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.