I would like to construct a genus $g$ surface regularly tiled by triangles (for example by 238 triangles). Edmunds-Ewing-Kulkarni prove that the only obstruction to doing this is Euler characteristic considerations.

However, I'd like to ensure that the fundamental domain for my tiling contains a ball of some radius. Ideally, I'd like a result of the form: given a radius $R>0$, there is some genus $g_N$ such that for every genus $g \geq g_N$ a surface of genus $g$ can be regularly tiled by triangles using a fundamental domain that contains a ball of radius $R$.