Distribution of orbits of geometrically finite groups acting on null vectors

TL;DR

This paper investigates how non-discrete orbits of geometrically finite groups distribute in certain geometric spaces, providing asymptotic and quantitative results using horospherical flow equidistribution.

Contribution

It introduces new asymptotic and quantitative descriptions of orbit distributions for geometrically finite groups acting on hyperbolic spaces and related quotients.

Findings

Derived asymptotic formulas for orbit distribution.

Established quantitative equidistribution results under additional conditions.

Extended understanding of orbit behavior in hyperbolic geometry contexts.

Abstract

We study the distribution of non-discrete orbits of geometrically finite groups in $\operatorname{SO}(n,1)$ acting on $\mathbb{R}^{n+1}$, and more generally on the quotient of $\operatorname{SO}(n,1)$ by a horospherical subgroup. Using equidistribution of horospherical flows, we obtain both asymptotics for the distribution of orbits for the action of general geometrically finite groups, and we obtain quantitative statements with additional assumptions.