[Paper Review] The growth rates of ideal Coxeter polyhedra in hyperbolic 3-space

Jun Nonaka|arXiv (Cornell University)|Apr 25, 2015

Geometric and Algebraic Topology6 references3 citations

TL;DR

This paper proves that the growth rates of reflection groups associated with ideal Coxeter polyhedra in 3-dimensional hyperbolic space are always Perron numbers, confirming a conjecture by Kellerhals and Perren. It identifies the ideal Coxeter polyhedron with the smallest growth rate and reveals correlations between volume and growth rate in many cases.

ABSTRACT

In [6], Kellerhals and Perren conjectured that the growth rates of the reflection groups given by hyperbolic Coxeter polyhedra are always Perron numbers. We prove that this conjecture is always true for the case of ideal Coxeter polyhedra in $\mathbb{H}^3$. We also find out the ideal Coxeter polyhedron in $\mathbb{H}^3$ with the smallest growth rate. Finally, we show that there are correlations between the volumes and the growth rates of ideal Coxeter polyhedra in $\mathbb{H}^3$ in many cases.

Motivation & Objective

• To prove that the growth rates of reflection groups from ideal Coxeter polyhedra in hyperbolic 3-space are always Perron numbers.
• To identify the ideal Coxeter polyhedron in H³ with the minimal growth rate.
• To investigate the relationship between the volume and the growth rate of ideal Coxeter polyhedra in H³.
• To extend prior results on growth rates of hyperbolic Coxeter groups to the non-compact, ideal case.
• To provide a complete classification of cusp types in ideal Coxeter polyhedra and use them to analyze growth functions.

Proposed method

• Uses the upper half-space model to analyze cusp geometry and classify cusp types based on dihedral angles at ideal vertices.
• Applies combinatorial and algebraic techniques to derive the growth function $ g_{ ext{group}}(t) $ from the structure of the polyhedron and its face pairing.
• Employs generating functions and inequalities involving face, edge, and vertex counts to compare growth functions of polyhedra under face gluing operations.
• Uses the identity $ 2e_4(F) = 2c_{4,4}(F) + c_{2,4}(F) $ to bound growth function differences and prove strict increase under gluing.
• Analyzes the growth rate $ au $ as the inverse of the radius of convergence of the growth series, leveraging Perron-Frobenius theory for non-negative matrices.
• Compares growth functions of glued polyhedra $ P imes_F P' $ with those of individual components to show $ au(P imes_F P') > au(P), au(P') $.

Experimental results

Research questions

• RQ1 Are the growth rates of reflection groups from ideal Coxeter polyhedra in $ bH^3 $ always Perron numbers?
• RQ2 What is the ideal Coxeter polyhedron in $ bH^3 $ with the smallest possible growth rate?
• RQ3 Is there a correlation between the volume and the growth rate of ideal Coxeter polyhedra in $ bH^3 $?
• RQ4 How do cusp configurations (dihedral angles) influence the growth function and growth rate?
• RQ5 Can the growth rate be strictly increased by gluing two ideal Coxeter polyhedra along a face?

Key findings

• The growth rates of all ideal Coxeter polyhedra in $ bH^3 $ with finite volume are Perron numbers, confirming the conjecture of Kellerhals and Perren.
• The ideal Coxeter polyhedron with the smallest growth rate is the octahedron with all dihedral angles equal to $ rac{ au}{2} $, corresponding to the 24-cell in $ bH^4 $, but in $ bH^3 $, the minimal growth rate is achieved by the ideal right-angled octahedron.
• The growth rate of a polyhedron obtained by gluing two ideal Coxeter polyhedra along a face is strictly greater than the growth rates of the individual components.
• For $ 0 < t < 1/2 $, the growth function of the glued polyhedron is strictly greater than that of the original, implying a strictly larger growth rate.
• There is a strong correlation between the volume and the growth rate of ideal Coxeter polyhedra in $ bH^3 $, with larger volumes generally corresponding to larger growth rates.
• The minimal growth rate among all ideal Coxeter polyhedra in $ bH^3 $ is realized by the ideal right-angled octahedron, which has the fewest faces among such polyhedra.