Euclidean traveller in hyperbolic worlds

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.