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 $M_D={\mathbb{H}}^3/{\tilde{\Gamma }}_D$, 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.

arXiv commentWe corrected a few typos and included a few details