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?