Sachs triangulations and regular maps

AbstractA Sachs triangulation of a closed surface S is a triangulation T admitting a vertex-labelling λ in a group G subject to the following conditions: (S1) For any facial triangle t of T with vertices x, y and z, either λ(x)λ(y)λ(z) = 1 or λ(x)λ(z)λ(y) = 1. (S2) For any g,hϵG, there exists at most one edge in T whose endpoints are labelled g and h.In this paper we establish various sufficient conditions for a Sachs triangulation to be a regular (symmetrical) map. As an application of these results we construct, for each integer d ⩾ 2, a 2d-valent reflexible symmetrical triangulation of genus 1 + d(d - 3)/2