Bias in the distribution of holonomy on compact hyperbolic 3-manifolds (Q6634486)

Closed geodesics on a hyperbolic \(3\)-manifold are parametrized by their hyperbolic length and their holonomy (measuring the angle of rotation from parallel transport along the geodesic). The prime geodesic theorem due to \textit{P. Sarnak} [Acta Math. 151, 253--295 (1983; Zbl 0527.10022)] gives precise asymptotics for the count of primitive geodesics of bounded length in terms of the eigenvalues of the hyperbolic Laplacian; \textit{P. Sarnak} and \textit{M. Wakayama} [Duke Math. J. 100, No. 1, 1--57 (1999; Zbl 0939.22008)] also showed that holonomy equidistributes and gave asymptotics for the count of primitive closed geodesics of bounded length and holonomy constrained to an interval. In the setting of orientable compact hyperbolic \(3\)-manifolds, the author and \textit{D. Milićević} [Int. Math. Res. Not. 2023, No. 1, 588--635 (2023; Zbl 1521.53034)] gave refined asymptotics with a good error term, giving effective equidistribution as well as equidistribution in sufficiently slowly shrinking intervals of holonomy. Here the count of geodesics is smoothed by a weight function giving stronger asymptotics, and the second or error term is studied. In particular, a function on holonomy that vanishes on average is chosen and the asymptotic properties of the second term are studied and shown to exhibit bias controlled by the number of zero spectral parameters arising from the unitary representation theory of the space of square-integrable functions on the locally homogeneous space corresponding to the hyperbolic \(3\)-manifold.