TRIANGULATIONS OF UNORIENTABLE SURFACES

Abstract

We consider the question of when an unorientable surface without boundary can be tiled by an $(l,m,n)$ triangle. We desire that the automorphism group of the surface preserving the tiling be “substantial”. We use the well known theory of quasiplatonic surfaces and symmetries of Riemann surfaces to propose a classification algorithm, entirely by finite group calculations. A comprehensive analysis of symmetries of quasiplatonic surfaces in low genus, by hand and by computer, is carried out, yielding triangulations of unoriented surfaces as a byproduct.