Picture of the Universal Cover of the Hyperbolic Pair of Pants

Question

Does anyone have a good picture of the universal cover of the hyperbolic pair of pants with geodesic boundary in either upper half space or the disk? I'm also wondering how the picture changes as you change the geodesic boundary lengths of the pair of pants. Thanks.

Answer 1

Basically what you are asking for requires knowing the whole subject of Fuchsian groups acting on the hyperbolic plane $\mathbb H^2$. I'll write a very brief answer, but my answer will have several significant undefined terms (limit set; convex hull of limit set), and so in order to understand the answer you should probably just crack open a book on this subject, where you'll learn a lot more than I can write in the context of a math.stackexchange answer.

Let $P$ denote your hyperbolic pair of pants. Let $\Gamma$ denote the fundamental group of $S$, a rank $2$ free group. There is a properly discontinuous, free, isometric action of $\Gamma$ on $\mathbb H^2$ with limit set $\Lambda$ such that, if we let $\overline{\mathcal{H}}$ denote the convex hull of $\Lambda$ in $\overline{\mathbb H}^2 = \mathbb H^2 \cup \partial \mathbb H^2$, and if we let $\mathcal H = \overline{\mathcal H} \cap \mathbb H^2$ denote the convex hull in $\mathbb H^2$, then the universal cover of $P$ is identified with $\mathcal H$ up to isometry.

The set $\Lambda$ is a Cantor set embedded in $\partial \mathbb H^2$. The set $\mathcal H$ is a closed subset of $\mathbb H^2$, it is a subsurface with totally geodesic boundary, and its boundary $\partial\mathcal H$ consists of a countable collection of geodesic lines, one for each component of $\partial \mathbb H^2 - \Lambda$, connecting the endpoints of that component.

So, you can say that the universal cover of $P$ is obtained from the hyperbolic plane $\mathbb H^2$ by removing a collection of open hyperbolic half planes, in such a way that the closure of any two of those planes are disjoint, and such that what's left over after this removal limits on a Cantor subset of the circle at infinity.

Very roughly, the way that the picture changes as you vary lengths is that the distances between components of $\partial\mathcal H$ varies.