Pseudomodular Group

Answered the open problem "Are there infinitely many commensurability classes of pseudomodular groups"

Problem

Modular group is a fundamental concept that is especially important for hyperbolic geometry. It has a clean definition (isomorphic to a specific group) and a nice property (its orbit is all rational numbers). For long, people have been speculating property <=> definition, until the first counter-example, known as pseudomodular group, that also has the nice property despite not modular by definition. But how many pseudomodular groups are there? Most importantly, are there finitely many, or infinitely many?

Answer

In short, yes. We came up with a construction method that can give infinitely many commensurability classes that are not modular. Each of them has the nice property of having an orbit of all rational numbers.

For full details, see our two papers (Lou et al., 2018) and (Lou et al., 2021).

Huge shout-out to Prof. Ser Peow Tan who carried us through this long journey.

References

2021

  1. IMRN
    Hyperbolic Jigsaws and Families of Pseudomodular Groups II
    Beicheng Lou, Ser Peow Tan, and Anh Duc Vo
    International Mathematics Research Notices, 2021
    Abs HTML

2018

  1. Geom. Topol.
    Hyperbolic jigsaws and families of pseudomodular groups, I
    Beicheng Lou, Ser Tan, and Anh Duc Vo
    Geometry & Topology, 2018
    Abs HTML