I have been trying to get a broad idea of modular forms. As I understand it, a modular form of weight $k$ is a holomorphic function $f:\mathbf{H} \to \mathbb{C}$ that is holomorphic also holomorphic at the limit $i\infty$ and satisfies a functional equation of the form $$f(\gamma z) = (cz+d)^k f(z)$$ for every $$\gamma=\begin{pmatrix} a & b\\ c & d \end{pmatrix} \in \Gamma$$ Let us ignore the "holomorphic at $i\infty$" part for the moment and focus on the functional equation.
What the functional equation tells us is that $f$ is completely described by the values it takes on orbit representatives of the action of the modular group $\Gamma$ on $\mathbf{H}$. This means that $f$ is can be studied based on its values on representatives of orbits of $\Gamma$.
The conventional approach at this point is to define the Poincare metric on $\mathbf{H}$ so as to make it a model of the hyperbolic plane. This also makes $\mathbf{H}$ a topological space. Since the group of isometries of $\mathbf{H}$ equipped with the Poincare metric is the group of Mobius transformations $PSL(2,\mathbf{R})$, we can then conclude that $\Gamma$ is a discrete group and the $\mathbf{H}/\Gamma$ is a quotient topological space that can be visualized as the fundamental domain as seen in the following diagram
My question is: is it necessary to introduce the Poincare metric here?
I mean, all we are interested in is the fundamental domain of $\Gamma$, which is simply a set of representatives of every orbit of its action. Is a metric and a topology even essential in this regard? Moreover, why the Poincare metric? Why are we even concerned with the fact that $\Gamma$ is a group of isometries for the metric defined? Even if that is necessary, is the Poincare metric the only metric with respect to which $\Gamma$ is an isometry group?
Clearly I am missing the picture here. Sorry if the question is a bit broad and imprecise. My aim is to understand the motivations clearly.