I'm studying the paper The Big Picard Theorem and Other Results on Riemann Surfaces by P. Arés-Gastesi and T. Venkataramana, which presents a new proof of the Big Picard Theorem using covering spaces rather than the classical one which uses Montel's Theorem. The theorem, as stated in the paper is
Theorem 3.2. If $f:\mathbb{D}^*\to X=\hat{\mathbb{C}}\setminus\{\infty,0,1\}$ is holomorphic then $f$ cannot have an essential singularity at the origin.
(Where $\mathbb{D}^*=\{z\in\mathbb{C}; 0<|z|<1\}$ and $\hat{\mathbb{C}}\cong\mathbb{C}\cup\{\infty\}$ is the Riemann Sphere)
I'm having trouble to understand the beginning of the proof, which I quote below:
Let $c$ be a generator of the fundamental group of $\mathbb{D}^*$ (we again make an abuse of notation and use the same letter for a path and its homology class) and let $\gamma$ be the element of $\Gamma(2)$ [the principal congruence subgroup of level 2] that corresponds to $f_*(c)$ under the convering defined by the modular function $\lambda$.
(The matrices of $\Gamma(2)$ act in $\mathbb{H}$ as Möbius transformations, and $\lambda:\mathbb{H}\to\mathbb{C}$ is a holomophic function such that $\lambda(\gamma(\tau))=\lambda(\tau)$ for any $\gamma\in\Gamma(2)$ and $\tau\in\mathbb{H}$. Furthermore, $\lambda:\mathbb{H}\to\hat{\mathbb{C}}\setminus\{\infty,0,1\}$ is a covering space).
I don't understand how a matrix ($\gamma$) would be correspondent in any sense to a homotopy class ($f_*(c)$). Could someone please explain this to me?