Suppose S is a genus g surface with n punctures satisfying the hyperbolicity condition 2g + n – 2 > 0. If n > 0 the fundamental group of the surface is a free group on 2g + n – 1 := m generators.
If we look the universal covers of different punctured surfaces with the same m (e.g., thrice-punctured sphere and once-punctured torus for m = 2) in, say the hyperbolic plane or the Poincare disc model, how do they differ? The "only" apparent difference is in the number of punctures which should give rise to a difference in the lifts of the punctures to the boundary of the disc. The fundamental groups are isomorphic, but they must act differently to produce quotient surfaces of different genera. How?
How does the set of lifts of punctures on the boundary relate to the standard Farey set?
Thanks a lot in advance!
Best Answer
The simplest case, where $g=0, n=3$ and $g = n = 1$ yield isomorphic groups (free of rank 2), can be written explicitly using a little $2 \times 2$ matrix calculation. We choose a fundamental domain in the upper half-plane made out of two vertical lines with real parts $-1$ and $1$, and two semicircles whose diameters are the real intervals $[-1,0]$ and $[0,1]$. Since the boundaries are geodesics, it suffices to find Möbius transformations that transform the endpoints appropriately.
For $g=0, n=3$, we choose generators $\begin{pmatrix}1& 2 \\ 0 & 1 \end{pmatrix}$ to glue the vertical lines together, and $\begin{pmatrix}1& 0 \\ 2 & 1 \end{pmatrix}$ to glue the semi-circles. If you like modular curves, this quotient is called $Y(2)$, and classifies isomorphism classes of elliptic curves equipped with an ordered list of all 2-torsion points.
For $g=1, n=1$, we choose $\begin{pmatrix}1& 1 \\ 1 & 2 \end{pmatrix}$ to glue the left vertical line to the right semicircle, and $\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ to glue the left semicircle to the right vertical line. This quotient is a leaky torus.
The above results form a special case of a general phenomenon (mentioned by Sam Nead), where the loops around punctures give unipotent (aka parabolic) generators of $\pi_1$, and handles give a pair of hyperbolic generators.
I don't have a good answer concerning the set of lifts of the boundary points - it will always be a disjoint union of $n$ orbits under the transformation group, but it can vary widely, since the transformation group has a continuous family of representations in $PSL_2(\mathbb{R}$.