Sachs triangulations and regular maps (Q1339874)

The authors are interested in constructing ``highly symmetrical'' triangulations of closed surfaces. They note that known results on obtaining such triangulations are based on groups. The authors exploit an idea of Sachs concerning a method of triangulating surfaces. The adaptation of this idea is to define a Sachs triangulation of a closed surface \(S\) as a triangulation \(T\) with a vertex labeling \(\lambda\) in a group \(G\) so that: (1) For any facial triangle \(t\) of \(T\) with vertices \(x\), \(y\), \(z\), either \(\lambda(x) \lambda(y) \lambda(z)= 1\) or \(\lambda(x) \lambda(z) \lambda(y)= 1\). (2) For any \(g\), \(h\) in \(G\), there exists at most one edge in \(T\) with end points labelled \(g\) and \(h\). The authors consider the highest degree of symmetry to occur when the triangulation becomes a regular map. A triangulation is a regular map if for any two edges \(e\) and \(f\) with arbitrarily assigned directions there exists a map-automorphism sending \(e\) to \(f\) and preserving directions. In this paper, new proofs of fundamental results about Sachs triangulations are obtained, with focus on conditions under which they become regular maps. Also, a characterization of the label sets is given.