Spectral rigidity and discreteness of 2233-groups

Abstract

AbstractIn this paper we describe methods for dealing with the trace spectrum of a subgroup of PSL(2,$\mathbb{R}$) generated by four elliptic elements α, β, γ, δ of respective orders 2, 2, 3, 3, satisfying αβγδ = 1. We give a parametrization and a fundamental domain in the parameter space of such groups. Furthermore we construct an algorithm that decides whether or not a given group is discrete and which moves the discrete groups into the fundamental domain. Our main result is that any two discrete such groups are isospectral if and only if they are conjugate in (2,$\mathbb{R}$).In the Appendix we consider pairs of subgroups of (2,$\mathbb{R}$) that arise from non-conjugate maximal orders in a quaternion algebra over a number field. We show that for the isospectrality of such pairs there is a peculiar exception in the case where the groups contain elements of both orders 2 and 3.