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

TL;DR

This paper investigates the distribution of probability measures supported on compact leaves of horocycle foliations in Bianchi 3-orbifolds, revealing equidistribution properties linked to Eisenstein series.

Contribution

It introduces a novel analysis of Lebesgue probability measures on compact leaves of horocycle foliations in Bianchi orbifolds, connecting their distribution to Eisenstein series and hyperbolic geometry.

Findings

Proves equidistribution of measures on compact leaves as parameters vary.

Establishes connections between horocycle dynamics and Eisenstein series.

Provides new insights into the geometry of Bianchi 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.