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VERTEX-TRANSITIVE MAPS on a TORUS Contents List of Figures 1 VERTEX-TRANSITIVE MAPS ON A TORUS ONDREJ SUCH· Abstract. We examine free, vertex transitive actions on semiregular tilings. By taking quotients by lattices we then obtain various families of Cayley maps on a torus, and describe the presentations of structure groups. Altogether there are 29 families, 5 arising from the orientation preserving wallpaper groups and 2 from each of the remaining wallpaper groups. We prove that all vertex- transitive maps on torus admit a Cayley map structure. Contents List of Figures 1 List of Tables 2 1. Introduction 2 1.1. Preliminaries 3 2. FVT actions of wallpaper groups on semiregular tilings 5 2.1. Actions of subgroups of p4 m 5 2.2. Actions of subgroups of p6 m 13 2.3. Nonexistence of actions 18 3. Families of toric groups 21 3.1. Quotients of p1 and p2 21 3.2. Quotients of pm, pg, pmm, pmg and pgg 22 3.3. Quotients of cm and cmm 22 3.4. Quotients of p3 ; p4 ; p6 22 3.5. Quotients of p4 m and p4 g 23 3.6. Quotients of p3 1m, p3 m1 and p6 m 23 3.7. List of presentations 23 3.8. Non-redundancy of families 28 4. Cayley and vertex-transitive maps on torus 28 4.1. Cayley maps on the torus 28 4.2. Vertex-transitive maps that are not Cayley 29 4.3. Proulx examples 29 5. Open questions 30 References 30 List of Figures 1 Semi-regular tilings 6 2 Subgroups of p4 m 7 Date: December 30, 2008. Key words and phrases. Torus, wallpaper group, vertex-transitive map, Cayley map, semireg- ular tiling. 1 2 ONDREJ SUCH· 3 Actions of the group p1 7 4 Actions of the group p2 8 5 Action of the group pm on the square lattice 8 6 Actions of the group pg 9 7 Action of group p4 9 8 Action of the group pmm 10 9 Actions of the group cm 11 10 Actions of the group pmg 12 11 Actions of the group pgg 13 12 Actions of the group pgg 14 13 Action of group p4 g 14 14 Actions of group cmm 15 15 Action of group p4 m on the truncated square tiling 15 16 Groups containing rotation by 2¼=3 15 17 Action of group p3 16 18 Action of the group p6 16 19 Action of group p3 1m 17 20 Action of the group p3 m1 17 21 Action of p6 m on the great rhombitrihexagonal tiling 18 22 Proulx toroidal Cayley graphs 30 List of Tables 1 Crosstable of FVT actions 19 2 List of 29 toric group families 24 1. Introduction This paper has expository nature. It is devoted to the nice interplay between various symmetric objects arising from the following diagram: semiregular tiling T with Cayley/vertex plane R2 o transitive action of wallpaper group ¡ ² ² a map T =¤ with Cayley/vertex torus R2=¤ o transitive action of ¯nite group G This diagram can be constructed in two ways. Starting from the top, one may choose a lattice ¤ invariant under ¡ and construct quotients. Alternatively, one can start at the bottom, and construct the universal cover space of the torus. All VERTEX-TRANSITIVE MAPS ON A TORUS 3 objects involved are very symmetric, thus they have been of continued interest in pure and applied mathematics. The torus, as the most symmetric Riemann surface, has enormous importance in the theory of modular forms, culminating in the proof of Shimura-Taniyama- Weil conjecture. As a consequence of this work, proof of celebrated Fermat's last theorem has been achieved. Semiregular tilings [WiTi] arise when studying analogues to Archimedean solids on the torus. Various de¯nitions can be given, isometric or topological. Interest- ingly on the torus the local notion of identical local type and the global notion of vertex-transitivity coincide. This can be contrasted with the case of sphere, where there is a solid with vertices of the same local type, which is not vertex transitive [WoGy]. Wallpaper groups [WiWa] classify symmetries of two-dimensional patterns. They have played great importance in crystallography so much so, they are also known as plane crystallographic groups. One way to understand structure of wallpaper groups is to construct their Cayley graphs or even better Cayley maps in which relation words can be read from closed walks in the graph. This approach is shown for instance in [CoMo]. Vertex-transitive maps on a torus received attention for various reasons. The underlying maps provide examples of several classes of ¯nite groups, as studied for instance in [CoMo]. Vertex transitive maps were also studied in contexts related to the Babai conjecture ([Thom], [Baba]). There is also continuing interest in symmetric toric maps from chemistry (see e.g. [MaPi]). Vertex-transitive maps on the torus are also used as the underlying structure for Kohonen neural networks [WiKo]. Let us make a brief survey of several related prior works. Well-known is the work of Burnside [Burn, 203-209] in which he describes the orientation preserving groups acting on a torus. Baker [Bake] in his work shows presentations of most groups with action on a torus. He omits some families of groups, for instance for the semiregular tiling of local type 3:4:6:4 he omits families p3 m13, p3 1m3. A classical work of Coxeter-Moser [CoMo] in Section 8.3 also provides lists of presentations. However, quotients of pmm and pmg listed there are in error. Proulx in her work [Prou] made a careful study of which Cayley graphs can be drawn on the torus. The list of groups she provides however does not give explicit presentations. A result in a similar vein can be found in [GrTu], Theorem 6.3.3. Thomassen in his work [Thom] gives a topological description of vertex-transitive maps on the torus. Our work is structured into three parts. In the ¯rst we describe wallpaper groups with various Cayley actions on semiregular tilings. We shall prove that except for the pairs (T ; ¡) desribed in Table 1, no other group actions exist (Theorem 2). The basis for the second part is a theorem of Tucker that allows us to explicitly enumerate all ¯nite groups acting on a torus. This explicit enumeration allows to show that there is no redundant family in our list. In the third part, we list all Cayley maps induced by Cayley actions of wallpaper groups on semiregular tilings. We prove that all vertex-transitive maps on the torus are in fact Cayley. This should be contrasted with the case of higher genus ([KaNe1], [KaNe2]). 4 ONDREJ SUCH· The author would like to thank R. Nedela for suggesting the problem, and J. Karab¶a·s,A. Rosa, V. Proulx, T. Kenney for their indispensable help during prepa- ration of the paper. 1.1. Preliminaries. 1.1.1. Lattices. By a lattice we mean a rank 2 Z submodule ¤ of C. If ! is a nonzero complex number then ! ¢ ¤ := f! ¢ zjz 2 ¤g is also a lattice. There are two special lattices that we will often consider. Firstly, the square lattice ¤¤ corresponding to Gaussian integers, that is to complex numbers of the form n1 + n2 ¢ i, where n1; n2 are integers. Secondly, the triangle lattice ¤4 is the lattice corresponding to complex numbers of form p ¡1 + 3 n + n ¢ ; 1 2 2 where again n1 and n2 are integers. These two lattices are distiguished by the following property. Proposition 1. Let ¤ be a lattice. Then the following are equivalent a) A lattice ¤ is invariant under a rotation by an angle Á with 0 < Á < ¼ b) ¤ = !¤4 or ¤ = !¤¤ for some nonzero complex number !. Let us now de¯ne several more classes of lattices ² a square-like lattice is a lattice of form ! ¢ ¤¤ with complex ! 6= 0, ² a triangle-like lattice is a lattice of form ! ¢ ¤4 with complex ! 6= 0, ² a real lattice is a lattice invariant under a reflection, ² a rectangular lattice is a lattice generated by a pair of orthogonal vectors, ² a rhombic lattice is a lattice generated by a pair of vectors of equal length. The latter two classes of lattices can be given other characterizations. The set N¤ := fkzk j z 2 ¤g is a discrete subset of the real line. Order the elements of N¤: 0 = c0 < c1 < c2 < : : : : If the norm c1 is achieved for six elements of ¤, then ¤ is triangle-like. If either of the norms c1, c2 is achieved by four elements, then ¤ is rhombic. If the four shortest nonzero vectors are §w1; §w2 with w1, w2 orthogonal, then the lattice is rectangular. Alternatively, one can use the modular j function from the theory of elliptic curves. The function associates to any lattice ¤ a complex number j(¤) that uniquely identi¯es lattices modulo the equivalence relation ¤ » !¤. One has equivalences: ² ¤ is square-like i® j(¤) = 1728, ² ¤ is triangle-like i® j(¤) = 0, ² ¤ is real i® j(¤) is real, ² ¤ is rectangular i® j(¤) is real with j(¤) ¸ 1728, ² ¤ is rhombic i® j(¤) is real with j(¤) · 1728. 1.1.2. Space groups. By a wallpaper group ¡ we mean a group of isometries of the plane that acts discretely with compact quotients. At present Wikipedia [WiWa] provides an excellent resource on the subject of wallpaper groups. There are sev- enteen isomorphism classes of groups. Their presentations can be found in the classical work of Coxeter and Moser [CoMo]. VERTEX-TRANSITIVE MAPS ON A TORUS 5 The simplest one of them is the group p1 , which is just the free abelian group on two generators. The subgroup of translations T (¡) in ¡ is always isomorphic to p1 , and its index in ¡ is · 12.
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