Bi-rotary Maps of Negative Prime Characteristic

Abstract

Bi-orientable maps (also called pseudo-orientable maps) were introduced by Wilson in the 1970s to describe non-orientable maps with the property that opposite orientations can consistently be assigned to adjacent vertices. In contrast to orientability, which is both a combinatorial and topological property, bi-orientability is only a combinatorial property. In this paper we classify the bi-orientable maps whose localorientation-preserving automorphism groups act regularly on arcs, called here bi-rotary maps, of negative prime Euler characteristic. Unlike other classification results for highly symmetric maps on such surfaces, we do not use the Gorenstein-Walter result on the structure of groups with dihedral Sylow 2-subgroups. Mathematics Subject Classification. 05C10, 05C15, 05C25.

Classifying edge-biregular maps of negative prime Euler characteristic

“…This, together with (3.1) and the facts summed up at the end of section 3, implies that |H| = νp where ν ∈ {4, 6, 8, 12}. Moreover, one may check that in our range for the even entries in the type of our biregular map M one has: ν(k, ) = 12 only for (k, ) = (4, 6); ν(k, ) = 8 only for (k, ) = (4, 8); ν(k, ) = 6 only when p = 2 and for types (6,6) and (4, 12); and finally ν(k, ) = 4 only for (k, ) equal to (6,12) or (8, 8), corresponding to the cases when p = 3 and p = 2 respectively.…”

“…Edge-biregular maps have been investigated in great detail in [10], including their classification on surfaces of non-negative Euler characteristic and also a classification of such maps with a dihedral automorphism group. Our aim here is to complement the results of [10] by deriving a classification of edge-biregular maps on surfaces with negative prime Euler characteristic, extending thereby also the existing classification results for fully regular maps [2] and bi-rotary maps [6] on these surfaces.…”

“…Smooth normal quotients of this subgroup give rise to orientably-regular maps in the terminology of [13], also known as rotary maps in [14]. The second type of biregular maps mentioned in the introduction, the bi-rotary maps of [6], arise as smooth quotients of the index-two subgroup R 0 , R 1 , R 2 of (2.1) containing R 0 but avoiding R 1 and R 2 , for even. Using the notation A = R 0 , Z = R 1 R 2 , and the square brackets for a commutator of two elements as usual, this subgroup admits a presentation…”

“…Proof. When the prime p = 3, we have the following possibilities: type (6,12) for |H| = 4 × 3; type (4, 8) for |H| = 8 × 3; and type (4, 6) for |H| = 12 × 3.…”

“…We note that orientability of a map is equivalent to admitting local orientations at vertices that are consistent when passing between incident vertices along an edge on the carrier surface of the map. An opposite phenomenon, namely, when a map of type (k, ) admits an assignment of local orientations at vertices which disagree when traversing along an arbitrary edge on the surface and, at the same time, exhibits the highest 'level of symmetry' with respect to this property, was studied in [6]. These maps, called bi-rotary in [6], arise as normal torsion-free quotients of a different index-two subgroup of the full (2, k, )-triangle group.…”