I am an undergraduate student looking into the quotients of hyperbolic space by triangle groups. For clarity, a triangle group is a group generated by reflections in the sides of a triangle.
For example, $$\Delta (2,3,7) = \; \langle a,b,c \mid a^2, b^2, c^2, (ab)^2, (bc)^7, (ca)^3 \rangle,$$ where $(2,3,7)$ refers to angles $\frac{\pi}{2}, \frac{\pi}{3}, \frac{\pi}{7}$ in a hyperbolic triangle.
My question is how can we determine which quotients of $\mathbb{H}^2$ by subgroups of $\Delta (p,q,r)$ (where $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} < 1$) form surfaces of certain genus $g$?
In particular, as a result of the Riemann-Hurwitz formula, we have that any finite quotient $G$ of $\Delta (p,q,r)$ satisfies $$ g = 1 + \frac{1}{2}|G|(1-(\frac{1}{p} + \frac{1}{q} + \frac{1}{r})). $$ Would this imply, for example, that there are no hyperbolic triangle groups which give rise to a genus 1 surface as a quotient of $\mathbb{H}^2$ by a subgroup of the triangle group?
Any help would be much appreciated.