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

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.