On volumes of hyperbolic Coxeter polytopes and quadratic forms

TL;DR

This paper calculates the covolume of certain quadratic form unit groups and applies these results to determine the volumes of associated hyperbolic Coxeter polytopes, advancing understanding of their geometric properties.

Contribution

It introduces a covolume formula for unit groups of specific quadratic forms and uses it to compute volumes of hyperbolic Coxeter polytopes.

Findings

Derived a covolume formula for unit groups of quadratic forms

Computed volumes of hyperbolic Coxeter polytopes associated with these forms

Extended previous work on Coxeter polytopes and quadratic forms

Abstract

In this paper, we compute the covolume of the group of units of the quadratic form f_d^n(x) = x_1^2 + x_2^2 + . . . + x_n^2 - d x_{n+1}^2 with d an odd, positive, square-free integer. Mcleod has determined the hyperbolic Coxeter fundamental domain of the reflection subgroup of the group of units of the quadratic form f_3^n. We apply our covolume formula to compute the volumes of these hyperbolic Coxeter polytopes.