Regular Maps of High Density

Discrete & Computational Geometry, Mar 2017

A regular map is a surface together with an embedded graph, having properties similar to those of the surface and graph of a platonic solid. We analyze regular maps with reflection symmetry and a simple graph with ratio between vertex-degree and number of vertices strictly exceeding \(\frac{1}{2}\). We conclude that all regular maps of this type belong to a family of maps naturally defined on the Fermat curves \(x^n+y^n+z^n=0\), excepting the one corresponding to the tetrahedron.

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Regular Maps of High Density

Rob H. Eggermont 0 1 Maximiliaan Hendriks 0 1 0 Zermelo Roostermakers , Schuttersveld 9, 2316 XG Leiden , The Netherlands 1 Department of Mathematics, University of Michigan , 1830 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043 , USA A regular map is a surface together with an embedded graph, having properties similar to those of the surface and graph of a platonic solid. We analyze regular maps with reflection symmetry and a simple graph with ratio between vertex-degree and number of vertices strictly exceeding 21 . We conclude that all regular maps of this type belong to a family of maps naturally defined on the Fermat curves x n +yn +zn = 0, excepting the one corresponding to the tetrahedron. Editor in Charge: János Pach Rob H. Eggermont Maximiliaan Hendriks Regular maps; Density; Fermat family 1 Introduction Objects of high symmetry have been of interest for a long time. Thousands of years ago, the Greeks already studied the platonic solids, regular convex polyhedra with congruent faces of regular polygons such that the same number of faces meet at each vertex. The Greeks proved that there are only five of them: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. One can consider the platonic solids to be orientable surfaces of genus 0 with an embedded graph satisfying certain properties. The natural question is then whether this concept can be generalized. This leads to the notion of regular maps. Regular maps have also been studied for many years (see for example [8]). In contrast to the succinct and complete list of platonic solids, a complete classification for regular maps seems far off, but at least a few families of regular maps are known. Investigations into these objects is of great interest, amongst other reasons because of their intriguing connection to algebraic curves. Regular maps are special cases of dessins d’enfants [7]. As such, the combinatorial structure of vertices, edges, and faces gives rise not only to a topological realisation, but even to a unique algebraic curve! Explicitly computing an ideal that defines some complex projective realisation of this algebraic curve for a given regular map is an ongoing area of research. Two happy examples of a whole family of regular maps for which the algebraic curves are known, are naturally defined on the Fermat curves x n + yn + zn = 0. For a given Fermat curve one has to consider the action on this complex algebraic curve (and hence a real surface) of its group of algebraic automorphisms, which turn out to be realizable by linear maps in C3. A map presentation (see Sect. 2) of these maps was described concisely by Coxeter and Moser (see [4]), although they did not hint at the link to the Fermat curves, and were perhaps not aware of it. We give a more detailed description of the Fermat family in Sect. 2. Regular maps can broadly be divided into two classes: chiral and reflexive. We will deal only incidentally with chiral maps. In low genus, all regular maps have been computed, with a list of reflexive maps up to genus 15 appearing in [3] and with more recent lists at [2], which runs up to genus 301 for reflexive maps at the time of writing. The combinatorial aspect of a regular map that is the focus of this paper is its (graph) density, defined in Sect. 2. One discovers from studying the maps of low genus, that having a high density is relatively rare. In fact, the only reflexive regular maps of low genus with simple graphs of density strictly exceeding 21 are members of the Fermat family (every vertex in its graph being connected to precisely two thirds of the vertices), the sole exception being the tetrahedral map, by which we mean the regular map corresponding to the natural embedding of the tetrahedron in the sphere. This naturally leads to the question whether this is true in higher genus as well. This paper answers this question by classifying all reflexive regular maps with simple graphs of density strictly exceeding 21 . Theorem 1.1 (Regular map density theorem) A regular map on v vertices with a simple underlying graph and degree strictly exceeding v2 is either chiral, or a member of the Fermat family, or the tetrahedral map. One interesting corollary of this theorem is the following. All complete graphs Kn can be embedded as regular maps for n ≥ 3. Whereas K3 can be embedded as the simplest Fermat map, and K4 yields the tetrahedral map, all regular embeddings for n ≥ 5 must be chiral. A more straightforward proof of that fact can also be found in [5]. Before proving this theorem in Sect. 3, we will give the definition of a regular map, show some properties of regular maps, and define the Fermat maps. 2 Background Let us now give a more formal definition of regular maps. Definition 2.1 A map M is an orientable surface together with an embedded connected finite graph with non-empty vertex set V and non-empty edge set E such that the complement of Γ in M is a finite disjoint union of open discs. Each of these open discs is called a face of Γ . If v ∈ V, e ∈ E and f is a face of Γ , we call any pair of these incident if one of the pair is contained in the closure of the other. A cellular homeomorphism of M is a homeomorphism of M that induces a graph automorphism of Γ . An automorphism of M is an equivalence class of cellular homeomorphisms under the equivalence relation of isotopy. We denote the set of automorphisms of M by Aut(M), and we write Aut+(M) for the set of orientationpreserving automorphisms of M. We call M chiral if Aut+(M) = Aut(M) and we call M reflexive otherwise. Below we will refer to reflexive maps with the symbol R to stress that this property is assumed. A map M is called regular if for all directed −→e, −→e ∈ E there is an orientationpreserving automorphism of M mapping −→e to −→e. Example 2.2 The platonic solids correspond to regular maps, simply by interpreting the traditional terms ‘vertices’ and ‘faces’ according to our definition above, and defining the embedded graph by rereading the term ‘edges’. The tetrahedral map, the one resulting from the tetrahedron, plays an important role in this article. Notation 2.3 If M is a map, we denote by Σ (M), Γ (M), V(M) and E(M) the corresponding surface, graph, vertex set and edge set. The set of faces of M is denoted F(M). Note that any cellular homeomorphism is uniquely determined up to cellular isotopy by the graph isomorphism it induces (although in general, not all graph isomorphisms are representable by a cellular homeomorphism). Proof Suppose e is an edge such that e and e share a common vertex. Let v be a vertex incident to both e and e and consider a local picture around v. The only way for φ to fix −→e and to preserve orientation is to act as the identity locally. In particular, this means it must fix −→e as well. Using connectedness of Γ (M), it follows that φ fixes all directed edges and hence is the identity as a graph isomorphism. This means φ = Id. As a direct corollary, any orientation-preserving automorphism of a map is uniquely determined by the image of a single directed edge. Similarly, any automorphism of a e1 Rotation around f f1 f2 e f2 f1 e f1 f2 e Fig. 1 Rotations and reflections. An indication of the action is given by curved arrows in the left and middle figure (rotations), and by the two-way arrows in the right figure (reflection) map is uniquely determined by the combination of its orientation and the image of a single directed edge. Regular maps have many other nice properties. For example, any face has the same number of edges, and any vertex has the same number of edges. Moreover, if a face f of a regular map M has p edges, and if we label the edges in counterclockwise order as e1, . . . , e p, then there is a unique orientation-preserving automorphism of M mapping e1 to e2. It fixes f by necessity. We can see it as a counterclockwise rotation around f of minimal degree. Likewise, if a vertex v of M has q edges, we can define a counterclockwise rotation around v of minimal degree. More details on these constructions can be found for example in [6]. Notation 2.5 Let M be a regular map. Let f be a face of M with counterclockwise orientation. We always use the letter p to denote the number of edges of f , and we denote the counterclockwise rotation around f of minimal degree as described above by R f . Similarly, given a vertex v of M, we always use the letter q to denote the number of edges of v. We write Sv for the counterclockwise rotation around v of minimal degree. We observe that R f has order p, and that any orientation-preserving automorphism of M fixing f is a power of R f . Likewise, Sv has order q, and any orientationpreserving automorphism of M fixing v is a power of Sv. Let M be a reflexive regular map, and let e be an edge of M. It can be shown that there are two orientation-reversing automorphisms that fix e as a non-directed edge. We will call these reflections. A visualization of rotations and reflections can be seen in Fig. 1. As an example, suppose v and w are adjacent vertices, and after fixing an orientation, let f be the face to the left of the oriented edge (−v−,−w→). The rotation Sv around v maps (−v−,−w→) to (−v−,−S−v−(−w→)), and the rotation R f around f maps the latter to (−w−−,→v). This means the automorphism R f ◦ Sv inverts the oriented edge (−v−,−w→). As a consequence, it has order 2. In particular, for any regular map M, if f is a face adjacent to a vertex v, then (R f ◦ Sv)2 = Id. Lemma 2.6 Let M be a regular map, let v be a vertex of M, and let f be a face of M incident to v. Then Sv and R f generate Aut+(M). This lemma can be proved by showing that Sv and R f can map a given directed edge to any other directed edge. A proof can be found in [5, Lem. 1.1.6]. For convenience, when we have v ∈ V(M) and f a face of M incident to v, we occasionally use S and R instead of Sv and R f . An important consequence of the lemma is that we can pin down a regular map M without explicitly talking about its topology. We can give a presentation of Aut+(M) in the generator pair (R, S). We call this a standard map presentation of M. Example 2.7 We have |Aut+(T)| = 12, where T denotes the tetrahedral map. Both the number of edges incident to a vertex and the number of edges incident to a face are 3. This means q = 3 and p = 3. Let v ∈ V(T) and f be a face incident to v. We find the obvious relations S3 = Id and R3 = Id. Moreover, we have the relation (S R)2 = 1, from the fact that S R reverses the edge (v, R−1(v)). It turns these are the only relators necessary to define a standard map presentation: Aut+(T) = R, S | R3, S3, (R S)2 . Remark 2.8 The isomorphism type of Aut+(M) is insufficient to determine a regular map M. Using Rg.n to denote the nth map in genus g according to the listing from [2], here are a few examples of regular maps with the same group of orientationpreserving automorphisms. To start with, the regular maps R3.1 and R10.9 both have Aut+(M) ∼= PSL(2, 7). Even fixing the triple (Aut+(M), p, q), and thereby also the genus, does not necessarily determine a unique regular map. The counterexamples, tuplets, are of special interest (a different story altogether). The first examples occur in genus 8 (the twins R8.1 and R8.2) and genus 14 (the first Hurwitz triplet, R14.1, R14.2, R14.3). A regular map is, however, completely determined by a standard map presentation, as is treated in [6]. Proposition 2.9 There is a categorical equivalence between regular maps M as cell complexes with cellular maps, and group presentations of the form R, S | R p, Sq , (R S)2, r1, . . . , rn , the rk being extra relators. The morphisms of the second are group homomorphisms that respect the conjugacy classes of R and S in two such presentations. A small theory of Aut+(M)-equivariant cellular morphisms between regular maps can be developed, as described in [5, Sect. 1.6]. The notion happily coincides with taking certain quotients of Aut+(M), and a result we will use later in Lemma 3.14 and Proposition 3.15 is the following. Proposition 2.10 Suppose M is a regular map and H a normal subgroup of Aut(M) that is contained in Aut+(M) and does not contain an automorphism that reverses some edge of M. Then Aut(M)/H is the automorphism group of a regular map M satisfying Γ (M) = Γ (M)/H and F(M) = F(M)/H . There is a branched cellular covering M → M with the fiber of a cell of M a coset of H . Each cell of M contains at most one ramification point, and each cell of M contains at most one branch point. These numbers only depend on the dimension of the cell. Here, by Γ (M)/H we mean the graph obtained by identifying vertices respectively edges of Γ (M) if they lie in the same H -orbit, and by F(M)/H we mean the set obtained by identifying faces of F(M) if they lie in the same H -orbit. The proposition follows from the work of [6]. A detailed proof can be found in [5]. Example 2.11 (Fermat maps) For n ∈ Z>0, let Gn = For each n, the group Gn is the group of orientation-preserving automorphisms of a regular map that we call the Fermat map Fer(n), obtained by considering the solutions of x n + yn + zn = 0, acted upon by its algebraic automorphism group. We omit the proof for this claim, but we do note that Fer(n) is a reflexive regular map, and that this is a remarkable property. The group structure can be described as Gn =∼ Zn2 Sym3. The graph Γ (Fer(n)) is the full tripartite simple graph Kn,n,n on 3n vertices. And since R has order 3, the map is a triangular embedding of Kn,n,n. Its genus turns out to be n −21 . A visualization of the first Fermat maps can be seen in Fig. 2. Definition 2.12 Let M be a regular map and let v, v ∈ V(M). Then the distance between v and v is the minimal length of a path in Γ (M) from v to v ; it is denoted by d(v, v ). The set of vertices at distance at most i from v is denoted D(v, i ). The set of vertices at distance precisely i from v is denoted by ∂D(v, i ). The density δ(M) of M, which is the central notion for the rest of this paper, is defined as Note that δ(M) does not depend on the choice of v by the regularity of the map. We will focus mainly on regular maps with simple graphs, in which case the density of the graph is simply the vertex-degree of any vertex divided by the total number of vertices. However, any regular map is a cellular cover of a map with simple graph, and the latter has the same density as the original map. Thus, while our main theorem only mentions reflexive regular maps with simple graph, it pins down the possibilities for density larger than 21 to cellular covers of the tetrahedral map and Fermat maps. Observe that if R is a regular map with simple underlying graph, then stating R has density strictly exceeding 21 is the same as stating that R has v vertices and degree strictly exceeding v2 . In order to prove our regular map density Theorem 1.1, we will proceed to show that the Fermat maps and the tetrahedral map are the only reflexive regular maps with simple graph of density strictly exceeding 21 . 3 The Regular Map Density Theorem Let R be a regular map. Let us write V = V(R) for convenience. We start with a rather technical but crucial lemma: 1. There is k ∈ Z such that Sv = Sk j . Moreover, k is well-defined and invertible j q v modulo gcd( j,q) . 2. Let g ∈ Aut(R). Suppose Sv = Sk j . Then Sg(v) = Sgk(jv ). Moreover, Svj fixes g(v) j j v if and only if it fixes g(v ) and in this case, if Svj = Sgl(v), then Svj = Sgk(lv ). 3. Let k ∈ Z be such that Svj = Svk j . For all i ∈ Z, we have Svj = SSkvij(v ) and Svj = SSjvi (v). 4. Let g1, . . . , gn ∈ Aut(R) and suppose there is vi ∈ V such that Svj fixes both vi and gi (vi ) for all i ∈ {1, 2, . . . , n}. Then Svj fixes gn gn−1 . . . g1(v ). j fixes g1(v1). The latter is true by assumption, so Sv fixes g1(v ). Claim 4 now follows by induction. While we formulated the previous lemma rather technically, the statements should be seen in a more geometrical and intuitive way. Any rotation Svj0 that fixes a vertex v1 must be a rotation around v1 of the same order (Claim 1). Moreover, such a situation translates well under conjugation (Claims 2 and 3). Finally, the set of fixed points of an orientation-preserving automorphism Svj is closed under the set of automorphisms g that map at least one fixed point of g to another fixed point of g (Claim 4). We will use this lemma often. From here on, we will work under the following assumption. Assumption 3.2 R is a reflexive regular map with a simple graph. An easy graph lemma gives us a point of departure to say something about regular maps with high density. Lemma 3.3 If Γ (R) has density δ(R) ≥ 21 , then Γ (R) has diameter at most 2. Proof Let v0, v1 ∈ V. Suppose that d(v1, v0) ≥ 2. Then ∂D(v0, 1) and ∂D(v1, 1) are contained in V − {v0, v1}, which has cardinality |V | − 2. Since we have |∂D(v0, 1)| + |∂D(v1, 1)| ≥ |V | > |V | − 2, we find that v0 and v1 share at least two common neighbors. In particular, this implies d(v0, v1) ≤ 2. In the following proposition, we will show that the faces of reflexive regular maps of high density are triangles ( p = 3). Proof Let v ∈ V and let f be a face incident to v. Consider v1 = R2f (v) and suppose v1 ∈/ ∂D(v, 1). Then the same holds for Svi(v1) for any i ∈ {0, 1, . . . , q − 1}, because it preserves distances. In particular, the size of the orbit under Sv of v1 is at most |V \ ∂D(v, 1)|. We have |V| < 2q because δ(R) > 21 . Therefore, |V \ ∂D(v, 1)| = |V | − q < 2q − q = q. So the size of the orbit of v1 under Sv is strictly smaller than q, and hence there is j ∈ {1, . . . , q − 1} satisfying Svj (v1) = v1. Note that on one hand, Svj does not fix any neighbor of v, and since δ(R) > 21 , j we find Sv has at most |V | − q < q fixed points. On the other hand, observe that for v = R f (v), we have v1 = Sv−1(v). By Claim 4 of Lemma 3.1, using gi = Sv−1, we find Svj fixes Svi (v) for any i , so Svj fixes all neighbors of v , meaning it has at least q fixed points, a contradiction. We conclude v1 ∈ ∂D(v, 1). Label the neighbors of v by the elements of Z/qZ, counterclockwise and let fi be the face on the left of (−v−, →i) (consistent with the orientation). Suppose S−1(v) = i . Since R is reflexive and the reflection that fixes the ori0 ented edge (−−v−,→0) maps i to −i , we find S0(v) = −i . Observe that this means that Si (v) = Svi S0 Sv−i (v) = 0. A visualization of the situation can be seen in Fig. 3. We now see two faces on the left of (−0−, →i), being f0 and fi−1. We conclude f0 = fi−1. Since v occurs only once on each face (using the assumption that Γ (R) is simple), we conclude i = 1 and hence p = 3, as was to be shown. −i f i−1 Fig. 3 Proof of Proposition 3.4 −1 −i −i fi−1 = f 0 −1 −i Fig. 4 Diagonal neighbors and diagonal alignment Definition 3.5 Assume we have a regular map with p = 3, and let v ∈ V. A diagonal neighbor of v is an element of V of the form Sw2(v) with w a neighbor of v. Let v ∈ V as well. We call v and v diagonally aligned if there is a sequence (v0, v1, . . . , vn) of elements of V satisfying v0 = v, vn = v and for all i ∈ {1, . . . , n} the vertex vi is a diagonal neighbor of vi−1. If v ∈ V is a vertex of a triangle vwu, then wu is an edge of precisely one other triangle, say wuv . In this case, v is a diagonal neighbor of v. All diagonal neighbors of v are of this form (Fig. 4). The observation that v is a diagonal neighbor of v if and only if v is a diagonal neighbor of v shows that being diagonally aligned is an equivalence relation. We write v ∼ v if v and v are diagonally aligned. It is easy to see that being diagonally aligned is preserved by graph isomorphisms. For v ∈ V, we write Vv = {v ∈ V : v ∼ v}. It is an easy exercise to see that if vwu is a face of Γ (V), then any element of V is diagonally aligned to at least one of v, w or u (and in fact, either Vv, Vw and Vu are pairwise disjoint or Vv = Vw = Vu = V). Definition 3.6 Let R be a regular map. We define J to be the minimal element of {1, 2, . . . , q} such that SvJ fixes all elements in Vv. We call J the primitive period of R. Note that J exists because Svq = Id. Moreover, J is independent of choice of v by Claim 2 of Lemma 3.1 (using the fact that v ∼ v if and only if g(v) ∼ g(v ) for all v, v ∈ V and all gq ∈ Aut(R)), so we are justified in not using a subscript. Thirdly, J divides q, since Sv = Id fixes Vv pointwise. Assumption 3.7 From this point on, we add to our previous assumptions (R is a reflexive regular map with simple graph) that R satisfies δ(R) > 21 . In particular, we will have p = 3 and q ≥ 2. Let v ∈ V. The following lemma shows that we can classify Vv as the set of fixed points of SvJ . 1. The element J is the minimal element in {1, . . . , q} such that SvJ fixes v . 2. We have J = q if and only if v ∈ ∂D(v, 1). 3. We have J = q if and only if Vv = V. 4. If w ∈ V is fixed by SvJ , then w ∈ Vv. 5. If J < q, then Vv ∩ ∂D(v, 1) = ∅, q is even, and δ(R) ≤ 23 . 6. If q is even, then J < q. Proof Claim 1 can be shown by induction. Suppose there is J such that SvJ fixes v . Then by Claim 3 of Lemma 3.1, it fixes any diagonal neighbor of v (since all of these are of the form Svi(v )), and analogously it fixes any diagonal neighbor of any diagonal neighbor of v, etcetera. Hence SvJ fixes Vv, and hence J divides J . Clearly, SvJ fixes v , so this shows Claim 1. For Claim 2, note that Sv induces bijections on the sets {v}, ∂D(v, 1) and ∂D(v, 2). Note that ∂D(v, 1) has cardinality q and both {v} and ∂D(v, 2) have cardinality less than q since δ(R) > 21 . In particular, if v ∈/ ∂D(v, 1), its orbit under the powers of Sv has cardinality strictly less than q, and hence J < q. Conversely, if v ∈ ∂D(v, 1), then it is fixed by S J , and therefore J = q. v For Claim 3, note that if Vv = V, then it contains a neighbor of v, and hence J = q. Conversely, if J = q, then v is a neighbor of v by the second part. Since Vv is the equivalence class of v under ∼, any rotation around v fixes it as a set. Hence all neighbors of v are elements of Vv. But then |Vv| ≥ q > 21 |V|, and hence Vv = V (using the fact that either |Vv| = 31 |V| or Vv = V). For Claim 4, observe that if SvJ fixes w, then it also fixes Vw. If Vw = Vv, then Vw contains a neighbor of v, and hence J = q. This gives a contradiction, since by Claim 3, Vv = V and hence w ∈ Vv. For Claim 5, suppose J < q. If Vv ∩ ∂D(v, 1) = ∅, then SvJ fixes a neighbor of v, which gives a contradiction. So Vv ∩ ∂D(v, 1) = ∅. Let w be a neighbor of v and consider the set Sw2i (v) : i ∈ 0, 1, . . . , q −21 of cardinality 1 + q −21 . All of these elements lie in Vv. If q is odd, then this set contains Swq−1(v), which is a neighbor of v, a contradiction. Hence q is even, and this set has cardinality q2 . In particular, we find |V \∂D(v, 1)| ≥ q2 , and hence δ(R) ≤ q+qq/2 = 23 . For Claim 6, suppose q is even and J = q. Then v ∈ ∂D(v, 1). Let w ∈ ∂D(v, 1) such that v = Sw2(v), and note that there is a directed edge (−−w−,−S−w−(−v→)). Consider the (unique) reflection that maps this edge to (−−S−w−(v−)−,−w→). Since q is even, this reflection does not fix any neighbor of v. On the other hand, it fixes v , a contradiction. We can now show that the only reflexive regular map of density greater than 21 with q odd is the tetrahedral map. After showing this, we can focus on the case where q is even. Proposition 3.9 Suppose δ(R) > 21 and the vertex-degree q of R is odd. Then R is the tetrahedral map, and δ(R) = 43 . Proof Let v ∈ V. By Lemma 3.8, the primitive period J of R is equal to q. Number the neighbors of v by 0, 1, . . . , q − 1 clockwise and let v = S02(v), a diagonal neighbor of v. It is a neighbor of v by the second part of Lemma 3.8. Now, consider the reflection that maps that maps (−−0,−q−−−−→1) to (−−q−−−−1−,→0). It fixes v and v . Moreover, it maps any neighbor i of v to −i − 1 mod q. We conclude v = q −21 . Observe that we have a face 0, q − 1, q −21 . Applying the rotation Sv(q+1)/2 yields the triangle q +21 , q −21 , 0. Because rotations preserve orientation, we conclude q +21 ≡ q −1 mod q. Since q > 2, we conclude q +21 = q − 1 and hence q = 3. We now easily deduce R is the tetrahedral map. From the above proposition, we conclude the following. Every regular map R satisfying the conditions in Assumption 3.7 will either be the tetrahedral map, or have even vertex-degree q. In the latter case, we can say a few things about R using Claims 5 and 6 of Lemma 3.8. Assumption 3.10 From here on, we’ll assume R is a reflexive regular map with simple graph satisfying δ(R) > 21 that is not equal to the tetrahedral map. In particular, we will have p = 3 and q is even. Moreover, we have J < q, δ(R) ≤ 23 , and for all v, no neighbor of v is diagonally aligned to v. Suppose vwu is a face of Γ (R). Then Sv maps Vw to Vu and vice versa. In particular, Sv2 is a bijection of Vw and Vu (and of Vv of course). This motivates us to give the following definition: Definition 3.11 Let j := lcm( J, 2). We call j the even period of R. Lemma 3.12 For any v, w ∈ V, we have [Svj , Swj ] = Id and [Svj , Sw4] = Id. Proof If v ∼ w, we have Svj is a rotation around w, and hence it commutes with Sw, which implies both of the relations. Suppose v w. Note that Svj fixes Vv pointwise and hence for any v ∈ Vv, we have [Svj , Swj ](v ) = Sv Sw Sv− j Sw− j (v ) = Sw Sw− j (v ) = v , using the fact that Sw j j j j fixes Vv as a set and Svj fixes Vv pointwise. Likewise, [Sv , Swj ] fixes Vw pointwise. j There is v ∈ Vv such that v is a neighbor of w. Now [Svj , Sw] fixes the edge (−−v−,−w→) j and hence it is Id. Also, we have [Svj , Sw4] = Svj SS−w4j(v). Let v = Sw2(v ). We see that both v and Sw4(v ) are diagonal neighbors of v . There is k such that Svj = Svk j by Claim 1 of Lemma 3.1. Note that k only depends on the fact that v is a diagonal neighbor of v . This means that SSw4(v ) = Svk j as well, since v is a diagonal neighbor of Sw4(v ). We j conclude SSw4(v ) = Svj and hence [Svj , Sw4] = Id. This implies [Svj , Sw4] = Id. j Let v ∈ V and let v be a diagonal neighbor of v. We have Svj = Svk j for some k, uniquely defined modulo qj . By Claim 2 of Lemma 3.1, we find k is independent of choice of v and v (as long as they are diagonal neighbors). In particular, we find Svj = Svk j and hence k2 ≡ 1 mod qj . We will show something stronger, namely the following. Lemma 3.13 Let v ∈ V and let v be a diagonjalj nejighbor of v. Then Svj = Svj . Moreover, for any face vuw of Γ (R), we have Sv Su Sw = Id. Proof Let w be a neighbor of both v and v such that Sw2(v) = v . Label the neighbors of w counterclockwise with the elements of Z/qZ with v0 = v. We have Svi = Svi+4m for any m ∈ Z using the fact that Svji commutes with Sw4 by j j the previous lemma. Note that Vv ∩∂D(w, 1) = {v2m : m ∈ Z} and Vv1 ∩∂D(w, 1) = {v2m+1 : m ∈ Z}. j We claim that Sv fixes Vv1 ∩ ∂D(w, 1) as a set. Note that it fixes Vv1 as a set since j is even. Suppose Svj0 (vi ) is not a neighbor of w for some i . Then also Svj4m (vi+4m ) is not a neighbor of w (because Svj4m (vi+4m ) is simply Sw4m (Sv0 (vi )), and rotations around w j preserve distance to w). Since Sv4m = Svj , we find that Sv0 (vi+4m ) is not a neighbor j j of w for any m. This means there are at least q4 distinct elements in Vvi at distance 2 of w. Moreover, there are q2 distinct elements in Vvi at distance 1 of w. Hence we find |Vvi | ≥ q2 + q4 = 43 q > 38 |V| > 13 |V| = |Vvi |, which gives a contradiction. This j shows that Sv fixes Vvi ∩ ∂D(w, 1) as a set for all i . Suppose Svj (v1) = v1+ j1 and Svj (v−1) = v−1+ j−1 . Note that Svj (v±1+4m ) = v±1+4m+ j±1 , using Sv = Sw4m Svj Sw−4m . j Observe that Sw− j1 Svj fixes v1, hence it is equal to Svx1 for some x . Consider the reflection σ that maps (−v−,−w→) to (−w−−,→v). It fixes v1. Conjugating the equality Sw− j1 Svj = Svx1 with σ gives Svj1 Sw− j = Sv−1x . Together, these give the equality Sw− j1 Svj Svj1 Sw− j = Id, or in other words, Svj+ j1 = Swj1+ j . This means that the rotation S j+ j1 fixes w, which is a neighbor of v, and hence it is Id. We conclude v j1 = − j modulo q. Analogously, we can show j−1 = − j modulo q, so we find Svj (v2m+1) = v2m+1− j for all m. Repeating this argument for Sv , we find Svj (v2m+3) = v2m+3− j for all m. This j means Sv = Svj , as both of these fix v and map v1 to v1− j . This shows the first part j of the lemma. For the second part, let u = v1 and note that the equality Sw− j1 Svj = Svx1 is simply j j the equality Sw Sv = Sux . Conjugating with the rotation around vuw that maps v to u yields Svj Suj = Swx. Together, these equations give Swj Swx Su− j = Sux , and hence Swj+x = Suj+x . We conclude x = − j modulo q as before, and hence Swj Svj = Su− j . j j j From here, we easily find Sv Su Sw = Id, as was to be shown. Lemma 3.14 Suppose j = q. Then R = Fer(1). With this lemma, the following proposition is now within our reach. Proposition 3.15 Suppose 21 < δ(R) ≤ 23 . Then δ(R) = 23 . Lemma 3.12 and the fact that Svx j S wyj = Id if and only if both x and y are 0 mod qj . Note that any conjugation of Svj or Swj is a rotation around some element of V of order qj . In other words, any conjugation of these elements is of the form Suk j . If u ∈ Vv or u ∈ Vw, this is an element of H by definition of j . If u ∈/ Vv ∩ Vw, we have Suk j = Sv−k j Sw−k j ∈ H by Lemma 3.13. We conclude that H is normal in Aut(R). Moreover, H is generated by any pair Svj , Swj with Vv , Vw distinct. By Proposition 2.10, Aut(R)/H corresponds to another regular map. Note that |H v| = qj since Svj fixes v and | Swj v| = qj . By the previous remark, |H v | = qj for all v ∈ V. Moreover, if two edges e, e are incident to v, then H e = H e if and only if e = Svk j e for some k ∈ Z. Let R be the regular map corresponding to Aut(R)/H . By the above remarks, we have |V(R)| = |V| qj and q(R) = j . Observe that R has a simple graph. Indeed, suppose that two edges H e, H e in Γ (R) have the same vertices, say H v1 and H v2. Then in the original graph, we have e = (v1, v2) and e = (v1, v2) with H v1 = H v1 and H v2 = H v2. Since H v1 = H v1, we may assume v1 = v1 by picking another representative if necessary. Since there is an edge from v1 to v2, we have Vv1 = Vv2 , and hence we have H v2 = Svj1 v2. This means that v2 = Svk1j (v2) for some k ∈ Z. Since Γ (R) is simple, we conclude e = Svk1j (e) as well, and hence H e = H e . We now find δ(R) = δ(R). Moreover, because we assumed |V| > 3, we have R = Fer(1) and hence j = q by Lemma 3.14. This means |V(R)| < |V|. By our induction hypotheses, we have δ(R) = δ(R) = 23 as desired. This concludes the proof. Proof We have ∂D(v, 1) ⊆ Vw ∪ Vu by Lemma 3.8 (since R is not the tetrahedral map, hence J < q), and |Vw ∪ Vu | = 23 |V|. As |∂D(v, 1)| = q = 23 |V| by the previous lemma, the result must hold. Proposition 3.17 Suppose δ(R) = 23 . Then j = 2. Proof Let v, w ∈ V be neighbors. Label the neighbors of w by 0, 1, . . . , q − 1 clockwise with v = 0. By our assumption on δ(R), the elements of Vv are precisely elements of the form 2i mod q. Let u = Sv(w). Suppose Sv(2i ) = 2i . Then Sv(2i + j ) = Sv(Swj (2i )) = Suj (Sv(2i )) = Suj (2i ) = 2i − j , using the fact that Suj Swj = Sv− j acts as the identity on Vv. This means Sv acts on congruence classes of 2i mod j . Moreover, we have Sv(2 + k j ) = −(2 + k j ) for any k ∈ Z because Sv(2) = Sv(Sw2(v)) = Sw−2(v) = −2. Let ι ∈ Z>0 be minimal such that Svι(2) ∼= 2 mod j . Since the number of even congruence classes modulo j is 2j , we have ι ≤ 2j . We now have Svι+1(2) = −Svι(2). Let σ be the reflection that fixes (−v−,−w→). It maps i to −i and moreover, it maps Svι(2) to Sv−ι+1(2). In particular, we can conclude Svι+1(2) = −Svι(2) = σ Svι(2) = Sv−ι+1(2). It follows Sv2ι fixes 2 and hence it fixes Vv pointwise. This means J |2ι and hence j |2ι. So ι ≥ 2j . This however means ι = 2j . In particular, 2, Sv(2), . . . , Svι−1(2) are 2j distinct even congruence classes modulo j , so it must be all of them. Note however that Sv fixes the congruence class 0 mod j . Since some power of Sv maps 2 to an element that is 0 mod j , we see that 0 must the only even congruence class mod j . This is only possible if j = 2, as was to be shown. We now restate and prove our main theorem. Theorem 3.18 Suppose R is a regular map with simple graph satisfying δ(R) > 21 . Then either R is chiral, or R is the tetrahedral map, or R = Fer(n) for some n ∈ Z>0. Proof Suppose R is reflexive, and suppose R is not the tetrahedral map. We have p = 3 by Proposition 3.4. By Lemma 3.8 and Proposition 3.9, we find q is even and δ(R) ≤ 23 . Write q = 2n. By Lemma 3.15, we have δ(R) = 23 and by Proposition 3.17, we find j = 2. Acknowledgements The first author is supported by Jan Draisma’s Vidi grant from the Netherlands Organisation for Scientific Research (NWO). 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Regular Maps of High Density, Discrete & Computational Geometry, 2017, DOI: 10.1007/s00454-017-9879-6