On fundamental domains and volumes of hyperbolic Coxeter-Weyl groups

TL;DR

This paper introduces a straightforward method to determine the shape and volume of fundamental domains for hyperbolic Coxeter-Weyl groups, simplifying previous complex calculations using only basic algebraic data.

Contribution

The authors provide a new simple formula and approach for computing fundamental domains and their volumes for hyperbolic Coxeter-Weyl groups based on minimal input data.

Findings

Derived a simple formula for fundamental domain volumes.

Reproduced known volume values efficiently.

Presented a method applicable to generalized modular groups.

Abstract

We present a simple method for determining the shape of fundamental domains of generalized modular groups related to Weyl groups of hyperbolic Kac-Moody algebras. These domains are given as subsets of certain generalized upper half planes, on which the Weyl groups act via generalized modular transformations. Our construction only requires the Cartan matrix of the underlying finite-dimensional Lie algebra and the associated Coxeter labels as input information. We present a simple formula for determining the volume of these fundamental domains. This allows us to re-produce in a simple manner the known values for these volumes previously obtained by other methods.