Hyperbolic structures on once punctured tori

Question

I've been working on a problem about billiards in ideal hyperbolic polygons and I was thinking about how the problem for ideal quadrilaterals relates to closed geodesics on once punctured tori.

My questions have to do with constructing a general hyperbolic punctured torus. I would like to think of it as the quotient of an ideal quadrilateral. I have not been able to find a description of the space of hyperbolic structures on the once punctured torus from this perspective.

The tori that come up in the problem are constructed as follows: Let $Q$ be an ideal quadrilateral. Let $L_1$ be the geodesic orthogonal to a pair of opposite sides $a_1$ and $a_2$. Let $L_2$ be the geodesic orthogonal to the other pair of opposite sides $b_1$ and $b_2$. Now let $A$ be the loxodromic isometry with axis $L_1$ taking $a_1$ to $a_2$ and let $B$ be the loxodromic isometry with axis $L_2$ taking $b_1$ to $b_2$. Let $G$ be the group of isometries generated by $A$ and $B$. Then the quotient of the hyperbolic plane by $G$ gives us a punctured torus. With this construction, there is a positive real number which determines the hyperbolic structure: the length, $y$, of the geodesic segment $L_1 \cap Q$, which determines the shape of $Q$. I understand that the space of hyperbolic structures on a once punctured torus has two real dimensions and can be identified with the upper half plane. This construction gives us hyperbolic structures parametrized by a single positive real number $y$. I guess that it corresponds to the positive imaginary axis in the upper half plane.

Question 1: Does the set of hyperbolic once punctured tori constructed as above have a name?

My next question has to do with what the other hyperbolic structures look like when thinking of them as the quotient of an ideal quadrilateral. Instead of just using the isometries $A$ and $B$ above, we can compose $A$ with a loxodromic isometry $C$ having axis $a_2$ (which is a side of $Q$). Call this composition $A'$. We want a loxodromic isometry $B'$ that identifies $b_1$ with $b_2$ such that the commutator of $A'$ and $B'$ is parabolic. My guess is that, given $A'$, there is a unique $B'$ such that the commutator of $A'$ and $B'$ is parabolic. This would mean that the hyperbolic structures could be parametrized by two real numbers $x$ and $y$. The number $y$ is a positive real number that determines the shape of $Q$ and the number $x$ is a real number that determines the translation length of the isometry $C$.

Question 2: Does this make sense and, if so, is it a reasonable way to think about the space of hyperbolic structures on once punctured tori?

Answer 1

Question 1: I would call them "rectangular", because they are conformally equivalent to gluing opposite sides of a Euclidean rectangle by Euclidean translation and then removing the point obtained by gluing the four corners.

Question 2: I believe this is true. The way to prove it would be to cut the quadrilateral into two ideal triangles. Each of side of each ideal triangle has a "foot" which is the base of the perpendicular ray going out to the opposite ideal vertex. For each of the three matched pairs of sides of this pair of triangles you have a "shearing coordinate" which is a real number giving the displacement from the foot of that side in one triangle to the foot of that side in the other triangle. This gives three shearing coordinates $a$, $b$, $c$, and the "completeness condition" or "parabolicity condition" is simply the equation $a+b+c=0$, and this is well known to be a parameterization of the Teichmuller space of the punctured torus by a plane in $a$, $b$, $c$ space (see for instance Penner's paper "The decorated Teichmüller space of punctured surfaces" which may contain this exact statement but if not contains the ideas needed to verify it).

Edit: Here's a bit more detail, since this seems pretty simple. Choosing the shape of the quadrilateral determines and is determined by one of the coordinates $a$. Then, having chosen that, choosing the translation along your side labelled $a_2$ determines and is determined by the second coordinate $b$. Finally, and the third coordinate $c=-a-b$ determines and is determined by the translation along your side labelled $b_2$.