On length sets of subarithmetic hyperbolic manifolds

In this paper, we study the set of lengths of closed geodesics (or equivalently, the set of traces of the fundamental group) of a hyperbolic manifold. By “subarithmetic,” we mean a manifold whose set of traces takes values in a ring of algebraic integers. For such, we formulate the “Asymptotic Length-Saturation Conjecture”, which states that, under certain natural conditions, there is an asymptotic local–global principle for the trace set. We prove the first instance of the conjecture for punctured, Zariski dense covers of the modular surface.