Vertex-transitive maps with Schläfli type{3,7}

Abstract

Among all equivelar vertex-transitive maps on a given closed surface S, the automorphism groups of maps with Schläfli types {3, 7} and {7, 3} allow the highest possible order. We describe a procedure to transform all such maps into 1-or 2-orbit maps, whose symmetry type has been previously studied. In so doing we provide a procedure to determine all vertex-transitive maps with Schläfli type {3, 7} which are neither regular or chiral. We determine all such maps on surfaces with Euler characteristic −1 ≥ χ ≥ −40.

Vertex-Transitive Polyhedra of Higher Genus, I

“…P is (geometrically) vertex-transitive if G acts transitively on the vertices of P . Since these polyhedra are free of non-trivial selfintersection, the problem at hand is distinct from the classification of the 75 uniform polyhedra by Coxeter, Longuet-Higgins, Miller [8], Sopov [35], and Skilling [34], and the enumeration of regular, uniform, or equivelar maps on surfaces [1,4,7,20,25,26,36] with its abundance of examples. In fact, we investigate in the spirit of [2,3,24,29,30,32,33,38], requiring a particular high geometric symmetry yet not necessarily combinatorial regularity.…”