ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.06
https://doi.org/10.26493/1855-3974.2503.f17
(Also available at http://amc-journal.eu)
Classification of thin regular map
representations of hypermaps*
Antonio Breda d’Azevedo , Domenico A. Catalano †
Department of Mathematics, University of Aveiro, Aveiro, Portugal
Received 12 December 2020, accepted 13 January 2023, published online 22 August 2023
Abstract
There are two well-known map representations of hypermaps, namely the Walsh and
the Vince map representations, that are dual to each other. They correspond to normal sub-
groups of index two of a free product Γ = (C2×C2)∗C2 which decompose as “elementary”
free product C2 ∗C2 ∗C2. However, Γ has three normal subgroups that decompose as “el-
ementary” free product C2 ∗ C2 ∗ C2, the third of these subgroups giving the less known
Petrie-path map representation. By relaxing the “elementary” free product condition to free
product of rank 3, and under the extra condition “words of smaller length” on the genera-
tors, we prove that the number of map representations of hypermaps increases to 15 (up to
a restrictedly dual), all of which are described in this paper.
Keywords: Map representation, hypermaps, maps, regularity, restricted regularity, orientably regular.
Math. Subj. Class. (2020): 05C10, 05C25, 05C65, 05E18, 20F65
1 Introduction
Using maps to describe hypermaps is not new. The well-known Walsh [8] bipartite map
representation uses a bipartite map M to describe a hypermap H by interpreting the two
monochromatic vertices of the map as hypervertices and hyperedges (respectively), and
the faces of M as the hyperfaces of H. The Vince 2-face bipartite map [7], a dual of a
bipartite map, also describes a hypermap by assigning the two monochromatic faces to
hyperedges and hyperfaces respectively, and vertices to hypervertices. These are two, out
*The authors thank the support given by the Portuguese funds through the CIDMA - Center for Research and
Development in Mathematics and Applications and the Portuguese Foundation for Science and Technology (FCT-
Fundação para a Ciência e a Tecnologia) within project UIDB/04106/2020.
†Corresponding author.
E-mail addresses: breda@ua.pt (Antonio Breda d’Azevedo), domenico@ua.pt (Domenico A. Catalano)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 24 (2024) #P1.06
of three, Θ-marked map representations realised by an index 2 normal subgroups Θ of the
free product
Γ = ∆(∞, 2,∞) = ⟨R0, R1, R2 | R20, R21, R22, (R0R2)2⟩ = C∞ ∗ (C2 × C2) ,
which are isomorphic to ∆ = ∆(∞,∞,∞) = ⟨S0, S1, S2 | S20 , S21 , S22⟩ (see [1] and
section 3). They are namely, Γ2.4 = ⟨RR01 , R1, R2⟩ and Γ2.1 = ⟨R0, R1, R
R2
1 ⟩. The third
subgroup of Γ of index 2 isomorphic to ∆ is Γ2.5 = ⟨RR01 , R1, R0R2⟩ (see Subsection 4.2).
This induces the third less known representation, succinctly described in [2], given by Γ2.5-
marked maps. In this representation Petrie-path-bipartite maps represent hypermaps by
assigning the two monochromatic Petrie polygons (closed zig-zag paths turning alternately
left and right) to hypervertices and hyperedges, and faces to hyperfaces.
More generally, a regular representation of hypermaps by maps is given by an epimor-
phism ρ from a finite index normal subgroup Θ of Γ to ∆.
This paper is inspired by the work of Lynne James on map representation of topological
categories (see [5]) and is organised as follow: In Section 2 we give an introduction to the
theory of hypermaps and maps focusing on restrictedly marked hypermaps and maps, a
theory developed in [1]. In particular, we focus on Θ-marked maps for normal subgroups
Θ of finite index in Γ. Section 3 is devoted to define the notion of clean and thin Θ-marked
representation of a hypermap by a map. As we will focus on Θ-marked representations for
rank 3 normal subgroups Θ of Γ, in Section 4 we derive a rank formula and classify the
rank 3 normal subgroups of Γ. The rank formula is derived using presentations for NEC
groups (see [3]). Last section is devoted to thin representations (given in Table 2) and its
geometric description by means of an example.
In what follows by “representation” we always mean “regular representation”. Note
that we use right notation, that is, we denote by xf the image of x by the function f .
2 Preliminaries
Hypermaps are 4-tuples H = (F ; r0, r1, r2) where F is a finite set and r0, r1, r2 are invo-
lutory permutations of F (r2i = 1) generating a transitive group on F . The elements of F
are called flags and the transitive group Mon(H) = ⟨r0, r1, r2⟩ is the monodromy group of
H. The orbits of the action of the subgroups of Mon(H) generated by {r0, r1, r2} \ {ri}
for i = 0, 1, 2 are respectively the hypervertices, hyperedges and hyperfaces of the hyper-
map H, called respectively 0-cells, 1-cells and 2-cells of H. The valency of an i−cell is
the length of the orbit of one of its flags by rjrk where {i, j, k} = {0, 1, 2}. If, for some
positive integers k, ℓ, m all hypervertices have valency k, all hyperedges have valency ℓ
and all hyperfaces have valency m, then we say that H is a uniform hypermap (of type
(k, ℓ,m)). In this case, (k, ℓ,m) = (|r1r2|, |r2r0|, |r0r1|), where |g| denotes the order
of g. If r0, r1 and r2 have no fixed point then we say that H has no boundary. Thus, a
uniform hypermap H = (F ; r0, r1, r2) without boundary has V = |F |2|r1r2| hypervertices,
E = |F |2|r2r0| hyperedges and F =
|F |
2|r0r1| hyperfaces.
A morphism or covering from the hypermap H1 = (E; r0, r1, r2) to the hypermap
H2 = (F ; s0, s1, s2) is a function ϕ : E → F satisfying
xriϕ = xϕsi,
for any x ∈ E and any i ∈ {0, 1, 2}. We say that the hypermap H1 covers the hypermap H2
if there is a covering from H1 to H2. It is straightforward to see that any covering is onto
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 3
and uniquely determined by the image of a flag. Injective coverings are therefore called
isomorphisms. An automorphism of H is an isomorphism from H to itself. We will denote
by Aut(H) the set of automorphisms of H, which is obviously a group under composition.
Topologically, a hypermap H can be seen as a triangulation of a compact surface S
with vertices labelled 0, 1 and 2 such that each triangle (a flag of H) has labels 0, 1 and 2
assigned to its vertices; the vertices labelled 0, 1 and 2 are respectively the hypervertices,
hyperedges and hyperfaces. For each x ∈ F the two triangles x and xri share the common
edge e opposite to the vertices labelled i if x ̸= xri; if x = xri, then the edge e is on
the boundary of S and so S is a bordered surface. This triangulation is a topological map
representation of hypermaps whose dual is the James topological map representation of
hypermaps [4]; here the faces are labelled 0 (grey faces), 1 (dotted faces) and 2 (white faces)
(see Figure 3). The hypermap H has (no) boundary if and only if S has (no) boundary. The
characteristic of H is the Euler characteristic of S. In particular, if H = (F ; r0, r1, r2) is a
uniform hypermap without boundary, then the Euler characteristic of H is
χ(H) = |F |
2
(
1
|r1r2|
+
1
|r2r0|
+
1
|r0r1|
− 1
)
.
Alternatively, a hypermap is a cellular embedding of a hypergraph in a compact connected
surface.
The monodromy group Mon(H) of a hypermap H is a quotient of the triangle group
∆. Hence we have an epimorphism π : ∆ → Mon(H) and an action
F ×∆ → F, (x, d) 7→ x(dπ)
of ∆ on the set F of flags of H. The stabiliser H of a flag under this action is a sub-
group of ∆ called a hypermap subgroup of H. As the action of ∆ is transitive, hyper-
map subgroups of H are conjugate. The hypermap H is then isomorphic to the hypermap
(∆/H;H∆R0, H∆R1, H∆R2), where ∆/H denotes the set of right cosets of a hypermap
subgroup H of H in ∆, H∆ is the normal core of H in ∆ and (Hd)H∆Ri = HdRi for
any d ∈ ∆ and any i ∈ {0, 1, 2} (see, for instance [1]).
Let Θ be a normal subgroup of finite index n in ∆ and let H be a hypermap with
hypermap subgroup H . Then Θ acts (as a subgroup of ∆) on the set F = ∆/H of flags
of H partitioning it into at most n orbits, called Θ-orbits; in fact, suppose that H is not a
subgroup of Θ and let b ∈ H \ Θ. Then Hb = H and bΘ ̸= Θ. Therefore the Θ-orbit
{Hbt : t ∈ Θ} is equal to the Θ-orbit {Ht : t ∈ Θ}, forcing the number of Θ-orbits being
at most n. The number of Θ-orbits is n if and only if H < Θ; in this case we say that
H is Θ-conservative. A Θ-conservative hypermap H is Θ-regular if the group AutΘ(H)
of automorphisms preserving Θ-orbits acts transitively on each Θ-orbit, or equivalently, if
H is normal in Θ. However, if H is normal in ∆, then H is a regular hypermap, that is,
∆-regular. We shall say that a hypermap H is restrictedly-marked if it is Θ-conservative for
some normal subgroup Θ of finite index in ∆. Ought to emphasise that not every hypermap
is restrictedly-marked (see [1] for examples).
A hypermap (F ; r0, r1, r2) satisfying (r0r2)2 = 1 is called a map. The hypervertices,
hyperedges and hyperfaces of a map are called vertices, edges and faces, since topologically
a map is a cellular embedding of a graph on a compact surface. The monodromy group of
a map M is then a quotient of the “right” triangle group Γ. This group acts on the set
of flags of M via the canonical projection π : Γ → Mon(M) sending Ri to ri. The
4 Ars Math. Contemp. 24 (2024) #P1.06
stabiliser of a flag under this action will be called a map subgroup of M. Keeping the
same notation as already used for hypermaps, we have that a map M is then isomorphic
to the map (Γ/M ;MΓR0,MΓR1,MΓR2), where M is a map subgroup of M. The theory
of restrictedly-marked maps unfolds in the same way as the theory of restrictedly-marked
hypermaps by taking finite index normal subgroups Θ of Γ instead of ∆. The group Γ is a
free product of C2 = ⟨R1⟩ with D2 = ⟨R0, R2⟩ and by the Kurosh’s Subgroup Theorem,
any normal subgroup Θ of Γ freely decomposes uniquely (up to a permutation of factors)
in a (indecomposable) free product (see [6] page 243 and 245)
C2 ∗ · · · ∗ C2 ∗D2 ∗ · · · ∗D2 ∗ C∞ ∗ · · · ∗ C∞ =
⟨A1⟩ ∗ · · · ∗ ⟨As⟩ ∗ ⟨B1, C1⟩ ∗ ⟨Bt, Ct⟩ ∗ ⟨Z1⟩ ∗ · · · ∗ ⟨Zu⟩
for a certain numbers s, t and u of factors ⟨Ai⟩ = C2, ⟨Bj , Cj⟩ = D2 and ⟨Zu⟩ = C∞
respectively, whereas s, t or u may be zero. Let m = s+ 2t+ u = rank(Θ) and let
{A1, . . . , As, B1, . . . , Bt, C1, . . . , Ct, Z1, . . . , Zu} = {X1, . . . , Xm}.
Then a Θ-conservative map M with map subgroup M can be represented by the Θ-marked
map
Q = (Ω;x1, . . . , xm),
where Ω = Θ/M is the set of right cosets of M in Θ and x1, . . . , xm are permutations of
Ω generating a group G acting transitively on Ω such that the function
X1 7→ x1 , . . . , Xm 7→ xm
extends to an epimorphism from Θ to G.
Any Θ-regular map M covers the regular map
TΘ = (Γ/Θ;ΘR0,ΘR1,ΘR2),
called the Θ-trivial map. As TΘ is a regular map, we have that
• any two vertices of TΘ have same valency, say k,
• any two edges of TΘ have same valency, say l ∈ {1, 2},
• any two faces of TΘ have same valency, say m.
The triple (k, l,m) is called the type of the regular map TΘ. As M is Θ-regular and covers
TΘ, we also have that:
• the vertices of M covering a vertex v of TΘ also have same valency, say kv (which
is a multiple of k),
• the faces of M covering a face f of TΘ also have same valency, say mf (which is a
multiple of m).
Denoting by V , E and F the sets of vertices, edges and faces of TΘ and assuming that M
has no boundary, then this together with Euler formula gives that the characteristic of M is
χ(M) = |Θ : M |
2
∑
v∈V
µv
k
kv
+
∑
e∈E
µe
l
2
+
∑
f∈F
µf
m
mf
− |Γ : Θ|
 , (2.1)
where µv = 1 or 2 according as the vertex v is on the boundary or not and similarly for µe
and µf . For details we refer the reader to [1].
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 5
3 Thin map representations of hypermaps
Let Θ be a finite index normal subgroup of Γ of rank 3 and let {X1, X2, X3} be a set of
generators of Θ. The pair R = (Θ, {X1, X2, X3}) will be called a Θ-marked representa-
tion (of hypermaps by maps) if the function
X1 7→ S0 , X2 7→ S1 , X3 7→ S2
extends to an epimorphism ρ from Θ onto ∆. We call ρ the canonical epimorphism of the
representation R. Two representations (Θ1, {X1, X2, X3}) and (Θ2, {Y1, Y2, Y3}) are to
be considered equal if Θ1 = Θ2 = Θ and their canonical epimorphisms ρ1, ρ2 : Θ → ∆
are such that ρ1 = ιρ2 for some inner automorphism ι of Θ. For example, since S0, S1, S2
are involutions, inverting one or more generators of R give the same representation.
Given a hypermap H with hypermap subgroup H , setting Ω = {(Hρ−1)t : t ∈ Θ} and
xi : Ω → Ω, (Hρ−1)t 7→ (Hρ−1)tXi, i = 1, 2, 3
we get a Θ-marked map (Ω;x1, x2, x3) called a Θ-marked map representation of H.
Remark 3.1. In fact, denoting by N the normal core of Hρ−1 in Θ, the group G =
⟨x1, x2, x3⟩ is isomorphic to Θ/N by an isomorphism φ mapping xi to NXi for any
i ∈ {1, 2, 3}. Hence πφ−1, where π : Θ → Θ/N is the canonical epimorphism, is an
epimorphism from Θ to G extending the function X1 7→ x1, X2 7→ x2, X3 7→ x2. Remark
also that ρ induces a bijection ρ̃ from Ω to {Hd : d ∈ ∆} which sends (Hρ−1)t to H(tρ)
and satisfies xi ρ̃ = ρ̃ ri−1, where ri−1 maps Hd to HdRi−1 for any i ∈ {1, 2, 3}. Thus,
we say that ρ̃ is an isomorphism from the Θ-marked map representation (Ω;x1, x2, x3) of
H to H.
A (Θ-marked) representation R = (Θ, (X1, X2, X3)) will be called clean if Θ is the
free product of the cyclic groups ⟨X1⟩, ⟨X2⟩, ⟨X3⟩, in which case we write
Θ = ⟨X1⟩ ∗ ⟨X2⟩ ∗ ⟨X3⟩ .
A clean representation is called thin if the sum of the lengths of its generators (as words in
the free group over {R0, R1, R2}) is minimal.
The number of rank 3 normal subgroups Θ of Γ is finite, but there are infinitely many
clean representations given by all possible sets {X1, X2, X3} such that Θ = ⟨X1⟩ ∗ ⟨X2⟩ ∗
⟨X3⟩. On the other hand the number of thin representations is finite (see Sections 4 and 5).
4 The rank 3 normal subgroups of Γ
4.1 Rank computation (see also [3])
In order to compute the rank of a normal subgroup Θ of finite index in Γ, we remark that Γ
acts as a group of isometries on the hyperbolic plane H, regarding its generators R0, R1,
R2 as the reflections on the geodesics given in Figure 1 in the Poincaré disk model.
6 Ars Math. Contemp. 24 (2024) #P1.06
R2
R0
R1
Figure 1: The generators of Γ as hyperbolic reflections.
The action of Θ on H gives rise to a quotient orbifold H/Θ which is a punctured
surface (with or without boundary) punctured at the vertices and at the face centers of the
regular map M = (Γ/Θ;ΘR0,ΘR1,ΘR2) with underlying surface S. If ΘR0 = ΘR2,
then the covering H → H/Θ is also branched at the edge centers of M. The group Θ,
being the fundamental group of H/Θ, has a presentation P with p + 2 − χ generators
X1, . . . , Xp, Y1, . . . , Y2−χ, where p is the total number of punctures and branching points
of H/Θ and χ is the characteristic of S. The presentation P has a relator S = X1 · · ·Xp ·
k∏
i=1
Yi · W (Yk+1, . . . , Y2−χ), where
⟨Y1, . . . , Y2−χ | W (Yk+1, . . . , Y2−χ)⟩
is a presentation of the fundamental group of the surface S with k boundary components
(setting
k∏
i=1
Yi = 1 if k = 0), and eventually e relators X21 , . . . , X
2
e if ΘR0 = ΘR2, where
e is the number of edges of M.
Hence rank(Θ) = p+ 2− χ− 1 = p+ 1− χ. More precisely:
• If S has no boundary (k = 0) and is non-orientable, then 2 − χ is the genus g of S,
W (Y1, . . . , Y2−χ) = W (Y1, . . . , Yg) =
g∏
i=1
Y 2i and therefore
S = X1 · · ·Xp ·
g∏
i=1
Y 2i .
• If S has no boundary and is orientable, then 2 − χ is even and the genus g of S is
2−χ
2 . Replacing (Y1, . . . , Y2−χ) by (A1, B1, . . . , Ag, Bg) we have
S = X1 · · ·Xp ·
g∏
i=1
[Ai, Bi] .
In the particular case of χ = 2 (sphere) the word W (Y1, . . . , Y2−χ) is empty and
therefore
S = X1 · · ·Xp.
• If S has boundary, then {R0, R1, R2}∩Θ ̸= ∅. Thus, any triangle of the triangulation
of S given by the flags of M has at least an edge on the boundary, since Θ is normal
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 7
in Γ. This shows that S is a closed disk, that is, a bordered surface on a sphere
with only one boundary component (k = 1). Hence χ = 2 − k = 1 and therefore,
setting Y = Y1 we have that
k∏
i=1
Y1 = Y , W (Yk+1, . . . , Y2−χ) is the empty word
and ⟨Y ⟩ ∼= C∞ is the fundamental group of S. Hence
S = X1 · · ·Xp · Y .
In particular, rank(Θ) = p in this case.
The next proposition relates the rank of Θ with its index n in Γ for n > 4. Relating rank
with all indices will give a clumsy formula which does not give more information about the
index bound for fixed rank.
Proposition 4.1. If Θ is a normal subgroup of finite index n > 4 in Γ, then n is even and
rank(Θ) =

1 + n if H/Θ has boundary and branching points;
1 + n2 if H/Θ has boundary and no branching point;
or H/Θ has no boundary but has branching points;
1 + n4 if H/Θ has no boundary and no branching point.
( in this case n is a multiple of 4).
Proof. Using the above notations and remarks we have the following:
If S has boundary, then ΘR1 ̸= Θ since |Γ/Θ| = n > 4. Hence Γ/Θ = ⟨ΘR1,ΘRj⟩ for
some j ∈ {0, 2}, that is, Γ/Θ is dihedral of even order n. The total number of vertices and
faces of the map M = (Γ/Θ;ΘR0,ΘR1,ΘR2) is then 1 + n2 . This gives
rank(Θ) = p =
{
1 + n2 if H/Θ has no branching points,
1 + n if H/Θ has branching points,
since in the case when H/Θ has branching points, M has n2 edges.
If S has no boundary, then n is a multiple of 4 and from Euler formula we have that
χ =
{
p− n4 if H/Θ has no branching points,
p− n2 if H/Θ has branching points.
Therefore
rank(Θ) = p+ 1− χ =
{
1 + n4 if H/Θ has no branching points,
1 + n2 if H/Θ has branching points.
Corollary 4.2. If rank(Θ) = 3, then the index n is 2, 4 or 8 and Γ/Θ is isomorphic to C2,
C2 × C2, C2 × C2 × C2 or D4.
Proof. Proposition 4.1 guaranties that n ∈ {2, 4, 8} if rank(Θ) = 3. The groups of order
2, 4 and 8 not listed in the statement are not generated by involutions.
8 Ars Math. Contemp. 24 (2024) #P1.06
4.2 The rank 3 normal subgroups of Γ
(1) n = 2: As mentioned in the introduction, there are seven epimorphisms from Γ to C2
having kernels Γ2.1, . . . ,Γ2.7. Only three of them have rank 3, as it is easily checked by
applying the Reidemeister-Schreier rewriting process. In this way, one gets that the rank 3
kernels Θ = ⟨X⟩ ∗ ⟨Y ⟩ ∗ ⟨Z⟩ are
Γ2.1 = ⟨R0⟩ ∗ ⟨R1⟩ ∗ ⟨RR21 ⟩ , Γ2.4 = ⟨R1⟩ ∗ ⟨R2⟩ ∗ ⟨R
R0
1 ⟩ and
Γ2.5 = ⟨R1⟩ ∗ ⟨R0R2⟩ ∗ ⟨RR01 ⟩ .
These three groups are isomorphic to the free product C2∗C2∗C2 and therefore isomorphic
to ∆. The remaining four epimorphisms have kernels
Γ2.2 = ⟨R0, R2⟩ ∗ ⟨RR10 , R
R1
2 ⟩ , Γ2.3 = ⟨R0⟩ ∗ ⟨R1R2⟩ , Γ2.6 = ⟨R2⟩ ∗ ⟨R0R1⟩ and
Γ2.7 = ⟨R0R2⟩ ∗ ⟨R1R2⟩ .
The group Γ2.2 has rank 4 and is isomorphic to the free product D2 ∗D2, while the other
three groups Γ2.3, Γ2.6 and Γ2.7 have rank 2 and are all isomorphic to C2 ∗ C∞.
(2) n = 4: Up to an automorphism of G = C2×C2 there are seven epimorphisms from
Γ to G with kernels Γ4.1, . . . ,Γ4.7. One can check that three of them have rank 3, namely
Γ4.1 = ⟨R0⟩ ∗ ⟨RR10 ⟩ ∗ ⟨(R1R2)2⟩ , Γ4.4 = ⟨R2⟩ ∗ ⟨R
R1
2 ⟩ ∗ ⟨(R0R1)2⟩ and
Γ4.5 = ⟨R0R2⟩ ∗ ⟨(R0R1)2⟩ ∗ ⟨(R0R2)R1⟩ .
These groups are all isomorphic to the free product C2∗C2∗C∞ so that ∆ is an epimorphic
image of each of them.
Remark 4.3. Γ4.1 = Γ2.3 ∩ Γ2.2 = Γ2.3 ∩ Γ2.1 = Γ2.2 ∩ Γ2.1 = Γ2.3 ∩ Γ2.2 ∩ Γ2.1 ,
Γ4.4 = Γ2.4 ∩ Γ2.2 = Γ2.4 ∩ Γ2.6 = Γ2.2 ∩ Γ2.6 = Γ2.4 ∩ Γ2.2 ∩ Γ2.6 ,
Γ4.5 = Γ2.7 ∩ Γ2.5 = Γ2.7 ∩ Γ2.2 = Γ2.5 ∩ Γ2.2 = Γ2.7 ∩ Γ2.5 ∩ Γ2.2 .
(3) n = 8, G = D4: Up to an automorphism of G there are six epimorphism from Γ to
G with kernels Γ8.1, . . . ,Γ8.6. Three of them have rank 3 and are all free groups, namely
Γ8.4 = ⟨R0R1R2R1⟩ ∗ ⟨R1R0R1R2⟩ ∗ ⟨(R0R1)2R0R2⟩ ,
Γ8.5 = ⟨(R1R2)2⟩ ∗ ⟨R2(R1R0)2⟩ ∗ ⟨R2(R0R1)2⟩ and
Γ8.6 = ⟨(R0R1)2⟩ ∗ ⟨R0(R1R2)2⟩ ∗ ⟨R0(R2R1)2⟩ .
Remark 4.4. Γ4.5 is the unique normal subgroup of index 4 containing Γ8.4, while Γ4.4
is the unique normal subgroup of index 4 containing Γ8.5 and Γ4.1 is the unique normal
subgroup of index 4 containing Γ8.6 .
(4) n = 8, G = C2 × C2 × C2: Up to an automorphism of G there is only one
epimorphism from Γ to G with kernel isomorphic to the rank 3 free group C∞ ∗C∞ ∗C∞,
namely
Γ8.7 = ⟨(R0R1)2⟩ ∗ ⟨(R1R2)2⟩ ∗ ⟨R0(R1R2)2R0⟩ .
Remark 4.5. Γ8.7 = Γ4.i ∩ Γ4.j for any distinct i, j ∈ {1, . . . , 7}.
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 9
The following table gives a overall description of Θ and the Θ-trivial map for each
normal subgroup Θ of Γ of index 2, 4, 6 and 8.
Θ index rank Free-Product dec. Type of T
Θ
surface χ fig
Γ2.1 2 3 C2 ∗ C2 ∗ C2 (2,2,1) border 1
Γ2.2 2 4 D2 ∗D2 (2,1,2) border 1
Γ2.3 2 2 C2 ∗ C∞ (1,2,2) border 1
Γ2.4 2 3 C2 ∗ C2 ∗ C2 (1,2,2) border 1
Γ2.5 2 3 C2 ∗ C2 ∗ C2 (2,1,2) border 1
Γ2.6 2 2 C2 ∗ C∞ (2,2,1) border 1
Γ2.7 2 2 C2 ∗ C∞ (1,1,1) orient. 2
Γ4.1 4 3 C2 ∗ C2 ∗ C∞ (2,2,2) border 1
Γ4.2 4 4 C2 ∗ C2 ∗ C2 ∗ C2 (2,2,2) border 1
Γ4.3 4 2 C∞ ∗ C∞ (2,2,1) orient. 2
Γ4.4 4 3 C2 ∗ C2 ∗ C∞ (2,2,2) border 1
Γ4.5 4 3 C2 ∗ C2 ∗ C∞ (2,1,2) orient. 2
Γ4.6 4 2 C∞ ∗ C∞ (1,2,2) orient. 2
Γ4.7 4 2 C∞ ∗ C∞ (2,2,2) nonori. 1
Γ6.1 6 4 C2 ∗ C2 ∗ C2 ∗ C∞ (3,2,2) border 1
Γ6.2 6 4 C2 ∗ C2 ∗ C2 ∗ C∞ (2,2,3) border 1
Γ6.3 6 4 C2 ∗ C2 ∗ C2 ∗ C∞ (3,1,3) orient. 2
Γ8.1 8 5 C2 ∗ C2 ∗ C2 ∗ C2 ∗ C∞ (4,2,2) border 1
Γ8.2 8 5 C2 ∗ C2 ∗ C2 ∗ C2 ∗ C∞ (2,2,4) border 1
Γ8.3 8 5 C2 ∗ C2 ∗ C2 ∗ C∞ ∗ C∞ (4,1,4) orient. 2
Γ8.4 8 3 C∞ ∗ C∞ ∗ C∞ (4,2,4) orient. 0
Γ8.5 8 3 C∞ ∗ C∞ ∗ C∞ (2,2,4) nonori. 1
Γ8.6 8 3 C∞ ∗ C∞ ∗ C∞ (4,2,2) nonori. 1
10 Ars Math. Contemp. 24 (2024) #P1.06
Γ8.7 8 3 C∞ ∗ C∞ ∗ C∞ (2,2,2) orient. 2
Table 1: Normal subgroups of indices 2, 4, 6 and 8 in Γ = C2 ∗D2.
5 Description of the thin map representations
In the previous section we computed all rank 3 normal subgroups Θ = Γi.j together with a
set of generators {X1, X2, X3} such that Γi.j = ⟨X1⟩∗⟨X2⟩∗⟨X3⟩ is a thin representation
Ri.j. Since some Γi.j gives rise to more than one thin representation, we label the corre-
sponding representations by Ri.ja, Ri.jb, etc. Note that the generators of a thin represen-
tation can be read out as fundamental group generators (written as words on {R0, R1, R2})
from the respective trivial map (Section 4). The classification is done up to a restrictedly
dual, that is, the generators of a Θ-marked representation are computed up to the usual map
dual if its restriction to Θ is an automorphism of Θ (see also Remark below). The following
table gives all the thin Θ-marked representations. Generators of Γi.j which are involutions
will be denoted by A, B, C and those which are not will be denoted by X , Y , Z.
Remark 5.1. The assignments
R0 7→ R2 , R1 7→ R1 , R2 7→ R0 and
R0 7→ R0R2 , R1 7→ R1 , R2 7→ R2
extend to automorphisms of Γ and give rise to the map dualities D (the usual map duality)
and P (the Petrie duality). Together they generate the outer automorphism group Out(Γ) =
⟨D,P ⟩ ∼= S3. The following diagram graphically pictures the action of Out(Γ) on the set
of rank 3 normal subgroups of Γ, where lines and dash lines represent the action of D and
P , respectively. Note that D, or P , fixes some Θ and therefore for those Θ’s it is a Θ-
¡2.1 ¡2.5¡2.4
¡4.1 ¡4.5¡4.4
¡8.7¡8.6¡8.4 ¡8.5
Figure 2: The actions of D and P on the Θ’s.
restrictedly duality. The Petrie duality is not a thin-preserving duality except in the case of
Γ8.7; here R8.7a and R8.7b are Petrie duals of each other. The duality D fixes Γ2.5, Γ4.5,
Γ8.4 and Γ8.7. These give rise to the restrictedly-dual representations given in the following
Table, but not listed in Table 2.
To illustrate each thin representation, we exhibit the Θ-marked map representation of
the toroidal regular hypermap H pictured in Figure 3 using the James hypermap represen-
tation [4], where hypervertices, hyperedges and hyperfaces of H are represented by simply
connected regions colored grey, dotted and white, respectively, and flags are the numbered
points. We note that Lynne James hypermap representation is actually the Γ6.1-marked
map representation sending (R2R1)3 to 1. Here Γ6.1 is the normal subgroup of index 6 of
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 11
Γ isomorphic to C2 ∗ C2 ∗ C2 ∗ C∞ generated by R0, RR10 , R
R1R2
0 and (R2R1)
3 (given
in Table 1). This representation is not listed in Table 2 because this restrictedly marked
representation is not thin (it is not even clean).
From Figure 3, H = (F ; r0, r1, r2) with F = {1, . . . , 6} and, up to permutation (coloring),
r0 = (1, 4)(2, 5)(3, 6) , r1 = (1, 2)(3, 4)(5, 6) , r2 = (1, 6)(2, 3)(4, 5) .
The hypermap H has one hypervertex, one hyperedge and one hyperface all of valency 3.
The monodromy group of H is G = ⟨r0, r1, r2⟩ ∼= S3. The Euler characteristic of a map
representation of H is given by (2.1) taking into account the Θ-trivial map given in Table 2
and using the isomorphism ρ̃ given in 3.1.
# Rep. Generators Epim. Θ-slice
1 R2.1 A=R0 B=R1 C=R2R1R2
A→r0
B→r1
C→r2
a
b
c
2 R2.4 A=R0R1R0 B=R1 C=R2
A→r0
B→r1
C→r2 a
b c
3 R2.5 A=R0R1R0 B=R1 C=R0R2
A→r0
B→r1
C→r2
a
b
c
4 R4.1 A=R0 B=R1R0R1 X=R1R2R1R2
A→r0
B→r1
X→r2 a
b c
d
5 R4.4 A=R2 B=R1R2R1 Z=R0R1R0R1
A→r2
B→r1
Z→r0
a
b
c
d
6 R4.5a A=R0R2 B=R1R0R2R1 Z=R0R1R0R1
A→r0
B→r2
Z→r1 a
b c
d
7 R4.5b A=R0R2 B=R1R0R2R1 X=R0R1R2R1
A→r0
B→r2
X→r1 a
b c
d
8 R8.4a X=R0R1R2R1 Y=R1R0R1R2 Z=R0R1R0R1R0R2
X→r0
Y→r1
Z→r2 x
a
b c
d
y
9 R8.4b X=R0R1R2R1 Y=R1R0R1R2 Z=R0R2R1R0R2R1
X→r0
Y→r1
Z→r2 x
a
b c
d
y
10 R8.5a X=R1R2R1R2 Y=R0R1R0R1R2 Z=R0R2R1R0R1
X→r0
Y→r1
Z→r2
a
d c
b
11 R8.5b X=R1R2R1R2 Y=R0R1R0R2R1 Z=R0R2R1R0R1
X→r0
Y→r1
Z→r2 a
d
c
b
e
12 R8.6a X=R0R1R2R1R2 Y=R0R1R0R1 Z=R0R2R1R2R1
X→r1
Y→r0
Z→r2 x
a
b c
d
y
13 R8.6b X=R0R2R1R2R1 Y=R0R1R0R1 Z=R1R0R2R1R2
X→r1
Y→r0
Z→r2 x
a
b c
d
y
14 R8.7a X=R1R2R1R2 Y=R0R1R2R1R2R0 Z=R0R1R0R1
X→r1
Y→r2
Z→r0
a
b
cd
15 R8.7b X=R1R2R1R2 Y=R0R2R1R2R0R1 Z=R0R1R0R1
X→r1
Y→r2
Z→r0
a
b
cd
Table 2: The 15 thin representations.
12 Ars Math. Contemp. 24 (2024) #P1.06
Θ-dual of Rep. Generators
R2.5 A = R2R1R2 B = R1 C = R0R2
R4.5a A = R0R2 B = R1R0R2R1 Z = R2R1R2R1
R4.5b A = R0R2 B = R1R0R2R1 Z = R2R1R0R1
R8.4a X = R2R1R0R1 Y = R1R2R1R0 Z = R2R1R2R1R0R2
R8.7a X = R1R0R1R0 Y = R0R2R1R2R0R1 Z = R2R1R2R1
Table 3: The dual representations.
a
a
b
b
c
c
2 1
6
5
4
3
¡  -Slice6.1 ¡  -Marked map rep. of H6.1 James rep. of H
a
a
b
b
c
c
2 1
6
5
4
3
Figure 3: The toroidal regular hypermap H.
As an example, we give a detailed construction of the thin representation R4.1 of H
following the generic description given in [1]:
The words R0, RR10 and (R1R2)
2, in this order, generate the subgroup Θ = Γ4.1 as a
free product C2 ∗ C2 ∗ C∞ (Table 2). A rooted Θ-slice can be obtained from a Schreier
transversal of Θ in Γ, or alternatively by a cut-opening of the trivial Θ-map (see Ta-
ble 1). The rooted Θ-slice we are taking here is the one given by the Schreier transver-
sal {1, R1, R2, R1R2}. Another Schreier transversal may lead to a different rooted Θ-
a
b c
d
Figure 4: The rooted Γ4.1-slice.
slice, and a choice of another flag as root corresponds to take another Schreier transver-
sal, and both will lead to “similar” Θ-marked maps, in the sense that the underlying map
is the same. The Θ-marked map representation of H is obtained by the isomorphism
ρ : Θ/Hρ−1 → ∆/H given by R0 7→ r0, RR10 7→ r1 and (R1R2)2 7→ r2. So we have
R0 = (1, 4)(2, 5)(3, 6), RR10 = (1, 2)(3, 4)(5, 6) and (R1R2)
2 = (1, 6)(2, 3)(4, 5). Now
we take 6 rooted Θ-slices labelled 1, 2, 3, 4, 5 and 6 and join them through their sides a, b,
c and d accordingly to the action of the words R0, RR10 and (R1R2)
2 on the root flag of the
slices. In this way, the word R0 joins the slices 1 and 4, 2 and 5, and 3 and 6, by their sides
labelled c, while RR10 joins the slices 1 and 2, 3 and 4, and 5 and 6, by their sides b. This
leaves to an incomplete picture:
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 13
b
c
e f
d
a
1
6
2
5
4
3
g
h
i
j
kl
Now (R1R2)2, which is an involution, says that the slices 1 and 6, 2 and 3, and 4 and 5,
are joined together through their sides a and d, that is, in the picture above we have the
following equality between labels: g = a and f = l, h = b and i = c, and d = j and
k = e. This lead to the final picture of R4.1 in Table 5.
In the following tables we illustrate the fifteen map representations Rep of the toroidal
regular hypermap H, we display the general Euler’s characteristic formula for the map
representation Rep of any hypermap, the actual Euler’s characteristic of Rep(H) and the
orientability (up to restricted dual) of Rep(H) - and when possible we record their overall
orientability behaviour in parenthesis.
H =
a
a
b
b
c
c
2
1
6
5
4
3
Euler characteristic of H = 0 oriented
Rep Rep(H) Euler characteristic of Rep(H) orient.
R2.1
a
a
b
b
c
c
2
1
65
4
3 |G|
(
1
2|AB| +
1
2|BC| +
1
2|CA| −
1
2
)
= 0 yes
R2.4
a
a
b
b
c
c
2
1
6
5
4
3 |G|
(
1
2|AB| +
1
2|BC| +
1
2|CA| −
1
2
)
= 0 yes
R2.5
a
a
b b
1
2
3
4
5
6
|G|
(
1
2|BCAC| +
1
2|BA| −
1
2
)
= 1 no
Table 4: The Θ-marked map representations of H for |Γ : Θ| = 2.
We discuss now orientability in more details. The first two thin representations R2.1
and R2.4 (Vince and Walsh representations) are the unique orientation-preserving repre-
sentations, that is, if they are orientable they represent orientable hypermaps and if they
are nonorientable they represent nonorientable hypermaps. However, the maps coming out
from the representations R4.5a, R4.5b, R8.4a, R8.4b, R8.7a and R8.7b are always ori-
entable, since the Θ-trivial maps for Θ ∈ {Γ4.5,Γ8.4,Γ8.7} are orientable. This poses the
question: when they represent non-orientable hypermaps? The same question hang over
the other representations with an additional hitch, both orientable and non-orientable maps
can represent orientable and nonorientable hypermaps. This means that for these represen-
tations we no longer have the clue given by R2.1 and R2.4, and for this reason we need
to make a local teste. In general, a Θ-marked map representation M = (Ω;x1, x2, x3)
14 Ars Math. Contemp. 24 (2024) #P1.06
is a representation of an orientable hypermap if and only if x1x2 and x2x3 act on the set
of Θ-slices with two orbits (Θ-orbits). As a hypermap H is orientable if and only if H
covers the orientably-trivial hypermap T +
H
(Figure 5), a thin map representation Rep(H)
of H represents an orientable hypermap if and only if Rep(H) covers the corresponding
representation Rep(T +
H
) of the orientably-trivial hypermap, call this representation RoriT-
map. In the cases of R2.1 and R2.4, the RoriT-map is spherical and so for any hypermap
2
1
Figure 5: The orientably-trivial hypermap T +
H
.
H the representations R2.1(H) and R2.4(H) are orientable if and only if H is orientable.
For the other cases, and specially the cases in which the representation R is always ori-
entable (R4.5a, R4.5b, R8.4a, R8.4b, R8.7a, R8.7b), the representation M = R(H) is
an orientable hypermap H if M covers the respective RoriT-map. Any RoriT-map has two
flags, so a Θ-marked map representation M = (Ω;x1, x2, x3) is a representation of an
orientable hypermap if and only if the triple (x1, x2, x3) induce a two blocks system on the
set of Θ-slices Ω (the two Θ-orbits) such that each xi permutes the two blocks exactly as
the permutation xi of the two flags of the RoriT-map does. That is, xi exchanges the two
blocks if and only if xi exchanges the two flags.
H =
a
a
b
b
c
c
2
1
6
5
4
3
Euler characteristic of H = 0 oriented
Rep Rep(H) Euler characteristic of Rep(H) orient.
R4.1 b
c
e f
d
a
1
6
2
5
4
3
a
b
c
d
ef
|G|
(
1
2|AB| +
1
2|AXBX| −
1
2
)
= 1 no
R4.4 a
cb b
d dc
2 3
4 5 6
1a |G|
(
1
2|BA| +
1
2|AZBZ| −
1
2
)
= 1 no
R4.5a
1
2 4
35
6
a
b
c
d
a d
b
c
|G|
(
1
|AZB| −
1
2
)
= 0 yes (always)
R4.5b
a
b
c
e
d
a b
d
e c
1
2
4
3
5
6
|G|
(
1
|AX| +
1
|XB| − 1
)
= −2 yes (always)
Table 5: The Θ-marked map representations of H for |Γ : Θ| = 4.
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 15
H =
a
a
b
b
c
c
2
1
6
5
4
3
Euler characteristic of H = 0 oriented
Rep Rep(H) Euler’s charac. form. on Rep(H) orient.
R8.4a
f
af
e
g
g
1 3
2 6
5
4a
b
b
c
c
d
d
e
|G|
(
1
|XZXY | +
1
|Y Z| − 2
)
= −4 yes (always)
R8.4b
1
6
2
3 5
4ab
e i
c
f
g
h
j
ae
c i b
d
f
g
hj
d
|G|
(
1
|XZY | +
1
|ZXY | − 2
)
= −6 yes (always)
R8.5a 1 5
2
3
6
4
aa
c
c
b
b
d d
e
f
f
e
g
g
|G|
(
1
|Y XZ| +
1
|ZY | −
3
2
)
= −4 no
R8.5b
1
4
2
5
3
6
b
c
e
i
a c d
f
g
h
h
a
e
b
f
d
g
i
|G|
(
1
|Y Z| +
1
|Y XZ| −
3
2
)
= −4 no
R8.6a 1
2
6
5
4
3
a
a
b
b
c
e
d
d
i
ci
e
h
ff
g
g hj k
j
k
|G|
(
1
|XZ| +
1
|ZYX| −
3
2
)
= −4 no
R8.6b
1
4
3
6
5
2
ab
e
d
i
c
f g
h
j
ae c
i b
d
f
g hj
|G|
(
1
|XY Z| +
1
|ZX| −
3
2
)
= −4 no
R8.7a
a
bce d fg
a
bc
defg
1
2
5
4
3
6 |G|
(
1
|Y ZX| −
1
2
)
= 0 yes (always)
R8.7b
a
bc
e
d
g
a
bd
e fg
c
f
1
2
5
4
3
6 |G|
(
1
|XY | +
1
|Y Z| − 1
)
= −2 yes (always)
Table 6: The Θ-marked map representations of H for |Γ : Θ| = 8.
Take for example the two map representations (always orientable) given by R4.5a and
R4.5b on the non-orientable hypermap H pictured in Figure 6 left, a non-regular, but uni-
form of type (3, 3, 3), the 6 flags hypermap with monodromy group generated by
r0 = (1, 5)(2, 4)(3, 6) , r1 = (1, 2)(3, 4)(5, 6) , r2 = (1, 6)(2, 3)(4, 5) .
It is simple to see that R4.5a(H) represents a non-orientable hypermap because by
having two vertices of valency 2 the map R4.5a(H) does not cover the uniform (regular)
toroidal map R4.5a(T +
H
) of type {4, 4}.
For the case R4.5b(H), the argument is not so simple as before because this map is uni-
form of type {6, 6} and the trivial oriented map R4.5b(T +
H
) is also uniform of type {2, 2}.
However, the word AXB = R0R2R0R1R0R1 fix the root flag 1 in the map R4.5b(H),
but does not fix any flag on the RoriT-map R4.5b(T +
H
).
16 Ars Math. Contemp. 24 (2024) #P1.06
Rep Rep(T +
H
)
R2.1
12
R2.4 1
2
R2.5
12
R4.1
1
2
R4.4
1
2
R4.5a
1
2
R4.5b 1 2
Rep Rep(T +
H
)
R8.4a a
b c
a
c
1 2
b
R8.4b a
b b
a
c
d
c
d
R8.5a
a
a
c
b
b
c12
R8.5b
b
c
d a
1
2
a
b
c
d
R8.6a a a
bc
1
2
bc
R8.6b
a a
b
c
b
c
d
d
1
2
R8.7a
a
a b
b
1
2
R8.7b
c
a
a
b
b
1
2
c
Table 7: The 15 RoriT-maps (thin representations of the orientably-trivial hypermap).
a
a
b
b
c
c
2 1
6
5
4
3
H
3
4
6
5
1
2
f
e
d
c
c
d
e
f
R4.5a(H)
e
b
c
1
6
4
b
c d
e
2 5
3
a
a
d
R4.5b(H)
Figure 6: A non-orientable hypermap H and its representations R4.5a(H) and R4.5b(H).
Alternatively, following the block system argument described above, painting by red
and blue the possible two blocks, we have for R4.5a(T +
H
)
1(red)
A−→ 5(blue) Z−→ 6(red) B−→ 1(blue)
and for R4.5b(T +
H
)
1(red)
A−→ 5(blue) X−→ 6(red) B−→ 1(blue)
In both cases such two block system does not exist.
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 17
ORCID iDs
Antonio Breda d’Azevedo https://orcid.org/0000-0002-7099-4704
Domenico A. Catalano https://orcid.org/0000-0002-1542-9614
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