings

• The best known general lower bound for the volume of hyperbolic $n$-orbifolds decreases super-exponentially with $n$, as given by the formula in equation (7) involving gamma functions and a trigonometric integral.
• For non-compact hyperbolic $n$-manifolds, the volume is bounded below by $m \cdot \frac{2n}{n(n+1)} \nu_n$, where $m$ is the number of cusps and $\nu_n \approx \frac{e\sqrt{n}}{n!}$, again yielding a super-exponentially decreasing bound.
• The construction of a quasi-isometric embedding of a regular tree into $\mathbf{H}^n$ via a $\Gamma$-action shows that $\mu_n \leq \epsilon$ if $\epsilon \geq \frac{C}{\sqrt{n}}$ for $C \approx 1.799$, under the condition $\cosh(b_n) \geq \cosh(2.303)$.
• The Kazhdan–Margulis theorem guarantees the existence of a uniform lower bound on the systole length in lattices, but the explicit quantitative version used here gives a bound that still allows for super-exponential decay.
• Despite extensive progress, a significant gap remains between the known super-exponentially decreasing lower bounds and the conjectured super-exponentially increasing minimal volumes in high dimensions.
• The problem of finding a better lower bound for $\mu_n$, especially for non-arithmetic lattices, remains open, with current methods failing to extend the quasi-isometric embedding argument beyond arithmetic cases.