庞加莱圆盘奇异等距的李群胚积分
Lie groupoid integration of singular isometries of the Poincaré disk
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中文总结 AI 辅助
针对庞加莱圆盘上带锥形奇点的无穷小等距变换，构造了积分其作用李胚的显式李群胚，该群胚边界限制为n重莫比乌斯作用。
中文摘要 AI 辅助
对每个整数n≥1，庞加莱圆盘上存在一个特殊的𝔰𝔩₂(ℝ)作用，该作用源于原点处阶数为n-1的锥形奇点的双曲度量的无穷小等距变换。当n=1时，此作用为标准无穷小莫比乌斯作用；当n>1时，这些向量场在原点处具有奇点且不完备，无法积分得到整体李群作用，但它们自然定义了带孔圆盘上的作用李胚Aₙ=𝔰𝔩₂(ℝ)⋉D̄*。我们构造了一个显式李群胚Gₙ以积分Aₙ，并将其与Severa–Weinstein群胚进行比较。尽管Gₙ不是作用群胚，但其在边界上的限制恢复了边界圆上的n重莫比乌斯作用。
英文摘要
For every $n \geq 1$ there is a distinguished $\mathfrak{sl}_2(\mathbb{R})$ 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 $\mathcal{G}_n$ 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 Möbius action on the boundary circle.
