Distribution of Holonomy about Closed Geodesics on Compact Hyperbolic 3-Manifolds
An important feature of compact hyperbolic 3-manifolds is their closed geodesics, which have both geometric and dynamical implications. These geodesics are parametrized by their length and holonomy, which describes the angle of rotation by parallel transport about the geodesic. First, we prove ambient prime geodesic theorems, which provide an asymptotic count of closed geodesics by their length and holonomy and imply e ective equidistribution of holonomy in shrinking intervals. These theorems result from a non-spherical Selberg's trace formula which relates spectral information to lengths and holonomies of closed geodesics. Then, we show that although holonomy is equidistributed, there is typically a bias in the finer distribution. In particular, we show that a smoothed bias count is distributed according to a probability distribution and compute the average value, which is typically negative but depends on the number of spectral parameters equal to zero. When all distinct non-zero spectral parameters are linearly independent over the rationals, we explicate the probability distribution function for this sum. In addition, we construct an example where the linear independence hypothesis is not met.
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