Computing arithmetic invariants for hyperbolic reflection groups

TL;DR

This paper introduces computer scripts in PARI/GP to compute key invariants of hyperbolic reflection groups, aiding in classifying their arithmetic properties and relationships.

Contribution

It provides new computational tools and a comprehensive set of examples for analyzing invariants of hyperbolic reflection groups, enhancing understanding of their arithmetic and commensurability.

Findings

Most groups considered are pairwise incommensurable.

Identical invariants imply commensurability for arithmetic groups.

Discovered unexpected pairs of commensurable groups with identical invariants.

Abstract

We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finite-volume polyhedra in $\mathbb{H}^3$, the commensurability invariants known as the invariant trace field and invariant quaternion algebra. Our scripts also allow one to determine arithmeticity of such groups and the isomorphism class of the invariant quaternion algebra by analyzing its ramification. We present many computed examples of these invariants. This is enough to show that most of the groups that we consider are pairwise incommensurable. For pairs of groups with identical invariants, not all is lost: when both groups are arithmetic, having identical invariants guarantees commensurability. We discover many ``unexpected'' commensurable pairs this way. We also present a non-arithmetic pair with identical invariants for which we cannot determine commensurability.