Lie groupoid integration of singular isometries of the Poincar\'e disk

Abstract

For every $n \geq 1$ there is a distinguished $\mathfrak{sl}_2(\mathbb{R})$ action on the Poincar\'e disk, arising as the infinitesimal isometries of a hyperbolic metric with conical singularity of order $n-1$ at the origin. For $n=1$ this is the standard infinitesimal M\"obius action, and for $n>1$ these vector fields have singularities at the origin and are incomplete, preventing integration to a global Lie group action. However, they naturally define an action Lie algebroid $\mathcal{A}_n=\mathfrak{sl}_2(\mathbb{R})\ltimes \Dbarstar $ over the punctured disk. We construct an explicit Lie groupoid $\mathcal{G}_n$ integrating $\mathcal{A}_n$ and compare it to the \v Severa--Weinstein groupoid. Although $\mathcal{G}_n$ is not an action groupoid, its restriction to the boundary recovers an $n$-fold M\"obius action on the boundary circle.