Isometries of Lorentz surfaces and convergence groups

TL;DR

This paper investigates the isometry groups of certain Lorentz surfaces, demonstrating their relation to convergence groups and classical groups like PSL(2,R), with implications for their actions on the circle.

Contribution

It establishes a connection between isometry groups of Lorentz surfaces and convergence groups, showing semi conjugacy to finite covers of PSL(2,R) and providing conditions for conjugacy in Homeo(S^1).

Findings

Isometry groups act on the circle via convergence groups.

Non proper actions relate to semi conjugacy with PSL(2,R) covers.

Additional assumptions yield conjugacy in Homeo(S^1).

Abstract

We study the isometry group of a globally hyperbolic spatially compact Lorentz surface. Such a group acts on the circle, and we show that when the isometry group acts non properly, the subgroups of $\mathrm{Diff}(\mathbb{S}^1)$ obtained are semi conjugate to subgroups of finite covers of $\mathrm{PSL}(2,\mathbb{R})$ by using convergence groups. Under an assumption on the conformal boundary, we show that we have a conjugacy in $\mathrm{Homeo}(\mathbb{S}^1)$.