Euclidean traveller in hyperbolic worlds

TL;DR

This paper explores the possible closures of Euclidean lines in various geometric spaces, revealing the diverse behaviors of such lines across different curved and infinite-volume manifolds.

Contribution

It provides a comprehensive analysis of Euclidean line closures in multiple geometric contexts, extending classical results to hyperbolic and infinite-volume manifolds.

Findings

Euclidean lines can have diverse closure behaviors in hyperbolic spaces.

The study connects classical and modern results on line closures across different geometries.

It highlights the role of group actions in determining line closures in complex manifolds.

Abstract

We will discuss all possible closures of a Euclidean line in various geometric spaces. Imagine the Euclidean traveller, who travels only along a Euclidean line. She will be travelling to many different geometric worlds, and our question will be "what places does she get to see in each world?". Here is the itinerary of our Euclidean traveller: In 1884, she travels to the torus of any dimension, guided by Kronecker. In 1936, she travels to the world, called a closed hyperbolic surface, guided by Hedlund. In 1991, she then travels to a closed hyperbolic manifold of higher dimension $n\ge 3$ guided by Ratner. Finally, she adventures into hyperbolic manifolds of infinite volume guided by Dal'bo in dimension $2$ in 2000, by McMullen-Mohammadi-Oh in dimension $3$ in 2016 and by Lee-Oh in all higher dimensions in 2019.