Regular triangulation of compact oriented hyperbolic space

Question

Is there a good way of explicitly constructing a regular triangulation of a compact orientable hyperbolic 2-manifold, ideally with any desired vertex degree $\ge 7$? I only need the topology, not any geometric information.

Answer 1

The meaning of regular triangulation varies in the literature; I suppose you mean degree-regular (all vertices have the same degree $d$). Then the Euler characteristic makes it impossible to get any desired $d$. Indeed, a triangulation with $n$ triangles and all vertices of degree $d$ must have $3n/2$ edges and $3n/d$ vertices, hence $n-3n/2+3n/g = 2-2g$. Rearranging as $$n=\dfrac{4\,d\,(g-1)}{d-6}$$ we see that certain triangulations are impossible just on the basis of divisibility: say, if $d=11$ and $g=2$. Besides the divisibility issue, the fact that you have only this many triangles to play with is a severe restriction. It makes finding degree-regular triangulations into a combinatorial search, rather than a constructive algorithm. The search has been carried out recently: see arXiv 0403433 and reference [6] therein. I don't know if they got all degree-regular triangulations.