#### Regular Maps of High Density

Discrete Comput Geom
Regular Maps of High Density
Rob H. Eggermont 0 1
Maximiliaan Hendriks 0 1
Editor in Charge: János Pach
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. Mathematics Subject Classification 05C25 · 05C42 · 05C75 · 20F65 · 57M15
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).
Lemma 2.4 Let M be a map. Suppose φ ∈ Aut+(M) fixes some directed edge −→e.
Then φ = Id.
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
υ
e2
eq
Sυ
e1
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 =
R, S | R3, S2n, (R S)2, [R, S]3 .
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
δ(M) :=
|∂D(v, 1)|
|V(M)|
.
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.
Example 2.13 The Fermat maps all have density 23 . The tetrahedral map has density
43 .
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:
Lemma 3.1 Let v, v ∈ V. Suppose we have j ∈ Z such that Svj fixes v . Then the
following claims hold.
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 ).
Proof For Claim 1, note that any orientation-preserving element that fixes v is a
rotation around v , say Svj = Svi . The order of these rotations must be equal, and
in this case is equal to the smallest positive integer x such that x j ≡ 0 mod q,
which is gcd(qj,q) , using the fact that (Svj )x = 1 precisely if x j is a multiple of q.
q
Analogously, we observe that the order of these rotations must be equal to gcd(i,q) ,
meaning gcd(i, q) = gcd( j, q). In particular, both i and j are integer multiples of
q
gcd( j, q), and moreover, these multiples must be invertible modulo gcd( j,q) (if not,
q
the order of i or j modulo q would be strictly smaller than gcd( j,q) ). Clearly, there
q
exists k, well-defined and invertible modulo gcd( j,q) , such that j k ≡ i mod q. This
shows Claim 1.
For Claim 2, we have the equalities Sgj(v) = g Sv± j g−1 = g Sv±k j g−1 = Sgk(jv ), where
the sign in the exponent corresponds to the orientation of g being positive or negative.
If Svj fixes g(v) respectively g(v ), it is a power of Sg(v) respectively Sg(v ) (which
j j
is itself a power of Sg(v)). In either case, Svj fixes both g(v) and g(v ). Moreover, if
j
Svj = Sgl(v), we have Svj = Sgk(lv ). This completes the proof of Claim 2.
The equality Sv = SSkvij(v ) is easily seen because of Claim 2, using g = Svi. The
j
equality Svj = SSjvi (v) follows by symmetry. This shows Claim 3.
To prove Claim 4, note that by Claim 2, we have Svj fixes g1(v ) if and only if it
j
fixes g1(v). Exchanging the roles of v and v1, we have Sv fixes g1(v) if and only if it
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).
Proposition 3.4 Suppose δ(R) > 21 . Then 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.
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 .
Lemma 3.8 Let v ∈ V and let v be a diagonal neighbor of v. Then the following
claims hold.
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
Svj (v2m+1) = v2m+1− j for all m.
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
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
).
q
Proof We have J < q because R is not the tetrahedral map. Hence we have J = 2
and J is odd. For all v, v ∈ V, we have Sv2 (v) ∈ Vv, and hence Sv2 SvJ Sv−2 is a rotation
2
of order 2 that fixes v, and hence it is S J . This means that [SvJ , Sv ] = Id for all
v
v, v ∈ V. Let v, w ∈ V be neighbors and consider H = SvJ , SwJ . Let u = SvJ (w);
note that u ∈/ Vv ∪ Vw. Now SvJ maps Vw to Vu and vice versa. Likewise, SwJ maps
Vv to Vu and vice versa.
It is easily verified that SvJ SwJ SvJ = SuJ = SwJ SvJ SwJ (where the latter part follows
from the fact that SuJ = SuJ for all u ∈ Vu ). We now easily verify that H ∼= Sym(
3
)
and H is a normal subgroup of Aut(R) (any conjugation of S J , SwJ is a rotation of
v
order 2 around some vertex, and all such rotations are elements of H ).
Observe that |H v| = 3; this follows from the fact that |H v| contains elements from
Vv, Vw and Vu and SvJ fixes v.
The argument now splits into two cases: either H v = {v, w, u} or w, u ∈/ H v.
Assume the latter holds. Because |H v| = 3 and SvJ (w) = u, the orbit of the edge
vw has size 6 and hence H reverses no edges; we may apply Proposition 2.10. This
tells us that Aut(R)/H is the automorphism group of a reflexive regular map R with
|V(R)| = |V| , q(R) = q2 with simple graph. However, this means δ(R) = 23 δ(R) > 43 ,
3
which is not possible. We conclude that H v = {v, w, u}.
Let u ∈ Vu such that vwu is a face of R with Su (v) = w. Observe that SuJ
interchanges v and w and hence the same is true for SuJ . We find Su ((−u−−,→v)) =
(−−u−,−w→) = SuJ ((−u−−,→v)). It follows J = 1 and hence q = 2. Together with p = 3, we
immediately see 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 .
Proof We apply induction to the cardinality of V. If R satisfies |V(R)| ≤ 3, then R
has density 23 simply because 23 is the only possible fraction dn satisfying 21 < dn ≤ 3
2
and d ≤ 3. Suppose |V| > 3 and assume the proposition is true for all R with
|V(R )| < |V|.
Consider H = Svj , Swj . Note that H is a subgroup of Aut(R) of order qj 2, using
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.
Corollary 3.16 If 21 < δ(R) ≤ 23 and v, w, u ∈ V with V = Vv ∪ Vw ∪ Vu , then
∂D(v, 1) = Vw ∪ Vu .
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.
Let v ∈ V and let w be a neighbor of v. Let f be the face on the left of (−v−,−w→). We
claim that [Sv , R f ]3 = Id.
Note that [Sv , R f ] = Sv Sw−1. We have Sv Sw−1(v) = Sv2(w). Note that
Sw−1(Sv2(w)) = Sw(Sv2(w)) = Sv−2(w) using the fact that Sv2(w) ∈ Vw. So
(Sv S−1)2(v) = Sv−1(w). We have Sv Sw−1 Sv−1(w) = v, showing [Sv , R f ]3 = (Sv Sw−1)3
w
fixes v.
On the other hand, we have [Sv , R f ]−3 = (Sw Sv−1)3. By symmetry of the situation,
it fixes w. But then [Sv , R f ]3 also fixes w. This immediately implies [Sv , R f ]3 = Id
as claimed.
It now follows that Aut+(R) is a quotient group of S, R | S2n = Id, R3 =
Id, (S R)2 = Id, [S, R]3 = Id using S = Sv and R = R f . So Aut+(R) is a quotient
group of Aut+(Fer(n)). On the other hand, we have |Aut+(Fer(n))| = 6n2 = 3n ·
2n = |V| · q = |Aut+(R)|. This means Aut+(R) = Aut+(Fer(n)) and hence R =
Fer(n), as was to be shown.
Acknowledgements The first author is supported by Jan Draisma’s Vidi grant from the Netherlands
Organisation for Scientific Research (NWO). This paper describes the parts of [5] that are joint work of the two
authors.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
1. Brehm , U. : Maximally symmetric polyhedral realizations of Dyck's regular map . Mathematika 34 ( 2 ), 229 - 236 ( 1987 ). doi: 10 .1112/S0025579300013474
2. Conder , M. : Non-orientable, chiral orientable and reflexible orientable regular maps . http://www.math. auckland.ac.nz/~conder/
3. Conder , M. , Dobcsányi , P. : Determination of all regular maps of small genus . J. Comb. Theory Ser. B 81 ( 2 ), 224 - 242 ( 2001 ). doi: 10 .1006/jctb. 2000 .2008
4. Coxeter , H.S.M. , Moser , W.O.J. : Generators and Relations for Discrete Groups . Ergebnisse der Mathematik und ihrer Grenzgebiete , vol. 14 , 4th edn. Springer, Berlin ( 1980 )
5. Hendriks , M. : Platonic Maps of Low Genus . PhD thesis . Technische Universiteit Eindhoven, Eindhoven ( 2013 ). doi: 10 .6100/96ec9f06-75df - 4a6f - 92f6-3f531b56155e
6. Jones , G.A. , Singerman , D. : Theory of maps on orientable surfaces . Proc. Lond. Math. Soc . 37 ( 2 ), 273 - 307 ( 1978 ). doi: 10 .1112/plms/s3- 37 .2. 273
7. Schneps , L . (ed.): The Grothendieck Theory of Dessins D'Enfants. London Mathematical Society Lecture Note Series , vol. 200 . Cambridge University Press, Cambridge ( 1994 )
8. Širánˇ , J.: Regular maps on a given surface: a survey . In: Klazar, M. et al. (eds.) Topics in Discrete Mathematics. Algorithms and Combinatorics , pp. 591 - 609 . Springer, Berlin ( 2006 ). doi: 10 .1007/ 3-540-33700-8_ 29