On the rate of equidistribution of expanding horospheres in finite-volume quotients of $\mathrm{SL}(2,\mathbb{C})$

TL;DR

This paper establishes an effective rate of equidistribution for expanding horospheres in finite-volume quotients of SL(2,C), with explicit error bounds, using unitary representation theory.

Contribution

It provides the first effective equidistribution results with explicit error terms for horospherical orbits in SL(2,C) quotients, advancing understanding of their dynamical properties.

Findings

Explicit error bounds for equidistribution rates

Application of unitary representation theory to dynamical problems

Enhanced understanding of horospherical orbit behavior in complex Lie groups

Abstract

Let $\Gamma$ be a lattice in $G=\mathrm{SL}(2,\mathbb{C})$. We give an effective equidistribution result with precise error terms for expanding translates of pieces of horospherical orbits in $\Gamma\backslash G$. Our method of proof relies on the theory of unitary representations.