Also available on http://amc.imfm.si
ARS MATHEMATICA CONTEMPORANEA 1 (2008) 51–65
Exotic Behaviour of Infinite Hypermaps
Gareth A. Jones
School of Mathematics, University of Southampton, Southampton, SO17 1BJ, UK
Received 10 August 2007, accepted 1 July 2008, published online 15 July 2008
Abstract
This is a survey of infinite hypermaps, and of how they can be constructed by using exam-
ples and techniques from combinatorial group theory, with particular emphasis on phenomena
which have no analogues for finite hypermaps.
Keywords: Hypermap, automorphism group, monodromy group.
Math. Subj. Class.: 05-02, 05C10, 20E06
1 Introduction
Hypermaps are natural generalisations of maps, representing surface embeddings of hyper-
graphs rather than graphs. In recent years, especially in the compact orientable case, they
have assumed considerable importance in the theory of dessins d’enfants, where they cor-
respond to algebraic curves defined over algebraic number fields. As in the case of maps,
most of the theory has been developed in terms of finite hypermaps (those on compact sur-
faces), often using techniques from finite group theory such as the classification of finite
simple groups, together with the classification of compact surfaces. By contrast, the group-
theoretic and topological tools available in the infinite case are much weaker, and the only
infinite hypermaps normally considered are the universal hypermaps of various types on the
euclidean and hyperbolic planes (see §6). Although much of the basic theory of hypermaps
extends without significant changes to the infinite case, there are a number of rather strange
phenomena which can arise only for infinite hypermaps.
The basic idea is that a hypermap H (which we will generally assume to be oriented)
can be regarded as a transitive permutation representation of a 2-generator group G, called
the monodromy group of H, and an orientably regular hypermap (one with maximum sym-
metry) can be regarded as the regular representation of G, which is then isomorphic to the
automorphism group of H. Any phenomenon which can happen inside a countable group
can happen inside a 2-generator group, and hence there must be a hypermap exhibiting some
E-mail address: G.A.Jones@maths.soton.ac.uk (Gareth A. Jones)
Copyright c© 2008 DMFA – založništvo
52 Ars Mathematica Contemporanea 1 (2008) 51–65
analogous behaviour. More precisely, Higman, Neumann and Neumann [17] have shown
that every countable group can be embedded in a 2-generator group, and hence in the auto-
morphism group of an orientably regular hypermap. The aim here is to give some examples
of 2-generator infinite groups with interesting (and occasionally surprising) properties, and
where possible to describe the corresponding hypermaps.
For simplicity of exposition, this paper concentrates mainly on oriented hypermaps and
orientation-preserving automorphisms, though these restrictions are removed in the last two
sections. Similarly, many of the results are valid in the special case of maps; indeed, any
hypermap can be regarded as a (bipartite) map on the same surface by means of the Walsh
construction described in §2.
2 Oriented hypermaps
Cori introduced the theory of finite oriented hypermaps in [7]; he and Machı̀ have given a
detailed survey in [8], and connections between the algebraic, combinatorial and topolog-
ical aspects of the theory are discussed by Corn and Singerman in [9]. Algebraically, an
oriented hypermap H (finite or infinite) can be regarded as a 2-generator permutation group
G = 〈x, y〉 acting transitively on a set Ω, or equivalently as a transitive permutation repre-
sentation θ : F2 → Sym Ω of the free group F2 of rank 2. The hypervertices, hyperedges
and hyperfaces of H are the cycles of x, y and z := (xy)−1 on Ω, with incidence given by
non-empty intersection. If y2 = 1 then H is a map. The natural cyclic order within each cy-
cle determines an orientation of the underlying surface associated with H. The monodromy
group Mon+H of H is the group G = θ(F2) of permutations of Ω induced by F2, and the
(orientation-preserving) automorphism group A = Aut+H is the centraliser of G in Sym Ω.
(We will introduce more general definitions of monodromy and automorphism groups, for
unoriented hypermaps, in §8.)
We say that H is orientably regular (or rotary) if A acts transitively on Ω; this is equiv-
alent to G acting regularly on Ω, in which case A and G can be regarded as the regular
representations of G on itself by left and right multiplication, and hence A ∼= G. In the case
of an orientably regular hypermap, where we take Ω = G, the hypervertices, hyperedges and
hyperfaces correspond to the cosets of the subgroups 〈x〉, 〈y〉 and 〈z〉 in G, with orienta-
tion corresponding to multiplication by the corresponding generator, and incidence given by
non-empty intersection.
There is a bijection between isomorphism classes of oriented hypermapsH and conjugacy
classes of subgroups H ≤ F2 : these are the stabilisers of points in Ω, called the hypermap
subgroups, and we have A ∼= NF2(H)/H . In this correspondence, orientably regular hyper-
maps correspond to normal subgroups H = ker θ of F2, with A ∼= F2/H ∼= G.
The Walsh map W = W (H) [32] is a useful way of visualising a hypermap H. This is
a bipartite map on the same surface asH, with black and white vertices corresponding to the
hypervertices and hyperedges of H, edges (corresponding to elements of Ω) indicating inci-
dences between them, and faces corresponding to the hyperfaces ofH. ThenH is orientably
regular if and only if the orientation- and colour-preserving automorphism group ofW acts
transitively on its edges. In this case, the map W is orientably regular if and only if there
is an automorphism of G transposing x and y (and thus inducing a duality of H), andW is
regular if and only if there is also an automorphism inverting x and y (and thus reversing the
orientations ofH andW).
A similar but more symmetric, and hence more elaborate, method of representingH is its
G. A. Jones: Exotic Behaviour of Infinite Hypermaps 53
barycentric subdivision B = B(H), a tripartite triangular map on the same surface. This has
one vertex corresponding to each hypervertex, hyperedge and hyperface of H, with incident
pairs and triples of these joined by edges and faces of B. Thus B is formed by stellatingW ,
putting a (red) vertex in each face of W , and joining it by an edge to each black or white
vertex incident with that face. Conversely, W can be reconstructed from B by deleting the
red vertices and their incident edges.
3 Group-theoretic constructions
Here we briefly summarise two important group-theoretic constructions, namely free prod-
ucts and HNN extensions. For full details, see [25, Ch. IV].
Let H and K be groups, assumed to be disjoint, with presentations H = 〈X | R 〉
and K = 〈Y | S 〉, where X and Y are sets of generators, and R and S are sets of defining
relations. The free product [25, §IV.1] ofH andK is the groupG = H ∗K with presentation
〈X ∪ Y | R ∪ S 〉; it is, up to isomorphism, independent of the choice of presentations of H
and K.
Each element g ∈ G can be written as a word
g1g2 . . . gn (n ≥ 0) (3.1)
where gi ∈ H ∪K for each i. A word (3.1) is reduced if gi 6= 1 for each i, and successive
terms gi and gi+1 do not belong to the same factor H or K: such a pair could be replaced
in (3.1) with their product in H or K, giving a representation of g by a shorter word, so it
follows that each g ∈ G can be represented by a reduced word, with the empty word (of
length n = 0) representing the identity element g = 1. The normal form theorem for free
products [25, Theorem IV.1.2] states that this reduced word is unique, and if it has length
n > 0 then g 6= 1. It follows from this that H and K are embedded isomorphically as
subgroups of G. The torsion theorem for free products [25, Theorem IV.1.6] states that the
torsion elements (those of finite order) in G are the conjugates of those in H or K.
An HNN extension [17], [25, §IV.2] is constructed from a group H and an isomorphism
φ : A→ B between two subgroups A and B of H; we define
G = 〈H, t | at = φ(a) for all a ∈ A〉,
meaning that G = F/N where F = H ∗ 〈t〉 is the free product of H and an infinite cyclic
group 〈t〉, and N is the normal closure in F of the elements atφ(a)−1 = t−1atφ(a)−1 where
a ∈ A. Equivalently, if H has a presentation 〈X | R 〉, then G has a generating set X ∪ {t},
and its defining relations consist of those in R together with the relations at = φ(a) for all
a ∈ A.
Each element g ∈ G can be written as a word
h1t
e1h2t
e2 . . . ten−1hn (n ≥ 0) (3.2)
where hi ∈ H and ei = ±1 for all i; the case n = 0 corresponds to the empty word,
representing g = 1. A word (3.2) is reduced if it contains no subword t−1hit with hi ∈ A or
thit
−1 with hi ∈ B: such a subword could be replaced with φ(hi) or φ−1(hi) respectively,
leading to a represention of g by a shorter word, so every element is represented by a reduced
word. The normal form theorem for HNN extensions [25, Theorem IV.2.1] states that this
reduced word is unique, and if n ≥ 1 then g 6= 1. Thus H and 〈t〉 are isomorphically
54 Ars Mathematica Contemporanea 1 (2008) 51–65
embedded as subgroups ofG, with conjugation by t inducing the isomorphism φ. The torsion
theorem for HNN extensions [25, Theorem IV.2.4] states that the torsion elements of G are
the conjugates in G of the torsion elements of H .
4 Counting hypermaps
For each n ∈ N there are only finitely many non-isomorphic oriented hypermaps H with
|Ω| = n; there is at least one such hypermap for each n (since the symmetric group Sn is
a 2-generator group), so the total number of finite oriented hypermaps (up to isomorphisms)
is ℵ0. In fact, if instead we use the regular representation of the cyclic group Cn we see
that the total number of finite orientably regular hypermaps is also ℵ0. When we consider
infinite hypermaps, however, we will see that the corresponding number is 2ℵ0 in each case.
To prove this, we first need some classic group-theoretic results. The proofs are outlined here
for completeness.
The following result is due to G. Higman, B. H. Neumann and H. Neumann [17]:
Theorem 1. Every countable group C can be embedded in a 2-generator group G, such that
G has an element of finite order n if and only if C has such an element.
Proof. In the free group F2 = 〈a, b〉 of rank 2, the elements ai = b−iabi (i ≥ 0) freely
generate a free subgroup A. Similarly, if C is generated by c1, c2, . . . then in the free product
H = F2 ∗ C the elements b0 = b and bi = cia−ibai (i ≥ 1) freely generate a free subgroup
B (since their images in F2 do, when we map C to 1). Using the isomorphism φ : A → B,
ai 7→ bi we can therefore form an HNN extension
G = 〈H, t | ati = bi (i ≥ 0)〉.
The group C is embedded in H and hence in G. The subgroup 〈a, t〉 of G generated by
a (= a0) and t also contains
b = b0 = at0 and ci = bia
−ib−1ai = atia
−ib−1ai = (b−iabi)ta−ib−1ai
for all i ≥ 1; the elements a, b, ci and t generate G, which is thus a 2-generator group. The
assertion about elements of finite order follows from the torsion theorems for free products
and HNN extensions. 
The first part of Theorem 1 immediately implies that every countable group can be em-
bedded in the automorphism group of some orientably regular hypermap, thus justifying the
claim made in §1. However, there is no single 2-generator group containing a copy of every
countable group, and hence there is no single orientably regular hypermap with this prop-
erty: a 2-generator group, beng countable, has only countably many 2-generator subgroups,
whereas B. H. Neumann [28] has proved the following result:
Theorem 2. There are 2ℵ0 mutually non-isomorphic 2-generator groups.
Proof. Let CP = ⊕p∈PCp where P is any set of prime numbers. This is a countable group
(since the elements of this direct sum have finite support), so by Theorem 1 it is embedded
in a 2-generator group GP which has an element of prime order p if and only if CP has one,
that is, if and only if p ∈ P . The groups GP , for distinct sets P , are therefore mutually non-
isomorphic, and there are 2ℵ0 of them since this is the number of sets P of prime numbers.

G. A. Jones: Exotic Behaviour of Infinite Hypermaps 55
Corollary 3. The free group F2 has 2ℵ0 conjugacy classes of subgroups, including 2ℵ0 nor-
mal subgroups.
Proof. Each of the 2ℵ0 2-generator groupsGP described above determines a normal subgroup
NP of the free group F2, with GP ∼= F2/NP , so we have at least 2ℵ0 conjugacy classes of
subgroups of the form {NP }. On the other hand, being countable, F2 has 2ℵ0 subsets, so it
has at most 2ℵ0 subgroups and hence at most 2ℵ0 conjugacy classes of subgroups. 
Corollary 4. There are 2ℵ0 mutually non-isomorphic oriented hypermaps, including 2ℵ0
orientably regular hypermaps.
Proof. There is one oriented hypermap for each conjugacy class of subgroups of F2, with
orientably regular hypermaps corresponding to normal subgroups. 
We can in fact prove a slightly stronger result than Corollary 4. As shown by James
[19], the group of all operations on oriented hypermaps (such as dualities and reflection) can
be identified with the outer automorphism group OutF2 = AutF2/InnF2 of F2 acting on
conjugacy classes of subgroups of F2. Since this action preserves normality of subgroups, it
preserves orientable regularity of hypermaps. Now OutF2 is a countable group (isomorphic
to GL2(Z), in fact, see [25, Ch. I, Prop. 4.5]), so we have:
Corollary 5. The group OutF2 of operations has 2ℵ0 orbits on oriented hypermaps, includ-
ing 2ℵ0 orbits consisting of orientably regular hypermaps.
In its action on normal subgroupsN of F2, OutF2 preserves their quotient groups F2/N ,
so it preserves the automorphism groups of the corresponding orientably regular hypermaps.
For example, for different sets P the normal subgroups NP in the proof of Corollary 3 have
mutually non-isomorphic quotient groups GP = F2/NP , so the corresponding orientably
regular hypermapsH lie in 2ℵ0 distinct orbits OP of OutF2, with Aut+H ∼= GP having an
element of prime order p if and only if p ∈ P .
There is a connection here with another topic from combinatorial group theory, namely
T-systems, introduced by B. H. Neumann and H. Neumann in [29]. If G is any group, then
the orientably regular hypermaps H with Aut+H ∼= G correspond to the normal subgroups
N of F2 with F2/N ∼= G. These are the kernels of the epimorphisms F2 → G, and two
such epimorphisms have the same kernel (and thus correspond to isomorphic hypermaps) if
and only if they differ by an automorphism of G. Thus these hypermaps H correspond to
the orbits of AutG on generating pairs for G. Now there is also a natural action of AutF2
on epimorphisms F2 → G, and hence on generating pairs for G, permuting the orbits of
AutG. A T-system of such pairs is an orbit of the permutation group generated by AutF2
and AutG, so two hypermaps H are equivalent under hypermap operations if and only if
they correspond to generating pairs in the same T-system for G.
The automorphism group G of any orientably regular hypermap, being a 2-generator
group, is countable, so there are at most ℵ0 epimorphisms F2 → G and hence at most
ℵ0 kernels N of such epimorphisms, or equivalently orientably regular hypermaps H with
automorphism group G. In some cases N and hence H are unique: for instance, if G = Z2
then N is the commutator subgroup F ′2 of F2, and H is the hypermap described later in
Example 6. In some other cases there are ℵ0 kernelsN and hence hypermapsH: for instance,
if G = Z then these subgroups N are the inverse images of the cyclic subgroups 〈(i, j)〉 =
〈(−i,−j)〉 of Z2 under the abelianising epimorphism F2 → F2/F ′2 ∼= Z2, with i and j
coprime integers; now GL2(Z) acts transitively on such vectors (i, j), so the corresponding
56 Ars Mathematica Contemporanea 1 (2008) 51–65
hypermaps H are all equivalent under hypermap operations. By contrast, it follows from
work of Dunwoody and Pietrowski [11] that if G is the group 〈x, y | x2 = y3〉 of the trefoil
knot, isomorphic to the three-string braid group B3 = 〈a, b | aba = bab〉, the corresponding
hypermaps form ℵ0 orbits under operations (see also Example 9).
In this section we have proved that there are uncountably many orientably regular hy-
permaps. In §6 we will show that this remains true if we restrict attention to hypermaps of
certain fixed types, and if we impose various algebraic conditions on their automorphism
groups. These results are not constructive, in the sense that they do not provide explicit de-
scriptions of the hypermaps or their groups, so first, in the next section, we will study a few
specific examples of infinite orientably regular hypermaps.
5 Baumslag-Solitar groups and hypermaps
The Baumslag-Solitar groups [1] are given by the presentation
G = BS(p, q) = 〈x, y | (xp)y = xq〉,
where p and q are non-zero integers. By inverting each side of the defining relation, if neces-
sary, we may assume that p ≥ 1.
These groups are examples of HNN extensions. In the notation of §3 we take H to be
an infinite cyclic group 〈x〉, with A and B the subgroups generated by xp and xq , we take
φ to be the isomorphism A → B defined by xp 7→ xq , and we let y play the role of t. The
resulting HNN extension G is BS(p, q), in which x and y are elements of infinite order.
Let H = H(p, q) be the orientably regular hypermap corresponding to (G, x, y), and let
W =W(p, q) be the corresponding Walsh map.
Example 6. Let G = BS(1, 1). This is the simplest of the Baumslag-Solitar groups, and
while it does not exhibit any particularly unusual behaviour, it does give rise to a number of
interesting maps and hypermaps.
An equivalent presentation for this group is
G = 〈x, y | xy = yx〉,
so G is a free abelian group of rank 2. Each element g ∈ G can be written uniquely in the
form g = xiyj where i, j ∈ Z, giving an isomorphism G → Z2, g 7→ (i, j). The Cayley
graph C of G with respect to the generators x, y can be identified with the 1-skeleton of
the standard square tessellation of R2, with edges labelled x and y pointing in the first and
second coordinate directions (to the right and upwards). The hypervertices, hyperedges and
hyperfaces of H correspond to the horizontal, vertical and diagonal lines x2 = c, x1 = c
and x1 − x2 = c (c ∈ Z). (Thus the underlying surface S of H is not the euclidean plane
on which we imagine C drawn.) Each hypervertex meets each hyperedge exactly once, so
the Walsh mapW is an embedding in S of the complete bipartite graph Kℵ0,ℵ0 of countably
infinite degree. This map is edge-transitive sinceH is orientably regular; in factW is regular
since there are automorphisms of G transposing and inverting x and y.
Every 2-generator abelian group is a quotient of G, so every orientably regular hypermap
with an abelian orientation-preserving automorphism group is a quotient ofH. For example,
if we reduce G mod (n) by factoring out the (normal) subgroup N generated by xn and yn,
the corresponding quotientH/N ofH is the Fermat hypermap of genus (n−1)(n−2)/2, with
automorphism group Z2n [23]; its Walsh map is the standard embedding of Kn,n, constructed
G. A. Jones: Exotic Behaviour of Infinite Hypermaps 57
by Biggs and White [3, §5.6.7] as a Cayley map for Z2n. More generally, taking the quotient
by 〈xn, ym〉 gives an edge-transitive embedding of Km,n.
The Cayley graph C ′ ofG with respect to the generators x, y and z can also be embedded
in the euclidean plane, namely as the 1-skeleton of the standard triangular tessellation, with
edges labelled x, y and z making angles 0, 2π/3 and 4π/3 with the first coordinate axis. In
this model, the hypervertices, hyperedges and hyperfaces can be identified with the corre-
sponding three families of parallel lines in the tessellation. Each pair from different families
meet exactly once, so the barycentric subdivision B ofH is a triangular embedding in S of the
complete tripartite graph Kℵ0,ℵ0,ℵ0 of countably infinite degree; again, this is a regular map.
Adding to G the relations xn = yn = 1 (which imply that zn = 1) we get a finite quotient of
this map, namely a regular (and minimum genus) embedding of Kn,n,n, the stellation of the
standard embedding of Kn,n considered earlier.
Example 7. Let p = 1 and q = −1, so
G = BS(1,−1) = 〈x, y | xy = x−1〉.
This is a semidirect product Z : Z of a normal subgroup X = 〈x〉 ∼= Z by a complement
Y = 〈y〉 ∼= Z, with y inverting X by conjugation. As in Example 6, each g ∈ G has the
unique form xiyj where i, j ∈ Z. The Cayley graph C for G with respect to x and y is as in
Example 6, except that x-edges in alternate horizontal lines x2 = c point to the right or left
as c is even or odd. The Walsh mapW is again an edge-transitive embedding of the complete
bipartite graph Kℵ0,ℵ0 ; in this caseW is not orientably regular, since no automorphism of G
can transpose x and y.
Example 8. Let
G = BS(1, q) = 〈x, y | xy = xq〉.
where q ≥ 2. Let xr denote xy
r
for each r ∈ Z, so by iterating the relation xy = xq we have
xr = xq
r
for all r ≥ 0. Thus
xr = xq
r−s
s
if r ≥ s ≥ 0, so conjugating by negative powers of y we see that this equation holds whenever
r ≥ s. This implies that xr and xs commute for all r and s, so the normal closure XG of
X = 〈x〉 in G, which is generated by the elements xr (r ∈ Z), is abelian. This group is
isomorphic to the additive group A = Z[ 1q ] = {aq
r | a, r ∈ Z} of q-adic rationals, with each
xr corresponding to qr. Using the equations yxr = xr−1y and y−1xr = xr+1y−1, each
word in x and y can be rewritten in the unique form xiyj where i ∈ A and j ∈ Z; here, if
i = aqr then xi denotes the element xar of X
G. Thus G is a semidirect product of XG ∼= A
by Y = 〈y〉 ∼= Z, with the action of y by conjugation on XG corresponding to multiplication
by q on A.
Let C be the Cayley graph of G with respect to x and y. The vertices xiyj can be
identified with the points (i, j) ∈ A×Z ⊂ R2. At each such vertex, there is an edge labelled
y from (i, j) to (i, j + 1), and (since xiyjx = xix−jyj) an edge labelled x from (i, j) to
(i + q−j , j). This is therefore not an embedding of C in the plane: instead the relation
xy = xq determines a set of overlapping rectangles [i, i + q−j ] × [j, j + 1] for i ∈ A and
j ∈ Z, with an edge labelled y along each of the the vertical sides, q edges labelled x along
the top, and one along the bottom.
The hyperedges of H correspond to the cosets gY = xiyjY = xiY of Y in G; distinct
elements xi lie in distinct cosets, so we can identify the hyperedges with the elements i ∈
58 Ars Mathematica Contemporanea 1 (2008) 51–65
A. Hypervertices correspond to cosets gX = xiyjX = yjxiq
j
X of X , with yjxiq
j
X =
yj
′
xi
′qj
′
X if and only if j = j′ and iqj−i′qj′ ∈ Z, so they can be identified with the elements
of the disjoint union of the groups A/〈q−j〉 for all j ∈ Z. Just as A is a dense subgroup of
the line R, each A/〈q−j〉 is a dense subgroup of the circle R/〈q−j〉, the circumference
q−j decreasing geometrically towards 0 as j increases. Hyperedges and hypervertices are
incident if and only if the corresponding group elements are related by the natural projection
A→ A/〈q−j〉.
Let us define Hr = 〈xr〉 = 〈xq
r 〉 ≤ G for each r ∈ Z, and Hr = H/Hr, a non-
regular quotient of H since the subgroup Hr is not normal in G. Now xr acts on H by left-
multiplication g 7→ x−1r g, so the hyperedges of Hr correspond to double cosets HrgY =
Hrx
iyjY = HrxiY , the hypervertices correspond to double cosets HrgX = HrxiyjX ,
and their incidences correspond to cosets Hrg = Hrxiyj .
One can form Hr from H by replacing the hyperedge set A of H with A/〈qr〉, and in
the hypervertex set replacing each A/〈q−j〉 with A/〈qr〉 if −j > r, so the large circles now
have constant circumference qr, while the small circles still have circumference q−j → 0.
As before, incidence is given by the natural projection.
The index q normal inclusions
· · · < H1 < H0 < H−1 < · · ·
give rise to q-sheeted regular coverings
· · · → H1 → H0 → H−1 → · · · .
SinceHyr = Hr+1 for all r, these hypermapsHr are all isomorphic to each other. (Of course,
simple counting arguments imply that a finite hypermap can never be isomorphic to a proper
quotient or covering of itself.)
For each r we have Aut+Hr ∼= NG(Hr)/Hr = XG/Hr ∼= A/〈qr〉. These groups are
all isomorphic to each other, and in particular they are isomorphic to the group Zq∞ , the
direct limit of the sequence of natural embeddings Zq → Zq2 → Zq3 → · · · .
Example 9. Let G = BS(2, 3). This is a well-known example of a non-Hopfian group. We
say that a group G is Hopfian if the only normal subgroup N with G/N ∼= G is the identity
subgroup, or equivalently, if every epimorphism G → G is an automorphism. To prove that
G is non-Hopfian, one first checks that x 7→ x2, y 7→ y defines an epimorphism θ from G to
G: it is a homomorphism since ((xθ)2)yθ = (x4)y = x6 = (xθ)3, and it is an epimorphism
since the image contains x2 and y, and thus contains (x2)y = x3 and hence also x = x3x−2;
now the commutator g = [xy, x] is in ker θ since (xθ)yθ = (x2)y = x3 commutes with
xθ = x2, and g 6= 1 by the normal form theorem for HNN extensions. (In fact, Baumslag
and Solitar showed that B(p, q) is Hopfian if and only if one of p and q divides the other or
they have the same prime factors.)
This implies that, although the orientably regular hypermapH corresponding to (G, x, y)
is not isomorphic to its regular quotient H = H/ ker θ, nevertheless Aut+H ∼= G ∼=
G/ ker θ ∼= Aut+H. By iterating this we get a sequence
H → H → H → · · ·
of mutually non-isomorphic orientably regular hypermaps, with
Aut+H ∼= Aut+H ∼= Aut+H ∼= · · · .
G. A. Jones: Exotic Behaviour of Infinite Hypermaps 59
In fact, Baumslag and Solitar showed that G and
G∗ := 〈x, y | (x2)y = x3, ([x, y]2y−1)2 = 1〉
are non-isomorphic groups, and each is an epimorphic image of the other. This implies that
there is a sequence
H0 → H∗0 → H1 → H∗1 → H2 → H∗2 → · · ·
of coverings of orientably regular hypermaps, where H0 and H∗0 are the hypermaps corre-
sponding to G and G∗, with Aut+Hi ∼= G and Aut+H∗i ∼= G∗ for all i.
Brunner [4] has shown that G is generated by x2
n
and y for each n ≥ 0, and that the
kernels of the corresponding epimorphisms F2 → G are inequivalent under AutF2, so the
group OutF2 of hypermap operations has ℵ0 orbits on orientably regular hypermaps with
automorphism group G (see §4). For another example of this phenomenon see [5].
6 Infinite quotients of triangle groups
The type of a hypermapH is the triple τ = (l,m, n) where l,m and n are the least common
multiples of the valencies of its hypervertices, hyperedges and hyperfaces; these can take
any values from {1, 2, 3, . . . ,∞}. We say that H has type dividing τ if these least common
multiples (and hence all the corresponding valencies) divide l,m and n; by convention, ev-
ery natural number divides ∞. For example, maps are simply hypermaps of type dividing
(∞, 2,∞).
An oriented hypermapH has type dividing τ if and only if the permutation representation
θ : F2 → Sym Ω in §2 can be factored through the triangle group
∆(τ) = 〈X,Y, Z | X l = Y m = Zn = XY Z = 1〉.
It follows that oriented hypermaps of type dividing τ correspond to conjugacy classes of
subgroups of ∆(τ), and orientably regular hypermaps correspond to normal subgroups of
∆(τ), with the quotient isomorphic to Aut+H.
One can realise ∆(τ) as the orientation-preserving automorphism group of the universal
hypermap U(τ) of type τ . Let S be the sphere, the euclidean plane or the hyperbolic plane
as l−1 + m−1 + n−1 > 1, = 1 or < 1, and let ABC be a triangle in S with internal angles
π/l, π/m and π/n. (If, for instance, l =∞ we take A to be an ideal vertex on the boundary
of S, and AB and AC to be parallel lines meeting at A with internal angle 0.) The extended
triangle group ∆[τ ] is the group generated by the reflections of S is the sides of ABC, and
∆(τ) is the subgroup of index 2 consisting of orientation-preserving isometries of S. Its
generators X,Y and Z are rotations around A,B and C through 2π/l, 2π/m and 2π/n. The
images of ABC under ∆[τ ] form a triangular tessellation of S, and this is the barycentric
subdivision of U(τ), with the images of A,B and C representing the hypervertices, hyper-
edges and hyperfaces. Any oriented hypermap H of type dividing τ is isomorphic to the
quotient of U(τ) by a subgroup of ∆(τ). This imposes on H a Riemann surface structure,
compact if and only if H is finite. Belyı̆’s Theorem [2] implies that the Riemann surfaces
obtained from finite hypermaps are those defined, as projective algebraic curves, over alge-
braic number fields, leading to Grothendieck’s theory of dessins d’enfants [15], [23], [34];
at present there is no corresponding characterisation of the non-compact Riemann surfaces
obtained from infinite hypermaps.
60 Ars Mathematica Contemporanea 1 (2008) 51–65
We call ∆(τ) and τ spherical, euclidean or hyperbolic as S is the sphere, the euclidean
plane or the hyperbolic plane. Spherical triangle groups are all finite, and are solvable apart
from ∆(2, 3, 5) ∼= A5, while euclidean triangle groups, although infinite, are abelian by
cyclic, so neither class have particularly complicated quotients. By contrast, hyperbolic tri-
angle groups are in some sense ‘large enough’ to have many interesting quotients, and these
correspond to interesting orientably regular hypermaps of various hyperbolic types. Here we
are able to demonstrate this for most hyperbolic triangle groups, but first we must describe
some possible exceptions.
We define a hyperbolic triple τ = (l,m, n), and the associated triangle group ∆(τ), to
have small periods if τ can be obtained from one of the following thirty triples by permuting
l,m and n:
(2, 3, n) for 7 ≤ n ≤ 15, (2, 4, n) for 5 ≤ n ≤ 10,
(3, 3, n) for 4 ≤ n ≤ 10, (3, 4, n) for 4 ≤ n ≤ 5,
(4, 4, n) for 4 ≤ n ≤ 5, (l, 5, 5) for 2 ≤ l ≤ 5.
The following result is proved in [21], as part of a more general result about Fuchsian
groups:
Theorem 10. Let ∆(τ) be a hyperbolic triangle group which does not have small periods.
Then every countable group can be embedded in
(1) a simple quotient of ∆(τ), and
(2) a non-Hopfian quotient of ∆(τ).
This shows that the simple and non-Hopfian quotients of ∆(τ) are arbitrarily compli-
cated. By imitating an argument due to P. M. Neumann [30], we can deduce that there are
uncountably many of them:
Corollary 11. Let ∆(τ) be a hyperbolic triangle group which does not have small periods.
Then ∆(τ) has
(1) 2ℵ0 non-isomorphic simple quotients, and
(2) 2ℵ0 non-isomorphic non-Hopfian quotients.
Proof. By Theorem 2 there are 2ℵ0 non-isomorphic 2-generator groups; each such group
is countable, so by Theorem 10 it can be embedded in a simple (respectively non-Hopfian)
quotient G of ∆(τ). Since ∆(τ) is finitely generated, G is countable and hence has only
countably many 2-generator subgroups, so ∆(τ) must have 2ℵ0 non-isomorphic simple (re-
spectively non-Hopfian) quotients. 
Theorem 10 shows that for each hyperbolic triple τ without small periods, the orientation-
preserving automorphism group Aut+H of an orientably regular hypermapH of type divid-
ing τ can be arbitrarily complicated, even if we require it to be simple or non-Hopfian. Corol-
lary 11 implies that in either case there are 2ℵ0 non-isomorphic orientably regular hypermaps
H of type dividing τ , each having no proper regular quotients or many regular quotients
respectively. Having mutually non-isomorphic automorphism groups, these hypermaps are
mutually inequivalent under hypermap operations.
The proof of Theorem 10 uses results from small cancellation theory [25, Ch. V], where
the required hypotheses are too restrictive to apply to the hyperbolic triangle groups with
G. A. Jones: Exotic Behaviour of Infinite Hypermaps 61
small periods. Apart from this, there is little evidence to suggest that the behaviour of such
hyperbolic triangle groups should be exceptional, so we conjecture that the above results
apply to all hyperbolic triangle groups. In support of this, we note that Wilson [33] has
shown that ∆(2, 3, n) satisfies Theorem 10(1) (and hence Corollary 11(1)) for all n ≥ 7; his
method, developed with Lucchini and Tamburini in [24] for ∆(2, 3, 7), uses matrix groups
over rings as quotients.
In [21], small cancellation theory is also used to prove:
Theorem 12. If ∆(τ) is a hyperbolic triangle group which does not have small periods, then
∆(τ) has ℵ0 non-isomorphic finitely presented quotients with unsolvable word problem.
Of course, there are (up to isomorphism) only countably many finitely presented groups,
so in this case one cannot hope to obtain 2ℵ0 non-isomorphic quotients, as we do in Corol-
lary 11. This result implies that for each corresponding triple τ there are ℵ0 non-isomorphic
orientably regular hypermaps of type dividing τ which are finitely presented but have an
unsolvable word problem. Here we say that an orientably regular hypermap H is finitely
presented if the corresponding normal subgroup of F2 in the normal closure of a finite subset
F ⊂ F2, so that any relation in H of the form αg = α for all α ∈ Ω (where g ∈ F2) is a
consequence of the finite set of relations αf = α for all α ∈ Ω where f ∈ F . The word
problem for H asks, for any given g ∈ F2, whether or not H has a relation αg = α for all
α ∈ Ω; it is solvable if there is a deterministic algorithm to decide this question. As before,
it is conjectured that Theorem 12 applies to hyperbolic triangle groups in general, not just
those without small periods.
7 Tarski monster groups and hypermaps
Given an infinite orientably regular hypermap H, one might expect to obtain coverings (not
necessarily regular)H → H = H/H of many different degrees |H| by choosing appropriate
subgroups H ≤ G = Aut+H. This is not always possible, as shown by Ol’shanskii’s
construction of so-called Tarski monster groups [31]:
Theorem 13. For each sufficiently large prime p there is an infinite group G in which every
proper subgroup has order p.
Here it is sufficient to take p > 1075. Any two elements x, y ∈ G which are not powers
of each other must generateG, since they cannot be contained in a proper subgroup; now x, y
and xy must have order p, so G is a quotient of the triangle group ∆(p, p, p), and is therefore
the automorphism group of an orientably regular infinite hypermap H of type (p, p, p). The
only proper quotients of H are the p-sheeted coverings H → H = H/H where 1 < H < G
and |H| = p; there is one (up to isomorphism) for each conjugacy class of such subgroups
H . Since H is a maximal subgroup of G,H has no proper quotients.
8 Unoriented hypermaps
Our final examples are based on Grigorchuk’s group, first introduced in [12] as an explicit
example of a finitely generated infinite periodic group. This group has many other remarkable
properties, and as shown in [20], it arises as the automorphism group of an infinite orientable
map. Since some of its elements reverse the orientation, we first need to extend our general
remarks about hypermaps in §2 to remove the restriction that orientation should be preserved,
or even that it should exist.
62 Ars Mathematica Contemporanea 1 (2008) 51–65
An orientably regular hypermapH is regular (or reflexible) if it is isomorphic to its mirror
image, with the reverse orientation, or equivalently its orientation-preserving automorphism
group Aut+H has an automorphism inverting its canonical generators x and y. The corre-
sponding hypermap subgroup N ≤ F2 is then a normal subgroup of the group
∆ = ∆[∞,∞,∞] = 〈r0, r1, r2 | r2i = 1〉 ∼= C2 ∗ C2 ∗ C2, (8.1)
which contains F2 as its even subgroup ∆(∞,∞,∞) of index 2 generated by r1r2 and
r2r0; the full automorphism group AutH of H, including orientation-reversing elements,
is isomorphic to ∆/N , and the automorphism of Aut+H inverting x and y is induced by
conjugation by the image of r2.
More generally, the theory of unoriented hypermaps (possibly non-orientable or with
boundary, see [18]) is similar to that for oriented hypermaps outlined in §2, with the above
group ∆ replacing F2, so that hypermaps now correspond to transitive permutation repre-
sentations of ∆ acting on flags, or equivalently to conjugacy classes of subgroups H ≤ ∆.
The elements r0, r1 and r2 permute the flags of a hypermap H by changing the hypervertex,
hyperedge or hyperface of each flag while preserving the other two constituents; the mon-
odromy group MonH is the group of permutations of the flags which they generate, and the
automorphism group AutH is its centraliser. The hypervertices, hyperedges and hyperfaces
of H are identified with the orbits of the dihedral groups 〈r1, r2〉, 〈r2, r0〉 and 〈r0, r1〉, and
incidence corresponds to non-empty intersection. Regular hypermaps, with AutH transitive
on flags, correspond to regular permutation groups MonH, or equivalently to normal sub-
groups H of ∆. Subgroups H contained in F2 correspond to orientable hypermaps without
boundary.
9 The Grigorchuk map
If a group G has a finite generating set X = X−1 then let γX(n) denote the number of
elements of G which can be represented as words of length at most n in the elements of
X . Although the values of the function γX depend on the choice of X , its asymptotic rate of
growth as n→∞ is independent ofX , so we call this the rate of growth ofG. For many years
it was conjectured that every finitely generated group has polynomial or exponential growth
[27]; Grigorchuk’s group [12], [13], [14] was the first counterexample, having intermediate
growth (faster than polynomial but slower than exponential). There are excellent surveys of
the construction and properties of this group in [6] and [16, Ch.VIII].
It is convenient to define Grigorchuk’s groupG as a group of automorphisms of the binary
rooted tree T whose vertices are the words w of length l ≥ 0 in the alphabet {0, 1}, with each
w adjacent to wi for i = 0, 1. Automorphisms a, b, c and d of T are defined to act as follows,
fixing any vertices not specified:
• a transposes 0w and 1w for all w;
• b transposes 1m00w and 1m01w for all m ≡ 0, 1 mod (3);
• c transposes 1m00w and 1m01w for all m ≡ 0, 2 mod (3);
• d transposes 1m00w and 1m01w for all m ≡ 1, 2 mod (3).
Then G is the subgroup of AutT generated by a, b, c and d. By inspection,
a2 = b2 = c2 = d2 = bcd = 1,
G. A. Jones: Exotic Behaviour of Infinite Hypermaps 63
so G has generators a, b and c which satisfy
a2 = b2 = c2 = (bc)2 = 1. (8.2)
It follows from (8.1) and (8.2) that there are epimorphisms ∆ → ∆[∞, 2,∞] → G
with r0 7→ b, r1 7→ a and r2 7→ c, so G is the monodromy group (and hence also the full
automorphism group) of an unoriented regular map G, the Grigorchuk map, with a, b and c
acting on flags of G by changing their edge, vertex and face respectively. These flags can
be identified with the elements of G, permuted regularly. In their actions as automorphisms,
a, b and c are reflections of G, changing the edge, vertex and face of one specific flag. The
vertices correspond to the cosets of 〈a, c〉 in G, and their valency is the order of ac, namely
8. Similarly, the faces of G are all 16-gons, since this is the order of ab, so G is a hypermap
of type (8, 2, 16), i.e. a map of type {16, 8} in the notation of Coxeter and Moser [10, Ch. 8].
The Petrie polygons (closed zig-zag paths) have length equal to the order of abc = ad, namely
4, so G is a quotient of the map {16, 8}4, the universal map of type {16, 8} with Petrie length
4 [10, §8.6]. The group G is not finitely presented, but Lysionok [26] has given a recursive
presentation in which all the relations have even length in the generating set {a, b, c}, so G is
an orientable map without boundary.
In any map one can define the distance between two vertices to be their distance in the
embedded graph, that is, the least number of edges in any path in the graph joining them.
One can then define the asymptotic rate of growth of a map to be that of the function which
counts the vertices at distance at most n from a given vertex. If the map is vertex-transitive
this is independent of the choice of base vertex. Most familiar examples of infinite maps,
such as the universal maps of particular types on the euclidean or hyperbolic planes (see §6),
have polynomial or exponential growth. However, since G has finite type its rate of growth is
equivalent to that of G, so G has intermediate growth; it is perhaps the first known example
of such a map. See [20] for more on rates of growth of maps.
It is well known that a finite map (or hypermap) is regular if and only if its automorphism
group and monodromy group are isomorphic. In the infinite case, this condition is necessary
but insufficient, and what is required is that these two groups should be isomorphic, not just
as abstract groups, but as permutation groups on the flags. The following example, based on
Grigorchuk’s group and taken from [22], illustrates the problem.
Example 14. The generator a of G transposes the subtrees Ti (i = 0, 1) of T spanned by the
vertices iw, while b, c and d preserve them, so G has a subgroup N = 〈b, c, d, ba, ca, da〉 of
index 2 preserving them. Under the isomorphism T0 → T, iw 7→ w, these six generators of
N act on T0 as a, a, 1, c, d and b act on T , so this isomorphism induces an epimorphismN →
G; its kernel K0 is the subgroup of N fixing T0. Similarly the action of N on T1 induces an
epimorphism N → G with kernel K1 = Ka0 , the subgroup fixing T1. Each Ki is normal in
N but not in G (for instance d ∈ K0 but da 6∈ K0), so it has normaliser NG(Ki) = N . The
map Gi = G/Ki therefore has automorphism group AutGi ∼= NG(Ki)/Ki = N/Ki ∼= G.
The core K∗i of each Ki in G is K0 ∩K1, which fixes T0 and T1, so K∗i = 1 and hence Gi
has monodromy group MonGi ∼= G. Thus AutGi ∼= MonGi, but Gi is not regular since Ki
is not normal in G: for instance, AutGi has two orbits on edges of Gi, and two on flags.
Further properties of G are considered in [20]. For instance, G is residually finite, mean-
ing that its normal subgroups of finite index have trivial intersection, and it has no non-trivial
normal subgroups of infinite index, so G has infinitely many regular quotients, all of them
(except G itself) finite. By modifying the conditions on m in the definitions of b, c and d, one
64 Ars Mathematica Contemporanea 1 (2008) 51–65
can construct uncountably many groups, and hence uncountably many maps, with properties
similar to those described here.
References
[1] G. Baumslag and D. Solitar, Some two-generator one-relator non-Hopfian groups,
Bull. Amer. Math. Soc. 68 (1962), 199–201.
[2] G. V. Belyı̆, On Galois extensions of a maximal cyclotomic field, Math. USSR Izvestija 14 (1980),
247–256.
[3] N. L. Biggs and A. T. White, Permutation Groups and Combinatorial Structures, London
Math. Soc. Lecture Note Ser. 33, Cambridge Univ. Press, Cambridge, 1979.
[4] A. M. Brunner, Transitivity systems of certain one-relator groups, in Proc. Conf. Canberra 1973,
Lecture Notes in Math. 372, Springer, Berlin–Heidelberg–New York, 1974, pp. 131–140.
[5] A. M. Brunner, A group with an infinite number of Nielsen inequivalent one-relator presentations,
J. Algebra 42 (1974), 81–84.
[6] T. Ceccherini-Silberstein, A. Machı̀ and F. Scarabotti, Il gruppo di Grigorchuk di crescita inter-
media, Rend. Circ. Mat. Palermo 50 (2001), 67–102.
[7] R. Cori, Un code pour les graphes planaires et ses applications, Astérisque 27, Société
Mathématique de France, Paris, 1975.
[8] R. Cori and A. Machı̀, Maps, hypermaps and their automorphisms: a survey. I, II, III. Exposi-
tion. Math. 10 (1992), 403–427, 429–447, 449–467.
[9] D. Corn and D. Singerman, Regular hypermaps, European J. Combin. 9 (1988), 337–351.
[10] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 3rd ed.,
Springer, Berlin–Heidelberg–New York, 1957.
[11] M. J. Dunwoody and A. Pietrowski, Presentations of the trefoil group, Canad. Math. Bull. 16
(1973), 517–520.
[12] R. I. Grigorchuk, On Burnside’s problem on periodic groups, Funct. Anal. Appl. 14 (1980), 41–
43; transl. of Funkt. Anal. Prilozh. 14 (1980), 53–54.
[13] R. I. Grigorchuk, On Milnor’s problem of group growth, Soviet Math. Dokl. 28 (1983), 23–26;
transl. of Dokl. Akad. Nauk SSSR 271 (1983), 31–33.
[14] R. I. Grigorchuk, Degrees of growth of finitely generated groups, and the theory of invariant
means, Math. USSR Izv. 25 (1985), 259-300; transl. of Izv. Akad. Nauk SSSR, Ser. Mat. 48 (1984),
939–985.
[15] A. Grothendieck, Esquisse d’un programme (French, with an English translation), in Geometric
Galois Actions 1, eds. L. Schneps and P. Lochak, London Math. Soc. Lecture Note Ser. 242,
Cambridge Univ. Press, Cambridge, 1997, pp. 5–48, 243–283.
[16] P. de la Harpe, Topics in Geometric Group Theory, University of Chicago Press, Chicago, 2000.
[17] G. Higman, B. H. Neumann and H. Neumann, Embedding theorems for groups, J. London
Math. Soc. 24 (1949), 247–254.
[18] M. Izquierdo and D. Singerman, Hypermaps on surfaces with boundary, European J. Combin. 15
(1994), 159–172.
[19] L. D. James, Operations on hypermaps, and outer automorphisms, European J. Combin. 9 (1988),
551–560.
[20] G. A. Jones, The Grigorchuk map, preprint.
G. A. Jones: Exotic Behaviour of Infinite Hypermaps 65
[21] G. A. Jones and J. M. Jones, Infinite quotients of Fuchsian groups, J. Group Theory 3 (2000),
199–212.
[22] G. A. Jones, J. M. Jones and J. Wolfart, On the regularity of maps, J. Comb. Theory, Ser. B 98
(2008), 631–636.
[23] G. A. Jones and D. Singerman, Belyı̆ functions, hypermaps and Galois groups, Bull. London
Math. Soc. 28 (1996), 561–590.
[24] A. Lucchini, M. C. Tamburini and J. S. Wilson, Hurwitz groups of large rank, J. London
Math. Soc. (2) 61 (2000), 81–92.
[25] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin, 1977.
[26] I. G. Lysionok, A system of defining relations for a Grigorchuk group, Math. Notes 38 (1985),
784–792; transl. of Mat. Zametki 38 (1985), 503–516.
[27] J. Milnor, Problem 5603, Amer. Math. Monthly 75 (1968), 685–686.
[28] B. H. Neumann, Some remarks on infinite groups, J. London Math. Soc. 12 (1937), 120–127.
[29] B. H. Neumann and H. Neumann, Zwei Klassen charakteristischer Untergruppen und ihrer Fak-
torgruppen, Math. Nachr. 4 (1951), 106–125.
[30] P. M. Neumann, The SQ-universality of some finitely presented groups, J. Australian Math. Soc.
16 (1973), 1–6.
[31] A. Yu. Ol’shanskii, Groups of bounded period with subgroups of prime order, Algebra i Logika
21 (1982), 553–618.
[32] T. R. S. Walsh, Hypermaps versus bipartite maps, J. Combin. Theory Ser. B 18 (1975), 155–163.
[33] J. S. Wilson, Simple images of triangle groups, Quart. J. Math. Oxford Ser. (2) 50 (1999), 523–
531.
[34] J. Wolfart, ABC for polynomials, dessins d’enfants, and uniformization—a survey, in Ele-
mentare und Analytische Zahlentheorie (Tagungsband), Proceedings ELAZ–Conference May
24–28, 2004 (eds. W. Schwarz, J. Steuding), Steiner Verlag, Stuttgart, 2006, pp. 313–345.
(http://www.math.uni-frankfurt.de/∼wolfart/).