Bias in the distribution of holonomy on compact hyperbolic 3-manifolds

TL;DR

This paper investigates the distribution of holonomy in closed geodesics on compact hyperbolic 3-manifolds, revealing a persistent bias influenced by spectral parameters and exploring its probabilistic distribution.

Contribution

It demonstrates the existence of a bias in the secondary term of geodesic counts and characterizes its distribution under spectral independence conditions.

Findings

Bias persists in the secondary term of geodesic counts.

Normalized bias follows a specific probability distribution.

Counterexample with dihedral forms lacking spectral independence.

Abstract

Ambient prime geodesic theorems provide an asymptotic count of closed geodesics by their length and holonomy and imply effective equidistribution of holonomy. We show that for a smoothed count of closed geodesics on compact hyperbolic 3-manifolds, there is a persistent bias in the secondary term which is controlled by the number of zero spectral parameters. In addition, we show that a normalized, smoothed bias count is distributed according to a probability distribution, which we explicate when all distinct, non-zero spectral parameters are linearly independent. Finally, we construct an example of dihedral forms which does not satisfy this linear independence condition.