Eisenstein series and equidistribution of Lebesgue probability measures on compact leaves of the horocycle foliations of Bianchi 3-orbifolds

Abstract

Inspired by the works of Zagier, we study the probability measures $\nu(t)$ with support on the flat tori which are the compact orbits of the maximal unipotent subgroup acting holomorphically on the positive orthonormal frame bundle $\mathcal{F}({M}_D)$ of 3-dimensional hyperbolic Bianchi orbifolds ${M}_D=\mathbb{H}^3/\widetilde{\Gamma}_D$, of finite volume and with only one cusp. Here $\Gamma_D=PSL(2, \mathcal{O})$, where $\mathcal{O}$ is the ring of integers of an imaginary quadratic field of class number one.