
Mathematics 2007
Computing arithmetic invariants for hyperbolic reflection groupsAbstract: We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finitevolume 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 nonarithmetic pair with identical invariants for which we cannot determine commensurability.
