Lie groupoid integration of singular isometries of the Poincaré disk
Abstract
For every n ≥ 1 there is a distinguished sl2(ℝ) action on the Poincaré 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 Möbius 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 𝒢n integrating 𝒜n and compare it to the \v Severa--Weinstein groupoid. Although 𝒢n is not an action groupoid, its restriction to the boundary recovers an n-fold Möbius action on the boundary circle.
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BibTeX
@misc{dalipi2026groupoid,
title = {{Lie groupoid integration of singular isometries of the Poincaré disk}},
author = {Rea Dalipi},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30077},
eprint = {2608.30077},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30077},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lie groupoid integration of singular isometries of the Poincaré disk
AU - Rea Dalipi
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.30077
UR - https://arxiv.org/abs/2608.30077
AB - For every n ≥ 1 there is a distinguished sl_2(ℝ) action on the Poincaré 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 Möbius 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 𝒢_n integrating 𝒜_n and compare it to the \v Severa--Weinstein groupoid. Although 𝒢_n is not an action groupoid, its restriction to the boundary recovers an n-fold Möbius action on the boundary circle.
ER -