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Edge-Transitive Maps of Low Genus∗

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Edge-Transitive Maps of Low Genus∗ Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 4 (2011) 385–402 Edge-transitive maps of low genus∗ Alen Orbanic´ Faculty of Mathematics and Physics, University of Ljubljana Jadranska 19, 1000 Ljubljana, Slovenia Daniel Pellicer Faculty of Mathematics and Physics, University of Ljubljana Jadranska 19, 1000 Ljubljana, Slovenia Tomazˇ Pisanski Faculty of Mathematics and Physics, University of Ljubljana Jadranska 19, 1000 Ljubljana, Slovenia Thomas W. Tucker Department of Mathematics, Colgate University, Hamilton, NY 13346, USA Received 6 November 2009, accepted 4 August 2011, published online 7 December 2011 Abstract Graver and Watkins classified edge-transitive maps on closed surfaces into fourteen types. In this note we study these types for maps in orientable and non-orientable surfaces of small genus, including the Euclidean and hyperbolic plane. We revisit both finite and infinite one-ended edge-transitive maps. For the finite ones we give precise description that should enable their enumeration for a given number of edges. Edge-transitive maps on surfaces with small genera are classified in the paper. Keywords: Edge-transitive map, edge-transitive tessellation, map on surface, symmetry type graph. Math. Subj. Class.: 52C20, 05B45, 05C12 ∗Dedicated to Branko Grunbaum¨ on the occasion of his 80th birthday. E-mail addresses: [email protected] (Alen Orbanic),´ [email protected] (Daniel Pellicer), [email protected] (Tomazˇ Pisanski), [email protected] (Thomas W. Tucker) Copyright c 2011 DMFA Slovenije 386 Ars Math. Contemp. 4 (2011) 385–402 1 Introduction In this paper we are interested in maps (see [16] or [5, Section 17.10] for definitions) whose automorphism group acts transitively on the edges. In addition to combinatorial finite maps on compact surfaces we consider also the (geometric) tessellations of the spherical, Euclidean or hyperbolic plane, where every automorphism can be extended to an isometry of the tessellation. We shall consider only locally finite, one-ended tessellations (see [7, 18,3]). For multi-ended planar maps, as well as for definitions of one- and multi-ended maps, see [6]. Additionally, unless explicitly specified, we will consider non-degenerate maps and tessellations, i.e. maps with all valences, co-valences and Petrie valences at least 3. An edge in a map has edge-symbol (f1 ·f2; v1 ·v2) if it is incident to vertices of valence v1 and v2 and faces of size f1 and f2 (see [9]). A map is called edge-homogeneous if all its edges have the same edge symbol (f1 · f2; v1 · v2). Note that this requires vertex valences to alternate between v1 and v2 around each face, and face sizes to alternate between f1 and f2 around each vertex. In particular, v1; v2; f1; f2 must satisfy the condition: if v1 or v2 is odd, then f1 = f2, and if f1 or f2 is odd, then v1 = v2. Call (f1 · f2; v1 · v2) admissible if it satisfies this condition with all numbers at least 3. An edge-transitive map is obviously edge-homogeneous. The converse is not true; for example, the map on the torus obtained by identifying opposite sides of an m × n chessboard has constant edge-symbol (4·4; 4·4), but it is not edge-transitive when m 6= n, since there is no map automorphism taking ‘horizontal’ edges to ‘vertical’ edges. On the other hand we have the following geometric result of Grunbaum¨ and Shepherd [10]: Theorem 1.1. Given any admissible edge-symbol (f1 · f2; v1 · v2), there is a unique (up to map isomorphism) universal tessellation M(f1 ·f2; v1 ·v2) of the sphere, Euclidean plane, or hyperbolic plane by congruent f1-gons and f2-gons, such that all map automorphisms are isometries and the map is edge-transitive with the given edge-symbol. The map has reflections with respect to any line joining a vertex and the center of an incident face. If f1 = f2 it also has a reflection with respect to the lines determined by the edges and if v1 = v2 it has reflections with respect to any line connecting centers of incident faces. The proof is mostly a matter of constructing an f1-gon and f2-gon with congruent sides and the appropriate internal angles. The theorem has the following consequences: Corollary 1.2. If the finite map M is homogeneous with edge symbol (f1 · f2; v1 · v2), there is a finitely generated group A of fixed-point free automorphisms of the universal tessellation M(f1 · f2; v1 · v2), such that M is isomorphic to the quotient M(f1 · f2; v1 · v2)=A. If A is a normal subgroup of the automorphism group of M(f1 · f2; v1 · v2) then the map M is edge-transitive. Proof. The covering transformations of the universal map M(f1 · f2; v1 · v2) of M are fixed-point free automorphisms. When A is normal, each automorphism of the universal map projects to an automorphism of M. Corollary 1.3. Any finite, edge-homogeneous map M is regularly covered by a finite edge- transitive map. Proof. Given a map M let A be any group such that M = M(f1 · f2; v1 · v2)=A. The group A is determined up to conjugacy and has finite index in the automorphism group Γ of M(f1 · f2; v1 · v2). Let B be the intersection of all conjugates of A in Γ. Then B is A. Orbanic´ et al.: Edge-transitive maps of low genus 387 normal of finite index in Γ. It follows that M(f1 · f2; v1 · v2)=B is an edge-transitive finite map regularly covering M. The proposition below follows directly from Theorem 1.1 and [9]. Proposition 1.4. Every edge-symbol that can be realized by a finite map on a closed surface with negative Euler characteristic is admissible for a tessellation of the hyperbolic plane. 2 Combinatorial vs. geometric edge-transitivity The notion of edge-transitivity of maps or tessellations may be understood with two dif- ferent meanings: combinatorial and geometric, depending on the choice of combinatorial automorphism or the geometric symmetry group. Our point of departure is the combinatorial meaning where the map M is considered as an abstract collection of flags together with three fixed-point free involutions that we denote by 0,1,2 (see [5]) with the product 02 being a fixed-point free involution, too. This is equivalent to identifying the map M with its 3-edge-colored flag graph G(M) in which the flags of M are the vertices of G(M) and the edges of color i are determined by the involution i. Clearly the only axiom of maps is observed as the fact that a 3-edge-colored 3-regular graph results from a map if and only if it is connected and the 02-two-factor of G(M) is ‘quadrilateral’, i.e. consists entirely of 4-cycles. Here automorphisms are defined as bijections of flags that preserve the colored adjacencies. However, they can also be viewed as color-preserving automorphisms of G(M). This is a general description of a finite map on a closed surface. We recall the following important facts: Theorem 2.1. Let M be a map with e edges and let AutM be its automorphism group or order m = jAutMj. Let k be the number of its flag-orbits. • There are 4e flags. • For any pair of flags there is at most one automorphism mapping one flag into the other. • Let a be a flag-orbit of M, let i be one of the three involutions and let b = i(a): Then b is a flag-orbit and i restricted to a is a bijection between a and b. • The group of automorphisms AutM partitions the set of flags into k equal sized orbits. The size of each orbit is equal to m, where km = 4e. However, if we work with tessellations T of the sphere, Euclidean plane or hyperbolic plane we may consider the additional structure of metric space and require the edges to be given as segments of geodesics. The transitivity in this case is considered with respect to the group of isometries fixing T . Note that any tessellation T of the hyperbolic plane H can be embedded as a combinatorial map on the Euclidean plane E, however the group of automorphisms of T that extend to isometries of H in general does not coincide with the group of automorphisms of T that extend to isometries of E. For example, it is possible to tessellate the hyperbolic plane with equilateral triangles, seven of them around each vertex. This tessellation, denoted by T3;7, has the Coxeter group 2 3 7 [3; 7] = hρ0; ρ1; ρ2 j (ρ0ρ2) = (ρ0ρ1) = (ρ1ρ2) = idi as its symmetry group; see [11]. In particular, the group [3; 7] acts transitively on the faces of T3;7. It is also possible to partition the Euclidean plane into triangles, seven of them 388 Ars Math. Contemp. 4 (2011) 385–402 around each vertex; however, in this case the tiling cannot be normal in the sense of [10, Section 3.2] (see also [18, p. 174]), and therefore, the symmetry group cannot be transitive on the faces. Following the discussion in the previous section, every edge-transitive map on the pro- jective plane is the quotient of a universal edge-transitive map on the sphere by the antipo- dal involution; each edge-transitive map on the torus is the quotient of a universal edge- transitive map on the Euclidean plane by the group generated by two linearly independent translations; and every edge-transitive map on the Klein bottle is the quotient of a universal edge-transitive map on the Euclidean plane by the group generated by two glide reflections with congruent translation component and parallel axes.
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