Benjamini-Schramm convergence of periodic orbits

TL;DR

This paper establishes a criterion for Benjamini-Schramm convergence of periodic orbits in Lie groups and applies it to homogeneous spaces and translation surfaces.

Contribution

It introduces a general criterion for convergence of periodic orbits and demonstrates its application to specific geometric spaces.

Findings

Criterion for Benjamini-Schramm convergence established

Applied to homogeneous spaces and translation surfaces

Provides a new tool for analyzing orbit distributions

Abstract

We prove a criterion for Benjamini-Schramm convergence of periodic orbits of Lie groups. This general observation is then applied to homogeneous spaces and the space of translation surfaces.