Counting problems for geodesics on arithmetic hyperbolic surfaces

Abstract

ABSTRACT. It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well known result of Reid, for instance, shows that the geodesic length spectrum of an arithmetic hyperbolic surface determines the surface's commensurability class. It is known, however, that non-commensurable arithmetic hyperbolic surfaces may share arbitrarily large portions of their length spectra. In this paper we investigate this phenomenon and prove a number of quantitative results about the maximum cardinality of a family of pairwise non-commensurable arithmetic hyperbolic surfaces whose length spectra all contain a fixed (finite) set of nonnegative real numbers.

Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds

“…We note that the hypothesis that π(V, S) → ∞ as V → ∞ is necessary as there exist subsets S for which π(V, S) is nonzero yet eventually constant. Examples of such sets were given in [7] in the context of hyperbolic surfaces using a construction that easily generalizes to hyperbolic 3-manifolds.We now state our main geometric result.…”