Derived Arithmetic Fuchsian Groups of Genus Two
Melissa L. Macasieb

TL;DR
This paper classifies all torsion-free derived arithmetic Fuchsian groups of genus two, establishes their relation to quaternion algebras, and provides explicit generators for these groups using computational methods.
Contribution
It provides a complete classification of such groups by commensurability class and explores their quaternion algebra origins, including explicit generator determination.
Findings
No such groups from quaternion algebras over fields of degree > 5
Existence of maximal orders in related quaternion algebras
Explicit generators for several examples
Abstract
We classify all torsion-free derived arithmetic Fuchsian groups of genus two by commensurability class. In particular, we show that there exist no such groups arising from quaternion algebras over number fields of degree greater than 5. We also prove some results on the existence and form of maximal orders for a class of quaternion algebras related to these groups. Using these results in conjunction with a computer program, one can determine an explicit set of generators for each derived arithmetic Fuchsian group containing a torsion-free subgroup of genus two. We show this for a number of examples.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic Geometry and Number Theory · Mathematical Dynamics and Fractals · Advanced Algebra and Geometry
