On the volume growth of the hyperbolic regular $n$-simplex

TL;DR

This paper establishes new bounds for the volume growth ratio of regular hyperbolic simplices, improving known bounds especially for the ideal case and in three dimensions, using novel coordinate methods.

Contribution

It introduces a new volume form based on hyperbolic orthogonal coordinates and provides improved bounds for volume growth ratios of regular hyperbolic simplices.

Findings

Upper bound for ideal regular simplex volume ratio is the best known.

Improved lower bound for n=3 in the ideal case.

New method using hyperbolic orthogonal coordinates.

Abstract

In this paper we give lower and upper bounds for the volume growth of a regular hyperbolic simplex, namely for the ratio of the $n$-dimensional volume of a regular simplex and the $(n-1)$-dimensional volume of its facets. In addition to the methods of U. Haagerup and M. Munkholm we use a third volume form is based on the hyperbolic orthogonal coordinates of a body. In the case of the ideal, regular simplex our upper bound gives the best known upper bound. On the other hand, also in the ideal case our general lower bound, improved the best known one for $n=3$.