Limit distributions for $\text{SO}(n,1)$ action on $k$-lattices in $\mathbb{R}^{n+1}$

TL;DR

This paper investigates the asymptotic behavior of lattice actions in hyperbolic space, showing convergence to specific measures except in special geometric cases, and resolves a conjecture by Sargent and Shapira.

Contribution

It provides a general result on the distribution of lattice orbits in hyperbolic space, including explicit measures and special case analysis.

Findings

Measures converge to a semi-invariant probability measure in general cases.

Exception identified for special 2-lattices in R^3 tangent to the light cone.

Resolves a conjecture of Sargent and Shapira.

Abstract

We study the asymptotic distribution of norm ball averages along orbits of a lattice $\Gamma \subset \text{SO}(n,1)$ acting on the moduli space of pairs of orthogonal discrete subgroups of $\mathbb{R}^{n+1}$ up to homothety. Our main result shows that, except for special $2$-lattices in $\mathbb{R}^3$ lying in hyperplanes tangent to the light cone, these measures converge to an explicit semi-invariant probability measure supported on the space of homothety classes of pairs of orthogonal lattices tangent to the light cone. Our main motivation is a conjecture of Sargent and Shapira, which is resolved as a special case of our general result.