关于Basmajian论文中裤对定义及测地边界裤对商空间的问询

Absolutely, yes — a pair of pants with geodesic boundaries can be constructed as the quotient of the hyperbolic upper half-plane $\mathbb{H}^2$ by a suitable torsion-free Fuchsian group, aligning perfectly with the definition from Ara Basmajian's paper.

Here's a breakdown of why this works:

• First, recall the core definition: a torsion-free Fuchsian group $G$ is called a "pair of pants" if $\mathbb{H}^2/G$ is topologically a sphere with three holes. When those holes correspond to geodesic boundaries (instead of cusps), we're exactly looking at a hyperbolic surface with three closed geodesic boundaries.
• The key is constructing the right Fuchsian group: this group will be generated by hyperbolic isometries of $\mathbb{H}^2$, whose axes correspond to the preimages of the geodesic boundaries in the quotient. By taking a fundamental domain (a hyperbolic hexagon with pairwise congruent, parallel geodesic sides) and identifying opposite sides via these isometries, we get the quotient space — which is precisely the pair of pants with geodesic boundaries.
• Unlike the "tight pair of pants" (which has one cusp), this version has all three boundaries as complete geodesics. The torsion-free condition ensures there are no cone singularities in the quotient, keeping the surface smooth in the hyperbolic sense.

To put it concisely: the quotient $\mathbb{H}^2/G$ for such a group $G$ is exactly the hyperbolic pair of pants with geodesic boundaries, so the answer is a definite yes.

内容的提问来源于stack exchange，提问作者user416933