An abstract spectral approach to horospherical equidistribution

TL;DR

This paper develops an abstract spectral method to establish effective equidistribution of expanding horospheres in hyperbolic manifolds, providing explicit error terms and demonstrating its applicability in various geometric settings.

Contribution

It introduces a novel spectral approach for horospherical equidistribution with explicit error bounds, applicable to hyperbolic and higher rank symmetric spaces.

Findings

Proves effective equidistribution in hyperbolic manifolds.

Extends results to higher rank symmetric spaces.

Provides explicit error estimates for the equidistribution.

Abstract

This paper introduces an abstract spectral approach to prove effective equidistribution of expanding horospheres in hyperbolic manifolds. The method, which is motivated by the approach to counting developed by (Lax-Phillips 1982), produces highly effective, explicit error terms. To exhibit the flexibility of this method we prove effective horospherical equidistribution theorems in $T^1(\mathbb{H}^{n+1})$ and in the higher rank setting, $\operatorname{SL}_n(\mathbb{R})/\operatorname{SO}_n(\mathbb{R})$.