Let $T$ be a torus, $T^* = T - \{p\}$ be the complement of a point $p$. Let's fix a pair of generators $x,y\in\pi_1(T^*)$. Their images in $\pi_1(T)$ also generate, and will also be denoted by $x,y$.
The Teichmuller space of $T$ can be defined in terms of lattices: A Teichmuller marking on $T$ is a homotopy class of complex structures, relative to which the monodromy of $x,y$ in the universal cover $\mathbb{C}$ of $T$ is given by translation by a pair of $\mathbb{R}$-linearly independent vectors $v,w$ in $\mathbb{C}$. Scaling by $u\in\mathbb{C}^\times$ results in the same Teichmuller marking, so we may assume that one of $v,w$ is 1 and the other lies in the upper half plane $H$, this identifies $Teich(T)$ with the upper half plane $H$.
The Teichmuller space of $T^*$ can be defined in terms of hyperbolic structures. This can be identified with discrete faithful representations $\rho : \pi_1(T^*) \to PSL(2,\mathbb{R})$ (up to $PGL(2,\mathbb{R})$ equivalence), such that a loop around the puncture maps to a unipotent element. Given such a representation, the $T^*$ can be identified with the quotient of $H$ by the image of $\rho$; since $\rho$ acts by isometries, the hyperbolic structure on $H$ then descends to one on $T^*$.
I'm trying to understand how to "read off" the hyperbolic structure from the period lattice. For example, given a pair of linearly independent $v,w\in\mathbb{C}$, can we write down a formula/expression that describes the corresponding $V,W\in PSL(2,\mathbb{R})$ (up to conjugation in $PGL(2,\mathbb{R})$)? If not, can we at least describe the traces of $V,W$? (i.e., the lengths of the corresponding hyperbolic geodesics?)
