Also available at http://amc.imfm.si
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 3 (2010) 87–98
Riemann surfaces and restrictively-marked
hypermaps
Antonio Breda d’Azevedo ∗
University of Aveiro, 3810-193 Aveiro, Portugal
Received 30 September 2007, accepted 17 March 2010, published online 9 April 2010
Abstract
If S is a compact Riemann surface of genus g > 1, then S has at most 84(g − 1)
(orientation preserving) automorphisms (Hurwitz). On the other hand, if G is a group of
automorphisms of S and |G| > 24(g − 1) then G is the automorphism group of a regular
oriented map (of genus g) and if |G| > 12(g − 1) then G is the automorphism group
of a regular oriented hypermap of genus g (Singerman). We generalise these results and
prove that if |G| > g − 1 then G is the automorphism group of a regular restrictedly-
marked hypermap of genus g. As a special case we also show that a marked finite transitive
permutation group (Singerman) is a restrictedly-marked hypermap with the same genus.
Keywords: Groups, Riemann surface, hypermaps, maps, restrictedly-marked, restrictedly regular.
Math. Subj. Class.: 05C15, 05C10, 30F10
1 Introduction
A compact Riemann surface S of genus g > 1 has at most 84(g−1) orientation-preserving
automorphisms (Hurwitz [5]). On the other hand, if a group G of automorphisms of a
compact Riemann surface S of genus g is sufficiently large, then it represents the auto-
morphism group of a regular oriented map (if |G| > 24(g − 1)) or hypermap (if |G| >
12(g − 1)), (Singerman [8]). Hypermaps can be seen as restrictedly-marked maps, in
this case restricted to the subgroup Υ0̂ = 〈R1, R2, RR01 〉 ∼= C2 ∗ C2 ∗ C2 of index 2 in
Υ = 〈R0, R1, R2 | R2i = (R2R0)2 = 1〉 (Jones and Breda [2]). As such they are rep-
resented by bipartite-maps (2-coloured maps called Walsh bipartite maps) where the hy-
permap’s automorphisms (similarly for homomorphisms) are bipartition-preserving map’s
automorphisms. Orientable maps (Υ+-conservative maps) give rise to the restricted forms
∗Research partially supported by UI&D “Matemática e Aplicações” of Universidade de Aveiro through the
Program POCTI of FCT, cofinanced by the European Community fund FEDER.
E-mail address: breda@ua.pt (Antonio Breda d’Azevedo)
Copyright c© 2010 DMFA Slovenije
88 Ars Math. Contemp. 3 (2010) 87–98
known as “oriented” maps (Υ+-marked maps), described by triples (Ω, R, L) consisting
of a finite set Ω and two Ω-permutations R and L, with L2 = 1, generating a transitive
group on Ω. Oriented hypermaps (∆+-marked hypermaps), the restricted forms of ∆+-
conservative (or orientable) hypermaps, are direct generalisations of that. More generally
yet are the restrictedly-marked hypermaps, the restricted forms of “Θ-conservative” hyper-
maps, where Θ is a normal subgroup of finite index of ∆ = C2 ∗ C2 ∗ C2. Often these
result in “multi-coloured” maps.
In this paper we generalise Singerman’s results and prove that if G is a group of auto-
morphisms of a compact Riemann surface of genus g > 1, and |G| > g − 1, then G is the
automorphism group of a regular restrictedly-marked hypermap of genus g.
1.1 Regular restrictedly-marked hypermaps
Algebraically, hypermaps correspond to finite transitive permutations representations ν :
∆ −→ G, Ri 7→ ri, where ∆ is the free product C2 ∗ C2 ∗ C2 generated by the reflections
in the sides of a hyperbolic triangle with zero internal angles in the hyperbolic plane. Let Θ
be a normal subgroup of finite index n. A hypermapH is Θ-conservative if its fundamental
subgroup H is a subgroup of Θ. In such case Θ acts on its flags uniformly dividing them
into Θ-orbits of equal length. A Θ-conservative hypermapH is Θ-regular if and only if the
group AutΘ(H), of the automorphisms of H preserving each Θ-orbit, acts transitively on
each Θ-orbit; and this happens if and only if H is normal in Θ. A hypermap is restrictedly-
marked if it is Θ-conservative, and is restrictedly-regular if it is Θ-regular, for some normal
subgroup Θ of finite index in ∆. Not every hypermap is restrictedly-regular, see [1].
By the Kurosh Subgroup Theorem (see [7], Corollary 4.9.1 and remarks after Corollary
4.9.2), Θ freely decomposes uniquely (up to a permutation of factors) in a free product
C2∗· · ·∗C2∗C∞∗· · ·∗C∞ = 〈X1, . . . , Xm | X2i = 1, i = 1 . . . s〉, for some 0 ≤ s ≤ m.
A Θ-conservative hypermap is combinatorially described by a (m+ 1)-tuple
Q = (Ω;x1, . . . , xm) (1.1)
where Ω is a finite set (the set of the “Θ-slices”), x1, . . . , xm are permutations of Ω gen-
erating a group G acting transitively on Ω such that the function ρ : Xi 7→ xi extends to
an epimorphism from Θ to G. Such (m + 1)-tuple is called a Θ-marked hypermap. For
example, if Θ is the even-word subgroup ∆+ ∼= F (2), a ∆+-conservative (i.e. orientable)
hypermap is described by a ∆+-marked (i.e. oriented) hypermap Q = (Ω;x1, x2), where
x1, x2 are usually denoted by the letters R, L or by ρ, λ. On the other hand, if Q is the fun-
damental Θ-marked subgroup ofQ, i.e. the stabiliser of some fixed ω ∈ Ω under the action
of Θ on Ω, then the “∆-form” hypermap
∆Q = (∆/rQ;Q∆R0, Q∆R1, Q∆R2), where Q∆
is the core of Q in ∆ and ∆/
r
Q is the set of the right cosets of Q in ∆, is Θ-conservative
and shares with Q the same underlying surface, the same underlying hypergraph, the same
set V of hypervertices, the same set E of hyperedges and the same setF of hyperfaces; only
the set Ω of Θ-slices of Q is just an orbit of the action of Θ on the set of flags F = ∆/
r
Q
of
∆Q. The sizes of Ω and F are |Θ : Q| and |∆ : Q| respectively. The permutations
x1, . . . , xm are restrictions to Ω of the permutations X1ν, . . . , Xmν ∈ Sym(F ), where
ν : ∆ −→ Mon(∆Q) is the canonical transitive permutation representation. A Θ-marked
hypermapQ is regular if its ∆-form ∆Q is Θ-regular. Moreover,Q has boundary if ∆Q has
boundary. That is, if Ri ∈ Qd for some d ∈ ∆ and i = 0, 1, 2. If Q has no boundary, then
the Euler characteristic of Q (we mean the Euler characteristic of its underlying surface,
A. Breda d’Azevedo: Riemann surfaces and restrictively-marked hypermaps 89
which is the Euler characteristic of
∆Q) is given by χ(Q) = |V|+ |E|+ |F| − |F |2 , where
|F | = n|Ω|.
Let (k; l;m) be the type of the trivial Θ-marked hypermap TΘ = (Θ/r Θ; ΘX1, . . . ,
ΘXm), with just one Θ-slice; geometrically a Θ-slice is the connected (polygonal) region
obtained by the elements of a fixed Schreier transversal for Θ in ∆ acting on a single flag
(a triangular region). The ∆-form of TΘ is the hypermap ∆TΘ = (∆/r Θ; ΘR0,ΘR1,ΘR2)
with n flags. If Q is a regular Θ-marked hypermap then Q regularly covers1 TΘ and, as a
consequence, its hypervertices (resp. hyperedges, hyperfaces) will be regularly partitioned
(or coloured) in qv parts, each part projecting to a hypervertex (resp. hyperedge, hyperface)
of TΘ . Hypervertices (resp. hyperedges, hyperfaces) of the same colour have the same
valency. This induces a sequence
(k1, . . . , kqv ; l1, . . . , lqe ; m1, . . . ,mqf )
called the Θ-type of Q, where k1, . . . , kqv are the common valencies of the hypervertices
in the same coloured-parts 1, . . . , qv , respectively, and similarly for the rest of the numbers.
Here qv , qe and qf are the numbers of hypervertices, hyperedges and hyperfaces of TΘ ,
respectively. The Euler characteristic of Q is then expressed by (cf §9.1 of [1])
χ(Q) = |Ω|
(
qv∑
i=1
µv k
2ki
+
qe∑
i=1
µe l
2li
+
qf∑
i=1
µf m
2mi
− n
2
)
(1.2)
where µv = 1 or 2 according as the hypervertex v of TΘ lies on the boundary or not, and
similarly for µe and µf . Note that n is the index of Θ in ∆. For further reading on the
subject, and for geometric illustrations, we refer the reader to [1].
2 Two special restrictedely-marked subgroups
Example 1. Let Θ be the subgroup ∆n in ∆ of index 2n generated by the n+1 generators
Z1 = R1R2 , Z2 = (R1R2)R0R2 , . . . , Zn = (R1R2)(R0R2)
n−1
and Zn+1 = (R2R0)n .
This group is normal in ∆ = 〈R0, R1, R2〉 = 〈Z1, R0R2, R2〉, for ZR0R2i = Zi+1, i =
1, 2, . . . , n−1,ZR0R2n = Z
Z−1n+1
1 , Z
R0R2
n+1 = Zn+1,Z
R2
1 = Z
−1
1 ,Z
R2
i = (Z
Zn+1
n−(i−2))
−1, i =
2, . . . , n and ZR2n+1 = Z
−1
n+1. The quotient ∆/∆n is a dihedral group Dn of order 2n.
By the Reidmaster-Schreier Rewriting Process, ∆n is isomorphic to a free product C∞ ∗
· · · ∗ C∞ (n + 1 times), that is, a free group of rank n + 1, and consequently any ∆n-
marked hypermap has the form Q = (Ω, z1, . . . , zn+1). As the generators lie in ∆1 =
∆+ = 〈R1R2, R2R0〉 (the subgroup of the even-length words of ∆), any ∆n-marked
hypermap is orientable. The trivial ∆n-marked hypermap T∆n is the regular hypermap of
type (1;n;n) on the sphere (pictured below for the case when n = 7) with n hypervertices,
one hyperedge and one hyperface. Any ∆n-conservative hypermap H covers T∆n and so
its hypervertices are n-coloured, which makesH a (n+ 2)-coloured map.
1A (hyper)mapM′ with fundamental subgroup M ′ regularly coversM with fundamental subgroup M if M ′
is a normal subgroup of M (Jones and Singerman [6]).
90 Ars Math. Contemp. 3 (2010) 87–98
Figure 1: The trivial ∆7-marked hypermap T∆7 .
The ∆n-type of Q is therefore (k1, . . . , kn; l;m) for some positive integers k1, . . . , kn,
l and m such that n divides both l and m. By (1.2), the Euler characteristic of a regular
∆n-marked hypermap of ∆n-type (k1, . . . , kn; l;m) is given by
χ(Q) = |Ω|
(
1
k1
+ · · ·+ 1
kn
+
n
l
+
n
m
− 2n
2
)
. (2.1)
IfQ = (Ω, z1, . . . , zn+1) is a ∆n-marked hypermap then each orbit of zi, for i = 1, . . . , n,
corresponds to a i-coloured hypervertex vi (denoting the i-coloured hypervertices by vi, wi,
etc.) while each orbit of zn+1 corresponds to a hyperedge (a white vertex in the picture
below) and each orbit of the product z1 . . . zn+1 correspond to a hyperface.
















	

	
	

	
	

	
Figure 2: Part of a zi-orbit (i ≥ 2) labeled 1,2,3,. . .
If Q is regular (equivalently, its ∆-form ∆Q = (F, r0, r1, r2) is ∆n-regular) the automor-
phism group Aut(Q), which is the ∆n-automorphism group of its ∆-form
∆Q, coincides
with the monodromy group Mon(Q) - only the actions of Mon(Q) and Aut(Q) on F
are different. Hence Aut(Q) is generated by z1, . . . , zn+1 (considered as automorphisms).
Each Zi ∈ ∆n, i = 1, . . . , n, is a conjugate of the product of two of the hyperbolic re-
flections R0, R1 and R2, so Z1, . . . , Zn are (parabolic) limit rotations about hypervertices,
Zn+1 is a n-step limit rotation about a hyperedge and the product Z1 . . . Zn is a n-step
limit rotation about a hyperface. They project via ρ to automorphisms zi ∈ Aut(Q). No-
tice that
∆Q may be not regular (that is, Q may be not ∆-symmetric); if this is the case,
A. Breda d’Azevedo: Riemann surfaces and restrictively-marked hypermaps 91
the size of the automorphism group Aut(
∆Q) is smaller than |F | and consequently at least
one of r0, r1 and r2 cannot be realised as an automorphism of
∆Q. However, Aut(∆Q)
acts regularly on the ∆n-orbit Ω and thus z1, . . . , zn can be realised as automorphisms of
∆Q. Now Q has n hypervertex-colours v1, v2, . . . , vn, only one hyperedge-colour e and
only one hyperface-colour f . These n hypervertex-colours appear around any hyperedge
as well as around any hyperface as shown on Figure 3. The generators z1, . . . , zn, being
the projections of Z1, . . . , Zn, are clearly rotations about the hypervertices v1, . . . , vn of
orders k1, . . . , kn respectively. The generator zn+1 is a rotation about e of order ln and the
product zn+2 = z1z2 . . . zn+1 = (r1r0)n is a rotation about f of order mn .
 
















Figure 3: z1, . . . , zn+2 seen as automorphisms of Q.
Formula (2.1) can then be rewritten as follows.
Lemma 2.1. If Q = (Ω, z1, . . . , zn+1) is a regular ∆n-marked hypermap then the Euler
characteristic of Q is given by
χ(Q) = |Ω|
(
1
|z1|
+ · · ·+ 1
|zn|
+
1
|zn+1|
+
1
|z1 . . . zn+1|
− n
)
.
If G is a group generated by g1, . . . , gm such that the function xi 7→ gi extends to
an epimorphism from Θ to G, then Q = (G, g1, . . . , gm) is a regular Θ-marked hypermap
(Theorem 22 of [1]). A slightly more general statement is obtained by taking the free group
Θ = ∆m−1 < ∆+ of rank m:
Lemma 2.2. If G is a group generated by g1, . . . , gm, then Q = (G, g1, . . . , gm) is a
regular ∆m−1-marked hypermap, and so, a regular restrictedely-marked hypermap.
Example 2. The subgroup Θ = K3 of ∆ of index 6 generated by A = R0R1R2 , B =
R0R2
R1 , C = R1R0R2 and D = R1R2R0 is normal in ∆, for AR0 = A−1, AR1 = DB,
AR2 = C−1, BR0 = B−1, BR1 = DA, BR2 = C−1B−1A, CR0 = BD, CR1 = C−1,
CR2 = A−1, DR0 = BC, DR1 = D−1 and DR2 = A−1D−1C. This group factors ∆
into a dihedral group D3 with 6 elements. By the Reidmaster-Schreier Rewriting Process,
92 Ars Math. Contemp. 3 (2010) 87–98
K3 is isomorphic to a free product C∞ ∗C∞ ∗C∞ ∗C∞ of rank 4. Therefore a regularK3-
marked hypermap has representative formQ = (G; a, b, c, d) whereG is a group generated
by a, b, c, d, without further restrictions. Moreover, as K3 is a subgroup of ∆+, any K3-
marked hypermap is orientable. The trivial K3-hypermap TK3 is the reflexible hypermap
(3, 3, 3)1,0 of type (3, 3, 3) on the torus whose Walsh map is the regular map {6, 3}1,0.





Figure 4: The trivial K3-hypermap TK3 .
This has 1 hypervertex, 1 hyperedge and 1 hyperface. Therefore theK3-type of any regular
K3-marked hypermap coincides with its topological type (or ∆-type) (k; l;m) for some
k = 0 mod 3, l = 0 mod 3 and m = 0 mod 3. By (1.2), Q has Euler characteristic
given by
χ(Q) = |G|
(
3
k
+
3
l
+
3
m
− 3
)
.
Since
k = |r1r2| = 3|(r1r2)3| = 3|a−1b|,
and similarly l = 3|c−1d| and m = 3|bca−1d−1|, we have:
Lemma 2.3. The Euler characteristic of a regular K3-marked hypermap Q = (G; a, b,
c, d) is given by
χ(Q) = |G|
(
1
|a−1b|
+
1
|c−1d|
+
1
|bca−1d−1|
− 3
)
.
3 Riemann surfaces
Let S be a Riemann surface of genus g ≥ 2. By the uniformization theorem S is a quotient
U/Γ, where Γ is a cocompact torsion-free discrete subgroup of Isom+(U) ∼= PSL(2,R),
the group of orientation-preserving isometries of the hyperbolic plane U (modelled on the
complex upper half-plane). The group Γ, the surface-group corresponding to S, is a Fuch-
sian group with signature (g;−); it is unique up to a conjugacy in PSL(2,R). Automor-
phisms of S lift to isometries of U normalising Γ, so if G is a group of automorphisms of
S then G = Λ/Γ, where Λ is a Fuchsian group containing Γ as a normal subgroup. Let
(λ;m1, . . . ,mr) be the signature of Λ. This means that Λ has presentation
〈x1, ..., xr, a1, b1, ..., aλ, bλ | xm11 = ... = xmrr =
r∏
i=1
xi
λ∏
i=1
[ai, bi] = 1〉 .
A. Breda d’Azevedo: Riemann surfaces and restrictively-marked hypermaps 93
This has measure µ(Λ) = 2π
(
2λ− 2 +
∑r
i=1(1−
1
mi
)
)
. The Riemann-Hurwitz formula
µ(Γ) = |Λ/Γ|µ(Λ) can be written as
2g − 2 = |G|
(
2λ− 2 +
r∑
i=1
(1− 1
mi
)
)
⇔ χ(S) = |G|
(
r∑
i=1
1
mi
− (r + 2λ− 2)
)
,
(3.1)
where χ(S) = 2 − 2g and the sum is considered empty in the case r = 0. Since the
order of the periods in the signature is irrelevant, we assume that m1 ≤ m1 ≤ · · · ≤ mr.
Using the same terminology as in [3], we say that a group G acts on genus g with signature
(λ;m1, . . . ,mr) if G = Λ/Γ where Λ and Γ are Fuchsian groups such that (1) Λ has
signature (λ;m1, . . . ,mr) and (2) Γ has signature (g;−) and is normal in Λ. In this case
G acts on the Riemann surface S = U/Γ as a group of automorphisms. The canonical
epimorphism from Λ to G with kernel Γ is called a surface-kernel epimorphism. Surface-
kernel epimorphisms are order-preserving, so the images of the generators x1, . . . , xr in G
have orders m1, . . . , mr respectively.
Theorem 3.1. Let S be a Riemann surface of genus g > 1 and let G be a group of auto-
morphism of S. If |G| > g− 1 then G is the automorphism group of a regular restrictedly-
marked hypermap Q of genus g. Moreover, G is the automorphism group of a regular
K3-marked hypermap if G acts with signature (1;m1, . . . ,mr), and of a regular ∆r−2-
marked hypermap if G acts with signature (0;m1, . . . ,mr).
Proof. LetG = Λ/Γ for some Fuchsian group Λ normalising Γ (the surface-group) and let
(λ;m1, . . . ,mr) be the signature of Λ. By the Riemann-Hurwitz formula,
|G| = 2(g − 1)
2λ− 2 +
∑r
i=1
(
1− 1mi
) .
Now |G| > g − 1 implies
0 < 2λ− 2 +
r∑
i=1
(
1− 1
mi
)
< 2 ,
and this inequality implies λ ≤ 1.
I : λ = 1. In this case 0 < r < 4.
(a) r = 3 Then Λ has signature (1;m1,m2,m3) and thus
G = 〈x1, x2, x3, a, b | xm11 = x
m2
2 = x
m3
3 = x1x2x3[a, b] = 1, . . . 〉 .
Fourth relation gives x2 = x−11 [a, b]
−1x−13 . Replacing x2 by its inverse we may take
x2 = x3[a, b]x1 to get
G = 〈x1, x3, a, b | xm11 = x
m3
3 = (x3[a, b]x1)
m2 = 1, . . . 〉 .
Equation (3.1) can then be written as
χ(S) = |G|
(
1
m1
+
1
m2
+
1
m3
− 3
)
.
94 Ars Math. Contemp. 3 (2010) 87–98
Now we will try to see G as the automorphism group of a regular K3-marked hyper-
map Q = (G;A,B,C,D). Taking into account Lemma 2.3, we will try to solve
|A−1B| = m3 , |C−1D| = m1 and |BCA−1D−1| = m2 ,
knowing that |x1| = m1, |x3| = m3 and |x3[a, b]x1| = m2.
We do this in two steps.
First, since m1 = |C−1D| = |DC−1| = |CD−1|, we set CD−1 = x1, from which
we get D = x−11 C. Then putting x3[a, b]x1 = BCA
−1D−1 = BCA−1C−1x1, we
get x3[a, b] = BCA−1C−1.
Second, since |A−1B| = |BA−1|, set BA−1 = x3. This gives B = x3A. Replacing
B in the above equation we get [a, b] = ACA−1C−1 = [A−1, C−1] .
Hence A = a−1, C = b−1, B = x3A = x3a−1 and D = x−11 C = x
−1
1 b
−1 is a
solution. Since they generate G,
Q = (G; a−1, x3a−1, b−1, x−11 b−1)
is a regular K3-marked hypermap with automorphism group G and Euler character-
istic
χ(Q) = |G|
(
1
|A−1B| +
1
|C−1D| +
1
|BCA−1D−1| − 3
)
= |G|
(
1
|ax3a−1| +
1
|bx−11 b−1|
+ 1|x3[a,b]x1| − 3
)
= |G|
(
1
m3
+ 1m1 +
1
m2
− 3
)
,
which gives genus(Q) = g = genus(S).
(b) r = 2 In this case G = 〈x1, x2, a, b | xm11 = x
m2
2 = x1x2[a, b] = 1, . . . 〉 = 〈x, a, b |
xm1 = ([a, b]x)m2 = 1, . . . 〉 . Equation (3.1) takes the form
χ = |G|
(
1
m1
+
1
m2
− 2
)
.
To see G as the automorphism group of a regular K3-marked hypermapQ = (G;A,
B,C,D) with χ = |G|( 1|A−1B| +
1
|C−1D| +
1
BCA−1D−1| − 3) we start by putting
A−1B = 1, that is, B = A. Comparing formulas we must now have |C−1D| =
m1 and |ACA−1D−1| = m2. Following a similar procedure as above, we eas-
ily get a solution A = a−1, B = A = a−1, C = b−1, D = x−1b−1 and so
Q = (G; a−1, a−1, b−1, x−1b−1) ( or (G; a, a, b, x−1b) ) is a regular K3-marked
hypermap with automorphism group G and Euler characteristic
χ(Q) = |G|
(
1
m1
+ 1m2 − 2
)
= χ(S).
(c) r = 1 If G acts with signature (1;m) then G has presentation 〈a, b | [a, b]m = · · · =
1〉. Expression (3.1) becomes
|G|
(
1
m
− 1
)
= χ(S) . (3.2)
A. Breda d’Azevedo: Riemann surfaces and restrictively-marked hypermaps 95
In this case the regular K3-marked hypermap Q = (G; a, a, b, b) has Euler charac-
teristic
χ(Q) = |G|
(
1
m
− 1
)
= χ(S) . (3.3)
II : λ = 0. In this case 2 <
∑r
i=1(1−
1
mi
) < 4 implies 2 < r < 8.
If G acts with signature (0;m1, . . . ,mr) then G has presentation 〈x1, . . . , xr−1 |
xm11 = · · · = x
mr−1
r−1 = (x1..xr−1)
mr = ... = 1〉 with r − 1 elliptic generators and
no parabolic generators. In this case the Riemann-Hurwitz formula yields
χ(S) = |G|
(
1
m1
+ · · ·+ 1
mr−1
+
1
mr
− (r − 2)
)
.
It is clear that the (canonical) regular ∆r−2-marked hypermapQ = (G;x1, . . . , xr−1) will
do. The product xr = x1 . . . xr−1 has order mr and so by Lemma 2.1,
χ(Q) = |G|
(
r∑
i=1
1
mi
− (r − 2)
)
= χ(S)
which shows that genus(Q) = g.
4 A note on finite marked permutations groups
Marked finite transitive permutation groups (MFTPG) are triples M = (G,Ω, D) con-
sisting of a finite set Ω and a set D = {z1, . . . , zn−1} of permutations of Ω generating
G and acting transitively on Ω (Singerman [9]). The genus of a marked finite transitive
permutation group M was defined by Singerman as being the genus of a certain Riemann
surface S := X/N “naturally” associated to M. Although not originally aimed to give
an insight on higher dimensional combinatorial structures associated to higher dimensional
manifolds, MFTPGs are related with Vince’s combinatorial maps [10], or the more general
Ferri’s connected (n + 1)-coloured graphs [4]. In fact, any finite combinatorial map G of
rank I is a marked finite transitive permutation group (G,V (G), D) where V (G) is the
set of vertices of G and D consists of |I| fixed-point free involutory permutations of V (G)
induced by the |I| edge-colours of G. The transitivity of G on V (G) is a consequence of
the connectivity of G and the fixed-point free action of the involutory permutations a con-
sequence of the non-existence of free edges in G (when realised as cell decompositions this
corresponds to boundary-free manifolds). These were all designed to describe cell decom-
positions of n-dimensional manifolds, n-polytopes and tessellations (n = |I|−1)), though
in general not all such combinatorial constructions of rank > 3 are realised in such a way.
Despite the name, a combinatorial map is just a graph and when its degree (rank) is 3 it
actually describes a 2-dimensional simplicial complex (best known as a hypermap).
In the special case when D = {z1, z2} and z1 is a fixed-point free involution, which
corresponds to an oriented map, Singerman showed that the associated “Riemann” genus
coincide with the genus of the map. In general Singerman did not associate MFTPGs to
embeddings of graphs on surfaces (of genus g), except in the above mentioned case. We
can now show that actually any marked finite transitive permutation group M of genus g
represents a restrictedly-marked hypermapQ of genus g. These “coloured” maps represent-
ing restrictedly-marked hypermaps are not related with Vince’s and Ferri’s edge-coloured
graphs.
96 Ars Math. Contemp. 3 (2010) 87–98
Let M = (G,Ω, D) be a marked finite transitive permutation group withD = {z1, . . . ,
zn−1}. If rank(M) = |D| = 1 then G is cyclic generated by one element and by the
transitivity of G on Ω we must have |Ω| = |G|. So M is a regular oriented (∆+-restricted)
mapM = (G; z1, 1).
For the rest of the paper let rank(M) > 1 (n > 2). Let k1, . . . , kn−2 be the orders of
z1, . . . , zn−2 respectively, l := n|zn−1| and m := n|zn|, where zn = z1 . . . zn−1. Let Γ
be a F-group with signature (0; k1, . . . , kn−2, ln ,
m
n ); this means that Γ has presentation
〈x1, . . . , xn−2, xn−1, xn | xk11 = · · · = x
kn−2
n−2 = x
l
n
n−1 = x
m
n
n =
n∏
i=1
xi = 1〉 .
This has measure given by µ(Γ) = 2π
(
n− 2− (Σn−2i=1 1ki +
n
l +
n
m )
)
. We have an
obvious epimorphism % : Γ → G defined by xi 7→ zi for i = 1, . . . , n − 1. Let
N = Stab
G
(w)%−1 /
q
Γ, where q = |Θ : Q| = |Ω|. This group N is also a F-group
and S := X/N is a Riemann surface, where X is the sphere S if µ(Γ) < 0, the Euclidean
plane C if µ(Γ) = 0 or the hyperbolic plane H if µ(Γ) > 0 (Singerman [9]). Singer-
man defined the genus of M to be the genus of the Riemann surface S. On the other
hand, M corresponds to an orientable ∆n−2-marked hypermap Q = (Ω, z1, . . . , zn−1),
where ∆n−2 = 〈Z1, . . . , Zn−1〉 /2(n−2) ∆ is the group generated by Z1 = R1R2, Z2 =
(R1R2)R0R2 , . . . , Zn−2 = (R1R2)(R0R2)
n−3
and Zn−1 = (R2R0)n−2. Let Zn be
the product Z1 . . . Zn−1 = (R1R0)n−2 and Q its ∆n−2-marked fundamental subgroup
. The orientable restricted marked hypermap Q has genus g(Q) = 2−χ2 where χ =
|V| + |E| + |F| − |F |2 is the Euler characteristic of the underlying surface (F is the set
of flags of its ∆-form
∆Q). Although natural, it is not yet clear whetherQ has genus g (see
theorem 6), since different Θ-marked hypermaps can be associated to a given group, with
different genus.
Theorem 4.1. A marked finite permutation group M = (G,Ω, D), where D = {z1, . . . ,
zn−1}, is an orientable ∆n−2-marked hypermapQ = (Ω, z1, . . . , zn−1) with genus(Q) =
genus(M).
Proof. Let g = genus(M). Decompose z1, . . . , zn in a product of cycles:
z1 = σ1,1 · · ·σ1,t1
· · ·
zn−2 = σn−2,1 · · ·σn−2,tn−2
zn−1 = α1 · · ·αs
zn = β1 · · ·βr .
Then the number of hypervertices |V|, the number of hyperedges |E| and the number of
hyperfaces |F| are given by |V| =
∑n−2
i−1 ti, |E| = s and |F| = r, respectively. The
number of flags |Ω| = |Θ : Q| = q is related to the above cycle decomposition as
|Ω| =
ti∑
j=1
|σi,j | =
s∑
i=1
|αi| =
r∑
i=1
|βi| .
Now the F-group N has signature (Singerman [9])(
g;
k1
δ1,1
, . . . ,
k1
δ1,t1
, . . . ,
kn−2
δn−2,1
, . . . ,
kn−2
δn−2,tn−2
,
l
n
a1
, . . . ,
l
n
as
,
m
n
b1
, . . . ,
m
n
br
)
,
A. Breda d’Azevedo: Riemann surfaces and restrictively-marked hypermaps 97
where δi,j = |σi,j |, ai = |αi| and bi = |βi|. The measures of the F-groups N and Γ are
related by µ(N) = |Ω|µ(Γ). This formula translates as
2g − 2 +
t1∑
i=1
(1− δ1,i
k1
) + · · ·+
tn−2∑
i=1
(1− δn−2,i
kn−2
) +
s∑
i=1
(1− ai
l
n
) +
r∑
i=1
(1− bim
n
) =
|Ω|(n− 2−
n−2∑
i=1
1
ki
− n
l
− n
m
) .
Since
∑ti
j=1 δi,j = |Ω|,
∑s
i=1 ai = |Ω| and
∑r
i=1 bi = |Ω|, we get
2g−2+
n−2∑
i=1
ti+s+r−
|Ω|
k1
−· · ·− |Ω|
kn−2
−|Ω|
l
n
−|Ω|m
n
= |Ω|(n−2)−
n−2∑
i=1
|Ω|
ki
−|Ω|(n
l
+
n
m
) .
Now replacing
∑n−2
i=1 ti by |V|, s by |E|, r by |F| and taking into account that |∆ : ∆n| =
2(n− 2), and thus |Ω|2(n− 2) = |Θ : Q||∆ : Θ| = |∆ : Q| = |F |, we get
|V|+ |E|+ |F| − |F |
2
= 2− 2g .
Thus χ(Q) = 2− 2g and hence genus(Q) = g.
Any hypermap H is a ∆-marked hypermap. As H may also be represented by some
Θ-marked hypermap Q, with flags in H being represented by Θ-slices in Q, the genus of
the ∆-marked hypermapHmay be different from the genus of the Θ-marked hypermapQ.
Theorem 4.1 just emphasises how natural is the choice of ∆n in the restrictedly-marked
subgroups considered earlier.
5 Acknowledgment
I would like to express my gratitude to the referees for spotting a mistake in an earlier
version of the paper.
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