The Space of Geometric Limits of One-generator Closed Subgroups of PSL2(R)

TL;DR

This paper characterizes the closure of the space of one-generator closed subgroups of PSL2(R) in the Chabauty topology, using explicit matrix computations and geometric limit analysis.

Contribution

It provides a complete description of the geometric limits of these subgroups, including explicit parametrizations and conditions for convergence.

Findings

Explicit matrices for automorphisms of the disk in PSL2(R)

Conditions for convergence of sequences of closed subsets

Geometric limits of PSL2(R) subgroups via Hausdorff limits

Abstract

We give a complete description of the closure of the space of one-generator closed subgroups of PSL2(R) for the Chabauty topology, by computing explicitly the matrices associated with elements of Aut(D) = PSL2(R), and finding quantities parametrizing the limit cases. Along the way, we investigate under what conditions sequences of maps transform convergent sequences of closed subsets of the domain into convergent sequences of closed subsets of the range. In particular, this allows us to compute certain geometric limits of PSL2(R) only by looking at the Hausdorff limit of some closed subsets of C.