I am learning hyperbolic geometry on my own . I studied about a proof that proves the upper half plane $\mathbb{H}$ as an universal cover of two holed torus (or genus $2$ surface) using tiling with the hyperbolic octagon.
But I am looking for a prove that the upper half plane $\mathbb{H}$ can be viewed as the universal cover of a genus $2$ Riemann surface throwing it in complex analysis without using any big theorem like, uniformization theorem.
Please help me. Thanking in advanced