On Length Sets of Subarithmetic Hyperbolic Manifolds

TL;DR

This paper formulates and proves a conjecture about the distribution of lengths of closed geodesics on certain hyperbolic manifolds with subarithmetic fundamental groups, using advanced number-theoretic tools.

Contribution

It introduces the Asymptotic Length-Saturation Conjecture for subarithmetic hyperbolic manifolds and proves it for a specific class of covers of the modular surface.

Findings

First proof of the conjecture for punctured, Zariski dense covers of the modular surface.

Development of methods combining the Orbital Circle Method with expansion in congruence towers.

Estimates for exponential sums and quadratic L-series in the context of hyperbolic geometry.

Abstract

We formulate the Asymptotic Length-Saturation Conjecture on the length sets of closed geodesics on hyperbolic manifolds whose fundamental groups are subarithmetic, that is, contained in an arithmetic group. We prove the first instance of the conjecture for punctured, Zariski dense covers of the modular surface. The tools involved include the Orbital Circle Method, expansion and counting in congruence towers of thin groups, estimates for exponential sums, bilinear forms, and quadratic L-series.