search

flatsurf / sage-flatsurf

Flat surfaces in Sage
https://flatsurf.github.io/sage-flatsurf/
GNU General Public License v2.0
10 stars 10 forks source link

Detecting rank 2 orbit closures and eigenform loci inside arithmetic rank 2 #136

Open videlec opened 2 years ago

videlec commented 2 years ago

Beyond coverings (see #54) there are finitely many known rank 2 GL(2,R)-orbit closures (see Eskin-McMullen-Mukamel-Wright). We should provide a way to detect whether a given surface belong to one of them.

Furthermore, inside an arithmetic rank 2 (ie field of definition is QQ), there is a countable family of proper subvarieties defined over quadratic number fields, the so-called "eigenform loci" in the language of Curtis McMullen (see his "genus 2" papers as well as his "Prym" paper). For a surface X that is known to belong to an arithmetic rank 2, checking whether it belongs to an eigenform locus should be a straightforward computation. I believe that the following is one way : We need to find a Thurston-Veech construction Y such that the GL(2,R)-orbit closures of X and Y coincide.

See also #134.

saraedum commented 1 year ago

@videlec what is #290 that you linked here?

videlec commented 1 year ago

@videlec what is #290 that you linked here?

No idea. I removed this reference.