Euclidean traveller in hyperbolic worlds

• Published in 2022
• Added on 2022-09-09

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.

Links

• http://arxiv.org/abs/2209.01306v1
• http://arxiv.org/pdf/2209.01306v1

Other information

BibTeX entry

@article{Euclideantravellerinhyperbolicworlds,
	key = {Euclideantravellerinhyperbolicworlds},
	type = {article},
	title = {Euclidean traveller in hyperbolic worlds},
	author = {Hee Oh},
	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.},
	comment = {},
	date_added = {2022-09-09},
	date_published = {2022-04-10},
	urls = {http://arxiv.org/abs/2209.01306v1,http://arxiv.org/pdf/2209.01306v1},
	collections = {easily-explained,fun-maths-facts,geometry},
	url = {http://arxiv.org/abs/2209.01306v1 http://arxiv.org/pdf/2209.01306v1},
	year = 2022,
	urldate = {2022-09-09},
	archivePrefix = {arXiv},
	eprint = {2209.01306},
	primaryClass = {math.GT}
}