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

TL;DR

This paper presents a method to compute bounds on the volume of non-obtuse hyperbolic polyhedra using combinatorial data, along with an algorithm for geometric decomposition of certain 3-orbifolds.

Contribution

It introduces a novel approach combining combinatorial and geometric techniques to estimate volumes and detect decompositions in hyperbolic 3-orbifolds.

Findings

Provides upper and lower volume bounds based on combinatorics

Develops an algorithm for geometric decomposition of 3-orbifolds

Extends previous volume bounds to non-obtuse hyperbolic polyhedra

Abstract

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, Schl\"afli's formula, and previous results of the author giving volume bounds for right-angled hyperbolic polyhedra.