Many Examples of Non-cocompact Fuchsian Groups Sitting in PSL(2,Q)

TL;DR

This paper constructs infinitely many noncommensurable, non-cocompact Fuchsian groups with finite covolume in PSL(2,Q), ensuring their hyperbolic fixed points include any specified finite set, advancing understanding of group embeddings.

Contribution

It introduces a method to produce infinitely many such groups with prescribed boundary fixed points, expanding the class of known Fuchsian groups in PSL(2,Q).

Findings

Constructed infinitely many noncommensurable non-cocompact Fuchsian groups in PSL(2,Q).

Ensured the hyperbolic fixed points include any finite set of boundary points.

Demonstrated the existence of groups with prescribed boundary behavior.

Abstract

We construct infinitely many noncommensurable non-cocompact Fuchsian groups $\Delta$ of finite covolume sitting in PSL(2,Q) so that the set of hyperbolic fixed points of $\Delta$ will contain a given finite collection of elements in the boundary of the hyperbolic plane.