ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P1.01 / 1–22
https://doi.org/10.26493/1855-3974.1840.6e0
(Also available at http://amc-journal.eu)
Realisation of groups as automorphism groups
in permutational categories
Gareth A. Jones *
School of Mathematical Sciences, University of Southampton,
Southampton SO17 1BJ, UK
Received 31 October 2018, accepted 18 July 2020, published online 10 August 2021
Abstract
It is shown that in various categories, including many consisting of maps or hypermaps,
oriented or unoriented, of a given hyperbolic type, or of coverings of a suitable topological
space, every countable group A is isomorphic to the automorphism group of uncountably
many non-isomorphic objects, infinitely many of them finite if A is finite. In particular, the
latter applies to dessins d’enfants, regarded as finite oriented hypermaps.
Keywords: Permutation group, centraliser, automorphism group, map, hypermap, dessin d’enfant.
Math. Subj. Class. (2020): 05C10, 14H57, 20B25, 20B27, 52B15, 57M10
1 Introduction
In 1939 Frucht published his celebrated theorem [15] that every finite group is isomor-
phic to the automorphism group of a finite graph; in 1960, by allowing infinite graphs,
Sabidussi [46] extended this result to all groups. Similar results have been obtained, re-
alising all finite groups (or in some cases all groups, or all countable groups) as automor-
phism groups of various other mathematical structures. Examples include the following, in
chronological order: distributive lattices, by Birkhoff [4] in 1946; regular graphs of a given
degree, by Sabidussi [45] in 1957; Riemann surfaces, by Greenberg [18, 19] in 1960 and
1973; projective planes, by Mendelsohn [37] in 1972; Steiner triple and quadruple systems,
by Mendelsohn [38] in 1978; fields, by Fried and Kollár [14] in 1978; matroids of rank 3,
by Babai [1] in 1981; oriented maps and hypermaps, by Cori and Machı̀ [10] in 1982; finite
volume hyperbolic manifolds of a given dimension, by Belolipetsky and Lubotzky [3] in
*The author is grateful to Ernesto Girondo, Gabino González-Diez and Rubén Hidalgo for discussions about
dessins d’enfants which motivated this work, and also to Ashot Minasyan, Egon Schulte, David Singerman and
Alexander Zvonkin for some very helpful comments.
E-mail address: g.a.jones@maths.soton.ac.uk (Gareth A. Jones)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
2005; abstract polytopes, by Schulte and Williams [47] in 2015 and by Doignon [12] in
2016. Babai has given comprehensive surveys of this topic in [1, 2].
In many of these cases, each group is represented as the automorphism group of not
just one object, but infinitely many non-isomorphic objects. The aim of this paper is to ob-
tain results of this nature for certain ‘permutational categories’, introduced and discussed
in [24]. These are categories C which are equivalent to the category of permutation rep-
resentation of some ‘parent group’ Γ = ΓC: thus each object O in C can be identified
with a permutation representation θ : Γ → S := Sym(Ω) of Γ on some set Ω, and the
morphisms O1 → O2 can be identified with the functions Ω1 → Ω2 which commute with
the actions of Γ on the corresponding sets Ωi. They include the categories of maps or hy-
permaps on surfaces, oriented or unoriented, and possibly of a given type. Other examples
include the category of coverings of a ‘suitably nice’ topological space; this includes the
category of dessins d’enfants, regarded as finite coverings of the thrice-punctured sphere,
or equivalently as finite oriented hypermaps.
The automorphism group AutC(O) of an object O in a permutational category C is
identified with the centraliser C := CS(G) in S of the monodromy group G := θ(Γ) of
O. Now O is connected if and only if G is transitive on Ω, as we will assume throughout
this paper. Such objects correspond to conjugacy classes of subgroups of Γ, the point-
stabilisers. An important result is the following:
Theorem 1.1. AutC(O) ∼= NG(H)/H ∼= NΓ(M)/M , where H and M are the stabilisers
in G and Γ of some α ∈ Ω, and NG(H) and NΓ(M) are their normalisers.
There are analogous results in various contexts, ranging from abstract polytopes to
covering spaces, which can be regarded as special cases of Theorem 1.1. Proofs of this
result for particular categories can be found in the literature: for instance, in [28] it is
deduced for oriented maps from a more general result about morphisms in that category;
in [29, Theorem 2.2 and Corollary 2.1] a proof for dessins is briefly outlined; similar results
for covering spaces are proved in [35, Appendix] and [39, Theorem 81.2], and for abstract
polytopes in [36, Propositions 2D8 and 2E23(a)]. Theorem 1.1 follows immediately from
the following ‘folklore’ result, proved in [26, Theorem 2(1)] (see also [44, Theorem 3.2]):
Theorem 1.2. Let G be a transitive permutation group on a set Ω, with H the stabiliser
of some α ∈ Ω, and let C := CS(G) be the centraliser of G in the symmetric group
S := Sym(Ω). Then C ∼= NG(H)/H .
Of course, finite objects in any category have finite automorphism groups. In most of
the permutational categories we will consider, the parent groups are finitely generated, so
by Theorem 1.1 the automorphism groups of connected objects are all countable. Let us
define a category C to be countably (resp. finitely) abundant if every countable (resp. finite)
group A is isomorphic to AutC(O) for some connected object (resp. finite connected ob-
ject) O in C. Let us define C to be countably (resp. finitely) superabundant if there are 2ℵ0
(resp. ℵ0) isomorphism classes of such objects O realising each A.
In the case of a permutational category C, these properties follow immediately from
Theorem 1.1 if the associated parent group Γ has the corresponding abundance properties,
namely that every countable group A is isomorphic to NΓ(M)/M for the required number
of conjugacy classes of subgroups M of Γ, and these can be chosen to have finite index in
Γ if A is finite. If Γ is finitely generated, then the cardinalities 2ℵ0 (resp. ℵ0) are the best
that can be achieved, since they are upper bounds on the number of conjugacy classes of
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 3
subgroups (resp. subgroups of finite index) in Γ, and hence on the number of isomorphism
classes of objects (resp. finite objects) available.
We will be mainly concerned with permutational categories consisting of maps and
hypermaps of various types (p, q, r), where p, q, r ∈ N ∪ {∞}. For these, the parent
groups are either extended triangle groups ∆[p, q, r], generated by reflections in the sides of
a triangle with internal angles π/p, π/q and π/r, or (for subcategories of oriented objects)
their orientation-preserving subgroups, the triangle groups ∆(p, q, r). We say that a triple
(p, q, r) is spherical, euclidean or hyperbolic as p−1 + q−1 + r−1 > 1,= 1 or < 1
respectively (where by convention we take ∞−1 = 0), so that these groups act on the
sphere, euclidean plane, or hyperbolic plane. We will call the triple cocompact if these
groups act cocompactly, or equivalently p, q, r ∈ N. We will use Theorem 1.1 to prove:
Theorem 1.3.
(a) If (p, q, r) is a hyperbolic triple, where p, q, r ∈ N∪{∞}, then the groups ∆(p, q, r)
and ∆[p, q, r], together with their associated categories of oriented hypermaps and
of all hypermaps of type (p, q, r), are finitely superabundant.
(b) If, in addition, (p, q, r) is not cocompact then these groups and categories are all
countably superabundant.
By contrast, if we take Γ to be a Tarski monster [42], an infinite group in which every
subgroup M ̸= Γ, 1 has order p for some (very large) prime p, then NΓ(M) = M for each
M ̸= 1, and hence the only groups realised as automorphism groups in the corresponding
category are 1 and Γ.
The spherical and euclidean triples must be excluded from Theorem 1.3 since the cor-
responding triangle groups are either finite or solvable, so the same restriction applies to
the automorphism groups of connected objects in the associated categories. By taking
p = r = ∞ and q = 2 or ∞ we see that the categories M and H of all maps and hy-
permaps, together with their subcategories M+ and H+ of oriented maps and hypermaps,
satisfy Theorem 1.3. In the case of M+ and H+, Cori and Machı̀ [9] showed in 1982 that
every finite group arises as an automorphism group; they considered only finite groups,
but their proof extends to countable groups. In fact, by Theorem 1.3(a) the category of
Grothendieck’s dessins d’enfants [20] of any given hyperbolic type is finitely superabun-
dant. Of course these categories are not countably abundant. Nevertheless, in §8 we will
prove a result, based on work of Conder [7, 8] on alternating and symmetric quotients of
triangle groups, to support the following conjecture:
Conjecture 1.4. The non-cocompactness condition can be omitted from Theorem 1.3(b),
so that the triangle groups ∆(p, q, r) and ∆[p, q, r] of any hyperbolic type, and their asso-
ciated categories, are countably superabundant.
The proof of Theorem 1.3 is divided into several cases, depending on the particular
group Γ = ΓC involved and whether we wish to realise countable or finite groups. In each
case we construct a primitive permutation representation of Γ, of infinite or unbounded
finite degree, such that a point stabiliser N has an epimorphism onto a free group of count-
ably infinite or unbounded finite rank, and hence onto an arbitrary countable or finite group
A. By arranging that the kernel M is not normal in Γ we see from the maximality of N
in Γ that NΓ(M)/M = N/M ∼= A, so Theorem 1.1 gives AutC(O) ∼= A for the object
O ∈ C corresponding to M . Variations in the constructions yield 2ℵ0 or ℵ0 conjugacy
4 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
classes of such subgroups M ≤ Γ, and hence that number of objects O realising A. These
objects are regular coverings, with covering group A, of the object N ∼= O/AutC(O) in C
corresponding to N and its conjugates.
In fact a deep result of Belolipetsky and Lubotzky [3, Theorem 2.1] implies finite su-
perabundance for every finitely generated group which is large, that is, has a subgroup of
finite index with an epimorphism onto a non-abelian free group. This applies to every non-
elementary finitely generated Fuchsian group, and in particular to every hyperbolic triangle
group, as in Theorem 1.3(a). However, the proof of [3, Theorem 2.1] is long, delicate and
non-constructive, so here we offer a shorter, more direct argument, specific to the context
of this paper in using maps and hypermaps.
One should not confuse countable abundance with the SQ-universality of a group Γ,
a concept introduced by P. M. Neumann in [41], and proved there for (among others) all
hyperbolic triangle groups and extended triangle groups: this requires that every countable
group is isomorphic to a subgroup of a quotient of Γ, that is, to N/M where M ≤ N ≤ Γ
and M is normal in Γ, so that NΓ(M) = Γ, while countable abundance requires that
NΓ(M) = N . In terms of permutational categories, SQ-universality of the parent group
Γ means that every countable group A is embedded in the automorphism group of some
regular object O (one with a transitive monodromy group G, so that M is normal in Γ and
AutC(O) ∼= Γ/M ∼= G), whereas countable abundance means that A is isomorphic to the
automorphism group of some object, not necessarily regular. Both properties mean that any
phenomenon exhibited by some countable group, no matter how exotic or pathological, can
be realised within Γ, and hence within C: see [23] for some examples where C = H+.
Soon after this paper was submitted, a very interesting paper [5] by Bottinelli, Grave
de Peralta and Kolpakov appeared on the arXiv. It independently introduces some of the
concepts and proves some of the results presented here: for instance their concept of a
‘telescopic group’ coincides with our notion of finite abundance, and they prove this for all
free products of cyclic groups (except C2 ∗ C2). However, their methods of construction
differ substantially from ours, and they obtain asymptotic estimates for the number of finite
objects realising a given finite group as their automorphism group, a topic not considered
here.
2 Permutational categories
Following [24], let us define a permutational category C to be a category which is equiv-
alent to the category of permutation representations θ : Γ → S := Sym(Ω) of a parent
group Γ = ΓC. We then define the automorphism group Aut(O) = AutC(O) of an object
O in C to be the group of all permutations of Ω commuting with the action of Γ on Ω;
thus it is the centraliser CS(G) of the monodromy group G = θ(Γ) of O in the symmetric
group S. In this paper we will restrict our attention to the connected objects O in C, those
corresponding to transitive representations of Γ. We will pay particular attention to those
categories for which the parent group Γ is either an extended triangle group
∆[p, q, r] = ⟨R0, R1, R2 | R2i = (R1R2)p = (R2R0)q = (R0R1)r = 1⟩,
or its orientation-preserving subgroup of index 2, the triangle group
∆(p, q, r) = ⟨X,Y, Z | Xp = Y q = Zr = XY Z = 1⟩,
where X = R1R2, Y = R2R0 and Z = R0R1. Here p, q, r ∈ N ∪ {∞}, and we ignore
any relations of the form W∞ = 1. We will now give some important examples of such
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 5
categories; for more details, see [24]. In what follows, Cn denotes a cyclic group of order
n, Fn denotes a free group of rank n, V4 denotes a Klein four-group C2×C2 and ∗ denotes
a free product.
1. The category M of all maps on surfaces (possibly non-orientable or with boundary) has
parent group
Γ = ΓM = ∆[∞, 2,∞] ∼= V4 ∗ C2.
This group acts on the set Ω of incident vertex-edge-face flags of a map (equivalently, the
faces of its barycentric subdivision), with each generator Ri (i = 0, 1, 2) changing the
i-dimensional component of each flag (whenever possible) while preserving the other two.
2. The subcategory M+ of M consists of the oriented maps, those in which the underlying
surface is oriented and without boundary. This category has parent group
Γ = ΓM+ = ∆(∞, 2,∞) ∼= C∞ ∗ C2,
the orientation-preserving subgroup of index 2 in ∆[∞, 2,∞]. This group acts on the
directed edges of an oriented map: X uses the local orientation to rotate them about their
target vertices, and the involution Y reverses their direction, so that Z rotates them around
incident faces. Here, and in the preceding example, ∆(p, 2, r) and ∆[p, 2, r] are the parent
groups for the subcategories of maps of type {r, p} in the notation of [10], meaning that
the valencies of all vertices and faces divide p and r respectively, so that Xp = Zr = 1.
(By convention, all positive integers divide ∞.)
3. Hypermaps are natural generalisation of maps, without the restriction that each edge is
incident with at most two vertices and faces which implies that Y 2 = 1. There are several
ways of defining or representing hypermaps. The most convenient way is via the Walsh
bipartite map [54], where the black and white vertices correspond to the hypervertices and
hyperedges of the hypermap, the edges correspond to incidences between them, and the
faces correspond to its hyperfaces. The category H of all hypermaps (possibly unoriented
and with boundary) has parent group
Γ = ΓH = ∆[∞,∞,∞] ∼= C2 ∗ C2 ∗ C2.
This group acts on the incident edge-face pairs of the bipartite map, with R0 and R1 pre-
serving the face and the incident white and black vertex respectively, while R2 preserves
the edge. As in the case of maps, ∆[p, q, r] is the parent group for the subcategory of
hypermaps of type (p, q, r).
4. For the subcategory H+ of oriented hypermaps, where the underlying surface is oriented
and without boundary, the parent group is the even subgroup
Γ = ΓH+ = ∆(∞,∞,∞) ∼= C∞ ∗ C∞ ∼= F2
of index 2 in ∆[∞, 2,∞]. This acts on the edges of the bipartite map, with X and Y
using the local orientation to rotate them around their incident black and white vertices,
so that Z rotates them around incident faces. Again ∆(p, q, r) is the parent group for the
subcategory of oriented hypermaps of type (p, q, r). Hypermaps of type (p, 2, r) can be re-
garded as maps of type {r, p} by deleting their white vertices; conversely maps correspond
to hypermaps with q = 2.
6 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
5. One can regard the category D of dessins d’enfants, introduced by Grothendieck [20],
as the subcategory of H+ consisting of its finite objects, where the bipartite graph is finite
and the surface is compact. The parent group is Γ = ∆(∞,∞,∞) ∼= F2, and its action is
the same as for H+.
Here we briefly mention two other classes of permutational categories where Theo-
rem 1.1 applies.
6. Abstract polytopes [36] are higher-dimensional generalisations of maps. Those n-
polytopes associated with the Schläfli symbol {p1, . . . , pn−1} can be regarded as transitive
permutation representations of the string Coxeter group Γ with presentation
⟨R0, . . . , Rn | R2i = (Ri−1Ri)pi = (RiRj)2 = 1 (|i− j| > 1)⟩,
acting on flags. For instance maps, in Example 1, correspond to the symbol {∞,∞}.
However, in higher dimensions, not all transitive representations of Γ correspond to abstract
polytopes, since they need to satisfy the intersection property [36, Proposition 2B10].
7. Under suitable connectedness conditions (see [35, 39] for example), the connected, un-
branched coverings Y → X of a topological space X can be identified with the transitive
permutation representations θ : Γ → S = Sym(Ω) of its fundamental group Γ = π1X ,
acting by unique path-lifting on the fibre Ω over a base-point in X . The automorphism
group of an object Y → X in this category is its group of covering transformations, the
centraliser in S of the monodromy group θ(Γ) of the covering. For instance, dessins (see
Example 5 above) correspond to finite unbranched coverings of the thrice-punctured sphere
X = P1(C)\{0, 1,∞} = C\{0, 1}, and hence to transitive finite permutation representa-
tions of its fundamental group Γ = π1X ∼= F2 ∼= ∆(∞,∞,∞). If we compactify surfaces
by filling in punctures, then the unit interval [0, 1] ⊂ P1(C) lifts to a bipartite map on the
covering surface Y , with black and white vertices over 0 and 1, and face-centres over ∞.
See [17, 29, 31] for further details of these and other properties of dessins.
3 Preliminary results
In this section we will prove some general results which ensure that certain groups have
various automorphism realisation properties.
Lemma 3.1. Let θ : Γ → Γ′ be an epimorphism of groups. If Γ′ is finitely or countably
abundant or superabundant, then so is Γ.
Proof. If A ∼= N ′/M ′ where M ′ ≤ N ′ ≤ Γ′ and N ′ = NΓ′(M ′), then A ∼= N/M
where M = θ−1(M ′) and N = θ−1(N ′) = NΓ(M), with |Γ : M | = |Γ′ : M ′|, so Γ
inherits finite or countable abundance from Γ′. Moreover, non-conjugate subgroups M ′ lift
to non-conjugate subgroups M , so the superabundance properties are also inherited.
Our basic tool for proving finite superabundance will be the following:
Proposition 3.2. Let Γ be a group with a sequence {Nn | n ≥ n0} of maximal subgroups
Nn of finite index such that for each a, d ∈ N there is some n with |Nn : Kn| > a, where
Kn is the core of Nn in Γ, and there is an epimorphism Nn → Fd. Then Γ is finitely
superabundant.
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 7
Proof. Any finite group A is an d-generator group for some d ∈ N, so there is an epimor-
phism Fd → A. By hypothesis, for some maximal subgroup N = Nn of Γ there is an
epimorphism N → Fd, and the core K of N satisfies |N : K| > |A|. Composition gives
an epimorphism N → A, and hence a normal subgroup M of N with N/M ∼= A. Then
NΓ(M) ≥ N , so the maximality of N implies that either N = NΓ(M) or M is a normal
subgroup of Γ. If M is normal in Γ then M must be contained in the core K of N , so that
|N : M | ≥ |N : K|. But this is impossible, since |N : M | = |A| and we chose N = Nn
so that |N : K| > |A|. Hence N = NΓ(M), as required. Moreover, given A we can
find such subgroups N with |N : K| arbitrarily large, so infinitely many of them are mu-
tually non-conjugate, and hence so are their corresponding subgroups M , since conjugate
subgroups have conjugate normalisers.
In order to deal with countable abundance or superabundance we need an analogue
of Proposition 3.2 for countable groups A. Here we have the advantage that, instead of
an infinite sequence of maximal subgroups, which are finitely generated if Γ is, a single
infinitely generated maximal subgroup is sufficient. However, when A is infinite we cannot
ensure that M is not normal in Γ simply by comparing indices of subgroups, since these
are not finite; a new idea is therefore needed.
Proposition 3.3. Let Γ be a group with a non-normal maximal subgroup N and an epi-
morphism ϕ : N → F∞. Then Γ is countably abundant. Moreover, each countable group
A ̸= 1 is realised as NΓ(M)/M by 2ℵ0 conjugacy classes of subgroups M in Γ with
NΓ(M) = N .
Proof. Given any countable group A there exist epimorphisms α : F∞ → A; composing
any of these with the epimorphism ϕ : N → F∞ gives an epimorphism ϕ ◦ α : N → A,
and hence a normal subgroup M = ker(ϕ ◦ α) of N with N/M ∼= A. As before, the
maximality of N implies that either N = NΓ(M), as required, or M is a normal subgroup
of Γ. In the latter case M is contained in the core K of N in Γ, so to prove the result we
need to show that we can choose α so that M ̸≤ K. Since N is not normal in Γ we have
N \K ̸= ∅, so choose any element g ∈ N \K, and define f := gϕ ∈ F∞. Then we can
choose α : F∞ → A so that all of the (finitely many) free generators of F∞ appearing in f
are in ker(α), and hence g ∈ M . Thus M ̸≤ K, so Γ is countably abundant.
If A ̸= 1 we can choose such epimorphisms α with 2ℵ0 different kernels, lifting back
to distinct subgroups M of Γ; these all have normaliser N , which is its own normaliser in
Γ, so they are mutually non-conjugate in Γ, giving us 2ℵ0 conjugacy classes of subgroups
M realising A.
Remark 3.4. Unfortunately, if A = 1 then α is unique, so that M = N , and the subgroup
N yields only one conjugacy class of subgroups realising A. In this case, in order to prove
that Γ is countably superabundant by this construction we would need to find not one but
2ℵ0 conjugacy classes of non-normal maximal subgroups N . For certain specific groups Γ
we will be able to do this.
4 Finite superabundance of hyperbolic triangle and extended triangle
groups
In this section we will use Proposition 3.2 to prove Theorem 1.3(a).
8 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
Case 1: Γ = ∆(p, q, r), cocompact. First assume that Γ = ∆(p, q, r), acting cocompactly
on the hyperbolic plane, that is, with finite periods p, q and r. By Dirichlet’s Theorem on
primes in an arithmetic progression, there are infinitely many primes n ≡ −1 mod (l),
where l := lcm{2p, 2q, 2r}. For each such n there is an epimorphism θn : Γ → PSL2(n)
sending the standard generators X,Y and Z of Γ to elements x, y and z of PSL2(n) of
orders p, q and r (see [16, Corollary C], for example). This gives an action of Γ on the
projective line P1(Fn), which is doubly transitive and hence primitive, so the subgroup Nn
of Γ fixing ∞ is a non-normal maximal subgroup of index n+1. Since p, q and r all divide
(n+ 1)/2, the elements x, y and z are semi-regular permutations on P1(Fn), with all their
cycles of length p, q or r. Thus no non-identity powers of X , Y or Z have fixed points, so
by a theorem of Singerman [48] Nn is a surface group
Nn = ⟨Ai, Bi (i = 1, . . . , g) |
g∏
i=1
[Ai, Bi] = 1⟩
of genus g given by the Riemann–Hurwitz formula:
2(g − 1) = (n+ 1)
(
1− 1
p
− 1
q
− 1
r
)
. (4.1)
This shows that g → ∞ as n → ∞. Now we can map Nn onto the free group Fg by sending
the generators Ai to a free basis, and the generators Bi to 1. The core Kn = ker(θn) of
Nn in Γ satisfies |Nn : Kn| = |PSL2(n)|/(n+1) = n(n− 1)/2, so Proposition 3.2 gives
the result.
Case 2: Γ = ∆(p, q, r), not cocompact. Now assume that Γ has k infinite periods p, q, r
for some k = 1, 2 or 3. We can adapt the above argument by first choosing an infinite set of
primes n ≥ 13 such that any finite periods of Γ divide (n+ 1)/2, as before. For each such
n we can map Γ onto a cocompact triangle group Γn, where each infinite period of Γ is
replaced with (n+1)/2. Since (n+1)/2 ≥ 7, the triangle group Γn is also hyperbolic, so as
before there is an epimorphism Γn → PSL2(n), giving (by composition) a primitive action
of Γ on P1(Fn). Again, no non-identity powers of any elliptic generators among X,Y and
Z have fixed points, but any parabolic generator (one of infinite order) now induces two
cycles of length (n+ 1)/2 on P1(Fn), so by [48] it introduces two parabolic generators Pi
into the standard presentation of the point-stabiliser Nn in Γ. We therefore have
Nn = ⟨Ai, Bi (i = 1, . . . , g), Pi (i = 1, . . . , 2k) |
g∏
i=1
[Ai, Bi].
2k∏
i=1
Pi = 1⟩,
a free group of rank 2g + 2k − 1, where the Riemann–Hurwitz formula now gives
2(g − 1) + 2k = (n+ 1)
(
1− 1
p
− 1
q
− 1
r
)
(4.2)
with 1/∞ = 0. Since k ≤ 3 we have g → ∞ as n → ∞, so Proposition 3.2 again gives
the result.
Case 3: Γ = ∆[p, q, r]. The proof when Γ is an extended triangle group ∆[p, q, r] of
hyperbolic type is similar to that for ∆(p, q, r). If Γ is cocompact then, as before, we
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 9
consider epimorphisms θn : Γ+ = ∆(p, q, r) → PSL2(n) for primes n ≡ −1 mod (l),
where now l = lcm{2p, 2q, 2r, 4}; the stabilisers of ∞ form a series of maximal subgroups
Nn of index n + 1 in Γ+. By an observation of Singerman [50] the core Kn of Nn in
Γ+ is normal in Γ, with quotient Γ/Kn isomorphic to PSL2(n) × C2 or PGL2(n) as
the automorphism of PSL2(n) inverting x and y is inner or not. Thus θn extends to a
homomorphism θ∗n : Γ → PGL2(n); in the first case its image is PSL2(n) and its kernel
K∗n contains Kn with index 2, and in the second case it is an epimorphism with kernel
K∗n = Kn. In either case the action of Γ
+ on P1(Fn) extends to an action of Γ, and the
stabiliser N∗n of ∞ is a maximal subgroup of index n + 1 in Γ, containing Nn with index
2. In order to apply Proposition 3.2 to these subgroups N∗n it is sufficient to show that they
map onto free groups of unbounded rank.
Now N∗n is a non-euclidean crystallographic (NEC) group, and Nn is its canonical
Fuchsian subgroup of index 2. We can obtain the signature of N∗n by using Hoare’s exten-
sion to NEC groups [22] of Singerman’s results [48] on subgroups of Fuchsian groups. As
before, Nn is a surface group of genus g given by (4.1). There are no elliptic or parabolic
elements in Nn, and hence none in N∗n. The reflections Ri (i = 0, 1, 2) generating Γ in-
duce involutions on P1(Fn), each with at most two fixed points. If Γ/Kn ∼= PSL2(n)×C2
these involutions are elements of PSL2(n), so they are even permutations by the simplicity
of this group, and hence they have no fixed points since n+1 ≡ 0 mod (4). Thus N∗n con-
tains no reflections; however, it is not a subgroup of Γ+, so it is a non-orientable surface
group
N∗n = ⟨G1, . . . , Gg∗ | G21 . . . G2g∗ = 1⟩
generated by glide-reflections Gi, with its genus g∗ given by the Riemann–Hurwitz formula
2− 2g = 2(2− g∗)
for the inclusion Nn ≤ N∗n, so g∗ = g + 1. Thus there is an epimorphism N∗n → Fd =
⟨X1, . . . , Xd | −⟩ where d = ⌊g∗/2⌋, given by G2i−1 7→ Xi and G2i 7→ X−1i for
i = 1, . . . , d and Gg∗ 7→ 1 if g∗ is odd. Since g → ∞ as n → ∞, we have d ∼ g/2 → ∞
also, so Proposition 3.2 gives the result.
Similar arguments also deal with the case where Γ/Kn ∼= PGL2(n). Since n ≡ −1
mod (4), each generating reflection Ri of Γ induces an odd permutation of P1(Fn) with
two fixed points, contributing two reflections to the standard presentation of the NEC group
N∗n. The Riemann–Hurwitz formula for the inclusion Nn ≤ N∗n then takes the form
2− 2g = 2(2− h∗ + s),
where h∗ = 2g∗ or g∗ as N∗n has an orientable or non-orientable quotient surface of genus
g∗ with s boundary components for some s ≤ 6. Thus h∗ ∼ g → ∞ as n → ∞. We
obtain an epimorphism N∗n → Fd with d ∼ h∗/2 as in the orientable or non-orientable
cases above, this time by mapping the additional standard generators of N∗n, associated
with the boundary components, to 1, so Proposition 3.2 again gives the result. Finally, in
the non-cocompact case, any periods p, q, r = ∞ can be dealt with as above for ∆(p, q, r).
This completes the proof of Theorem 1.3(a) .
Remark 4.1. It seems plausible that an argument based on the Čebotarev Density Theorem
would show that, given Γ = ∆[p, q, r], the cases Γ/Kn ∼= PSL2(n) × C2 and PGL2(n)
each occur for infinitely many primes n ≡ −1 mod (l), so that only one case would need to
10 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
be considered; however, the resulting shortening of the proof would not justify the effort.
Nevertheless this dichotomy, for general prime powers n, is interesting in its own right and
deserves further study.
Remark 4.2. The restrictions on the prime n in the above proof are partly for convenience
of exposition, rather than necessity. Relaxing them would allow X,Y and Z to have one
or two fixed points on P1(Fn), thus adding extra standard generators to Nn and N∗n and
extra summands to the Riemann–Hurwitz formulae used. However, these extra terms are
bounded as n → ∞, so asymptotically they make no significant difference. One advantage
of these restrictions is that since x, y and z are semi-regular permutations, the hypermaps
realising A in Case 1 are uniform, that is, their hypervertices, hyperedges and hyperfaces
all have valencies p, q and r. If we choose these periods so that ∆(p, q, r) is cocompact,
maximal (see [49]) and non-arithmetic (see [52]), then by a result of Singerman and Syd-
dall [51, Theorem 12.1] each hypermap (regarded as a dessin) has the same automorphism
group as its underlying Riemann surface. By [49, 52] these conditions apply to ‘most’
hyperbolic triples, such as (2, 3, 13), so we have the following:
Corollary 4.3. The category of compact Riemann surfaces is finitely superabundant.
In fact, Greenberg [19, Theorem 6′] showed in 1973 that, given a compact Riemann
surface S and a finite group A ̸= 1, there is a normal covering T → S with covering
group and Aut(T ) both isomorphic to A, while Teichmüller theory yields uncountably
many compact Riemann surfaces realising A. Since the Riemann surfaces realising a finite
group A in Corollary 4.3 are uniformised by subgroups of finite index in triangle groups, by
Grothendieck’s reinterpretation [20] of Belyı̆’s Theorem they are all defined (as algebraic
curves with automorphism group A) over number fields.
5 Countable abundance of non-cocompact hyperbolic triangle groups
We now turn to Theorem 1.3(b) and consider countable abundance, starting with the hyper-
bolic triangle groups Γ = ∆(p, q, r). We would like to show that Γ satisfies the hypotheses
of Proposition 3.3, that is, it has a non-normal maximal subgroup which has an epimor-
phism onto F∞. Given Γ, it is easy to find maximal subgroups of finite index by mapping
Γ onto primitive permutation groups of finite degree; however, such subgroups are finitely
generated, so they do not map onto F∞; a maximal subgroup of infinite index is needed,
and these seem to be harder to find. They certainly exist: by a result of Ol’shanskiı̆ [43],
Γ has a quotient Q ̸= 1 with no proper subgroups of finite index; by Zorn’s Lemma, Q
has maximal subgroups, which must have infinite index, and these lift back to maximal
subgroups of infinite index in Γ. These are not normal (otherwise they would have prime
index), but does one of them map onto F∞? Conceivably, they could be generated by ellip-
tic elements, which have finite order, in which case they would not map onto a free group
of any rank. As a first step we consider the case where Γ is not cocompact, that is, it has an
infinite period, so it is a free product of two cyclic groups. For simplicity of exposition we
first consider countable abundance, postponing superabundance until the next section.
Theorem 5.1. If Γ is a non-cocompact hyperbolic triangle group ∆(p, q, r), then Γ and
the corresponding category of oriented hypermaps are countably abundant.
Proof. By Proposition 3.3 it is sufficient to show that Γ has a non-normal maximal sub-
group N with an epimorphism N → F∞. Using the usual isomorphisms between triangle
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 11
groups, we may assume that r = ∞, and that p ≥ 3 and q ≥ 2, so that Γ ∼= Cp ∗ Cq with
p, q ∈ N ∪ {∞}.
Case 1: p = 3, q = 2. First we consider the case where p = 3 and q = 2, so that Γ is iso-
morphic to the modular group PSL2(Z) ∼= C3 ∗C2. We can construct a maximal subgroup
N of infinite index in Γ as the point stabiliser in a primitive permutation representation of
Γ of infinite degree. Since Γ is the parent group
∆(3, 2,∞) = ⟨X,Y, Z | X3 = Y 2 = XY Z = 1⟩
for the category C = M+3 of oriented trivalent maps, we can take N to be the subgroup of
Γ corresponding to an infinite map N3 in C.
−3−2−1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
4n− 2
4n− 1
4n
4n+ 1
1− n
F
F1 F2 F3 F4 Fn (n ≥ 3)
Figure 1: The trivalent map N3.
We will take N3 to be the infinite planar trivalent map shown in Figure 1, oriented with
the positive (anticlockwise) orientation of the plane. The monodromy group G = ⟨x, y⟩ of
this map gives a transitive permutation representation θ : Γ → G,X 7→ x, Y 7→ y, Z 7→ z
of Γ on the set Ω of directed edges of N3, with x rotating them anticlockwise around their
target vertices, and y reversing their direction. The vertices, all of valency 3, correspond
to the 3-cycles of x (it has no fixed points). The edges correspond to the cycles of y, with
three free edges corresponding to its fixed points and the other edges corresponding to its
2-cycles. The faces correspond to the cycles of z = yx−1, and in particular, the directed
edge α labelled 0 and fixed by y is in an infinite cycle C = (. . . , αz−1, α, αz, . . .) of z,
corresponding to the unbounded face F of N3; the directed edges αzi in C are indicated
by integers i in Figure 1. The unlabelled directed edges are fixed points of z, one incident
with each 1-valent face. The pattern seen in Figure 1 repeats to the right in the obvious
way. The ‘flowers’ Fn (n ≥ 1) above the horizontal axis continue indefinitely to the right,
with Fn an identical copy of F1 for each n ≥ 3; we will later need the fact that for each
n ≥ 2 the ‘stem’ of Fn (the vertical edge connecting it to the horizontal axis) carries two
directed edges in C, with only one of their two labels divisible by n.
Lemma 5.2. The group Γ acts primitively on Ω.
Proof. Suppose that ∼ is a Γ-invariant equivalence relation on Ω; we need to show that it
is either the identity or the universal relation. Since αy = α, the equivalence class E = [α]
containing α satisfies Ey = E. Since ⟨Z⟩ acts regularly on C we can identify C with Z by
identifying each αzi ∈ C with the integer i, so that Z acts by i 7→ i+ 1. Then ∼ restricts
to a translation-invariant equivalence relation on Z, which must be congruence mod (n) for
12 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
some n ∈ N∪{∞}, where we include n = 1 and ∞ for the universal and identity relations
on Z.
Suppose first that n ∈ N, so E ∩ C is the subgroup (n) of Z. If n = 1 then C ⊆ E.
Now Ex−1 is an equivalence class, and it contains αx−1 = 1; this is in C, and hence in E,
so Ex−1 = E. We have seen that Ey = E, so E = Ω since G = ⟨x−1, y⟩, and hence ∼ is
the universal relation on Ω.
We may therefore assume that n > 1. The vertical stem of the flower Fn is an edge
carrying two directed edges in C, with only one of its two labels divisible by n, so one
is in E whereas the other is not. However, these two directed edges are transposed by y,
contradicting the fact that Ey = E.
Finally suppose that n = ∞, so that all elements of C are in distinct equivalence
classes, and hence the same applies to Cy. In particular, since α ∈ C ∩ Cy we have
E ∩ C = {α} = E ∩ Cy. By inspection of Figure 1, Ω = C ∪ Cy and hence E = {α}.
It follows that all equivalence classes for ∼ are singletons, so ∼ is the identity relation, as
required.
We now return to the proof of Theorem 5.1. It follows from Lemma 5.2 that the sub-
group N = Γα of Γ fixing α is maximal. Clearly N is not normal in Γ, since G is not a
regular permutation group, so it sufficient to find an epimorphism N → F∞. One could
use the Reidemeister–Schreier algorithm to find a presentation for N : truncation converts
N3 into a coset diagram for N in Γ, and then deleting edges to form a spanning tree yields
a Schreier transversal. In fact a glance at Figure 1 shows that N is a free product of cyclic
groups: three of these, corresponding to the fixed points of y and generated by conjugates
of Y , have order 2, and there are infinitely many of infinite order, generated by conjugates
of Z and corresponding to the fixed points of z, that is, the 1-valent faces of N3, one in
each flower Fn for n ̸= 2. By mapping the generators of finite order to the identity we
obtain the required epimorphism N → F∞.
−1
0
1 2
3 4
p+ 2
t
t+ 1
t+ 3 t+ 4
2t− 1 nt
nt+ 1
nt+ 3 nt+ 4
(n+ 1)t− 1
F
F0 F1 Fn (n ≥ 1)
Figure 2: The p-valent map Np, with t := p+ 3.
Case 2: Finite p ≥ 4, q = 2. We now assume that Γ is a Hecke group Cp ∗ C2 for some
finite p ≥ 4. Let Np be the infinite p-valent planar map in Figure 2. Apart from F0, the
flowers are all identical copies of F1, with a ‘leaf’ growing out of its base and leading to
a vertex of valency 1, representing a fixed point of x. The ‘fans’ indicated by short dashed
lines represent however many free edges are needed in order that the incident vertex should
have valency p, that is, p − 3 free edges for vertices at the top of a stem, and p − 4 for
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 13
those at the base. As before, the elements αzi of the cycle C of z containing the directed
edge α = 0 are labelled with integers i; to save space in the diagram only a few labels are
shown. We define t = p + 3, since a translation from a flower Fn (n ≥ 1) to the next
flower Fn+1 adds that number to all labels.
The proof that the monodromy group G = ⟨x, y⟩ of Np is primitive is very similar to
that in Lemma 5.2 for N3. Any Γ-invariant equivalence relation ∼ on Ω restricts to C as
congruence mod (n) for some n ∈ N ∪ {∞}. The equivalence class E = [α] satisfies
Ey = E, so if n ̸= 1,∞ then the fact that y transposes the directed edges labelled nt and
nt + 1, with the first but not the second in E = (n), gives a contradiction. If n = 1 then
C ⊆ E, so both x and y preserve E and hence ∼ is the universal relation. If n = ∞ then
E ∩ C = {α}, and hence also E ∩ Cy = {α}; but Ω = C ∪ Cy and hence E = {α} and
∼ is the identity relation.
This shows that the subgroup N of Γ fixing α is maximal. As before, it is not normal,
and it is a free product of cyclic groups, now of order p, 2 or ∞, corresponding to the fixed
points of x, y and z (infinitely many in each case). Sending the generators of finite order to
the identity gives the required epimorphism N → F∞.
Case 3: Finite p, q ≥ 3. We modify the map Np in the proof of Case 2 by removing the leaf
attached to the base of each flower Fn (n ≥ 1), adding a white vertex to every remaining
edge (including one at the free end of each free edge), and finally adding edges incident
with 1-valent black vertices where necessary to ensure that all white vertices have valency
q or 1. The resulting map Np,q is shown in Figure 3.
−1
0
1
2
3 q
q + 1
t− 1
t t+ 1
t+ 2
t+ q
t+ q + 1
2t− 1
2t nt nt+ 1
nt+ 2
nt+ q
nt+ q + 1
(n+ 1)t− 1
F
F0
W1
F1
W2 Wn
Fn (n ≥ 1)
Figure 3: The bipartite map Np,q , with t := p+ 2q − 2.
Note that while the flowers Fn have grown since Case 2 was proved, small ‘weeds’
Wn (n ≥ 1) have grown between them. This bipartite map is the Walsh map for an oriented
hypermap of type (p, q,∞). Its monodromy group G is generated by permutations x and
y, of order p and q, which rotate edges around their incident black and white vertices. It
is sufficient to show that G acts primitively on the set Ω of edges, and that in the induced
action of Γ on Ω, the subgroup N fixing an edge has an epimorphism onto F∞.
The proof is similar to that for Case 2. The elements αzi of the cycle C of z containing
14 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
the edge α = 0 are labelled with integers i. (To save space in Figure 3, only a few signifi-
cant labels are shown.) Any Γ-invariant equivalence relation ∼ restricts to C as congruence
mod (n) for some n ∈ N ∪ {∞}. If E = [α] then since αy = α we have Ey = E. If
n ̸= ∞ then Ex = E since the edge β ∈ E labelled nt is fixed by x, so that EΓ = E
and hence E = Ω. Thus we may assume that n = ∞, so all elements of C are in distinct
conjugacy classes and hence E∩C = {α}. Similarly E∩Cy = {α}. But Ω = C∪Cy, so
E = {α} and ∼ is the identity relation. Thus G is primitive, so the subgroup N of Γ fixing
α is maximal. It is a free product of cyclic groups, of orders p, q and ∞, corresponding
to the fixed points of x, y and z. There are infinitely many of each, and mapping those of
finite order to the identity gives an epimorphism N → F∞.
Case 4: p or q = ∞. If p = ∞ or q = ∞ we can use the natural epimorphism from
Γ = ∆(p, q,∞) to a hyperbolic triangle group Γ′ = ∆(p′, q′,∞) with p′ and q′ both finite,
use Case 1, 2 or 3 to establish countable abundance for Γ′, and finally use Lemma 3.1 to
deduce it for Γ.
Remark 5.3. Constructions similar to those in the proof of Theorem 5.1 have been used
in [25] to prove that if Γ is a non-cocompact hyperbolic triangle group then Γ has uncount-
ably many conjugacy classes of maximal subgroups of infinite index. This strengthens
and generalises results of B. H. Neumann [40], Magnus [33, 34], Tretkoff [53], and Bren-
ner and Lyndon [6] on maximal nonparabolic subgroups of the modular group, and has
some overlap with work of Kulkarni [30] on maximal subgroups of free products of cyclic
groups.
6 Countable superabundance of non-cocompact hyperbolic triangle
groups
In order to prove countable superabundance for non-cocompact hyperbolic triangle groups
Γ = ∆(p, q, r), we need 2ℵ0 objects realising each countable group A. The proofs of
countable abundance for the various cases in Theorem 5.1 all used Proposition 3.3, and by
Remark 3.4 this yields the required number of objects in all cases except when A ̸= 1.
In fact, for any countable group A these proofs can be adapted (as in Remark 5.3) to
produce not just one but 2ℵ0 conjugacy classes of subgroups N satisfying the conditions of
Proposition 3.3. We thus obtain 2ℵ0 non-isomorphic objects N , each with 2ℵ0 coverings
O realising any countable group A ̸= 1, and with one covering (namely O = N ) realising
A = 1; each O has the property that O/Aut(O) ∼= N , so A is realised by 2ℵ0 non-
isomorphic objects.
We will give the required details for Case 1 of Theorem 5.1, where p = 3, q = 2 and
Γ is the modular group; the argument is similar in the other cases. We can modify the map
N3 in Figure 1 by adding ‘stalks’ between the flowers Fn, each consisting of a new vertex
on the horizontal axis, and a new free edge pointing upwards. Adding a stalk between Fm
and Fm+1 adds 2 to the value of all labels on flowers Fn for n > m. For the proof of
Theorem 5.1 to work we need to preserve the property that only one of the two labels on
the stem of each flower Fn (n > 1) is divisible by n. This can be done, in 2ℵ0 different
ways, by ensuring that for each n > 1 the total number of stalks added between F1 and Fn
is a multiple of n. The proof for Case 1 then proceeds as before, except that it now yields
2ℵ0 conjugacy classes of maximal subgroups N .
In the remaining cases of Theorem 5.1 we could use similar modifications to the maps
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 15
Np and Np,q in Figures 2 and 3, or alternatively add extra vertices and edges to those below
the horizontal axis, so that the non-negative labels above the axis are unaltered.
7 Countable superabundance of non-cocompact extended triangle
groups
We now consider countable superabundance for extended triangle groups Γ = ∆[p, q, r]
and their associated categories of unoriented hypermaps, again restricting attention to non-
cocompact groups. Earlier we realised countable groups A as automorphism groups in
various categories C+ of oriented hypermaps of a given type by constructing specific ob-
jects N = Np (p ≥ 3) or Np,q (p, q ≥ 3) in those categories, and then forming regular
coverings M of N , with covering group A, constructed so that M has only those auto-
morphisms induced by A. These objects M and N correspond to subgroups M and N of
the parent group Γ+ = ∆(p, q, r) for C+ with N = NΓ+(M). We can also regard M and
N as objects in the corresponding category C of unoriented maps or hypermaps of type
(p, q, r), for which the parent group is the extended triangle group Γ.
Lemma 7.1. For these objects M we have AutC(M) = AutC+(M) ∼= A.
Proof. We have AutC(M) ∼= NΓ(M)/M and AutC+(M) ∼= NΓ+(M)/M ∼= A by Theo-
rem 1.1, so it is sufficient to show that NΓ(M) = NΓ+(M). Clearly NΓ(M) ≥ NΓ+(M).
If this inclusion is proper then since NΓ+(M) = NΓ(M) ∩ Γ+ with |Γ : Γ+| = 2 we have
|NΓ(M) : NΓ+(M)| = 2, so the subgroup N = NΓ+(M) is normalised by some elements
of Γ \ Γ+. This is impossible, since in all cases the map or hypermap N corresponding
to N is chiral (without orientation-reversing automorphisms), by the proof of Theorem 5.1
and by inspection of Figures 1, 2 and 3. The same applies to the modified maps required to
produce 2ℵ0 such objects O.
Corollary 7.2. Each non-cocompact hyperbolic extended triangle group ∆[p, q, r] and its
associated category of all hypermaps of type (p, q, r) are countably superabundant.
This completes the proof of Theorem 1.3(b).
Remark 7.3. It would not have been possible to use Lemma 7.1 also in the proof of The-
orem 1.3(a) in §4, since the maximal subgroups Nn of Γ+ = ∆(p, q, r) constructed there
are normalised by orientation-reversing elements of Γ = ∆[p, q, r]. Instead of the natural
representation of PSL2(n), we could have used its representation on the cosets of a maxi-
mal subgroup H ∼= A5 for n ≡ ±1 mod (5), or H ∼= S4 for n ≡ ±1 mod (8): in both of
these cases there are two conjugacy classes of subgroups H , transposed by conjugation in
PGL2(n) (see [11, Ch. XII]) and corresponding to a chiral pair of hypermaps. However, in
either case the point stabilisers H have constant order as n → ∞, whereas Proposition 3.2
requires |Nn : Kn| = |H| to be unbounded, so we would need an alternative argument to
show that M is not normal in Γ+, as in the proof of Proposition 3.3.
8 Countable superabundance of some cocompact triangle groups
Theorem 1.3(b) proves countable superabundance only for non-cocompact hyperbolic tri-
angle groups ∆(p, q, r) and ∆[p, q, r]. We would like to extend to this property to the
cocompact case. The arguments we used to prove Theorem 1.3(b) depend on a standard
generator of ∆(p, q, r) (Z, without loss of generality) having infinite order, so that it can
16 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
have a cycle C of infinite length in some permutation representation, which is then proved
to be primitive by identifying C with Z. Clearly this is impossible in the cocompact case,
so a different approach is needed. The following is a first step in this direction.
Proposition 8.1. If one of p, q and r is even, another is divisible by 3, and the third is at
least 7, then the cocompact triangle groups ∆(p, q, r) and ∆[p, q, r] and their associated
categories are countably superabundant.
Proof. By permuting periods and applying Lemma 3.1 we may assume that p = 3, q = 2
and r ≥ 7. First suppose that r = 7. We will construct an infinite transitive permutation
representation of the group
Γ = ∆[3, 2, 7] = ⟨X,Y, T | X3 = Y 2 = T 2 = (XY )7 = (XT )2 = (Y T )2 = 1⟩
(where T = R2) in which the subgroup Γ+ = ∆(3, 2, 7) = ⟨X,Y ⟩ acts primitively, and
we will then apply Proposition 3.3 to a point-stabiliser in Γ+. This representation is con-
structed by adapting the Higman–Conder technique of ‘sewing coset diagrams together’,
used in [7] to realise finite alternating and symmetric groups as quotients of Γ+ and Γ. We
refer the reader to [7] for full technical details of this method. (Note that we have changed
Conder’s notation, which has X2 = Y 3 = 1, by transposing the symbols X and Y ; this
has no significant effect on the following proof.)
G H
Figure 4: The maps G and H.
Conder gives 14 coset diagrams A, . . . , N for subgroups of index n = 14, . . . , 108 in
Γ+, with respect to the generators X and Y ; these can be interpreted as describing transitive
representations of Γ+ of degree n. Each diagram is bilaterally symmetric, so this action
of Γ+ extends to a transitive representation of Γ of degree n, with T fixing vertices on the
vertical axis of symmetry, and transposing pairs of vertices on opposite sides of it. Although
Conder does not do this, in the spirit of the proof of Theorem 5.1 we can convert each of
his diagrams into a planar map of type {7, 3} (equivalently a hypermap of type (3, 2, 7))
by contracting the small triangles representing 3-cycles of X to trivalent vertices, so that
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 17
the cycles of X,Y and Z on directed edges correspond to its vertices, edges and faces.
(Warning: although Γ+ acts as the monodromy group of this oriented map, permuting
directed edges as described in Example 2 of §2, Γ does not act as the monodromy group
of the unoriented map, as in Example 1: the latter permutes flags, whereas T uses the
symmetry of the map to extend the action of Γ+ on directed edges to an action of Γ on
directed edges.)
We will construct an infinite coset diagram from Conder’s diagrams G and H of de-
gree n = 42; the corresponding maps G and H are shown in Figure 4. Conder defines
a (1)-handle in a diagram to be a pair α, β of fixed points of Y with β = αX = αT ,
represented in the corresponding map by two free edges incident with the same vertex
on the axis of symmetry. Thus G has three (1)-handles, while H has one. If diagrams
Di (i = 1, 2) of degree ni have (1)-handles αi, βi then one can form a new diagram,
called a (1)-join D1(1)D2, by replacing these four fixed points of Y with transpositions
(α1, α2) and (β1, β2), and leaving the permutations representing X,Y and T in D1 and D2
otherwise unchanged; the result is a new coset diagram giving a transitive representation
of Γ of degree n1 + n2. In terms of the corresponding maps Di, this is a connected sum
operation, in which the two surfaces are joined across cuts between the free ends of the
free edges representing the fixed points αi and βi; in particular, if Di has genus gi then
D1(1)D2 has genus g1+g2. This is illustrated in Figure 5, where the (1)-handle at the bot-
tom of G is joined to that at the top of H by two dashed edges to form G(1)H; these edges
can be carried by a tube connecting the two surfaces, showing the additivity of the genera
(both equal to 0 here). (Further details about this and more general joining operations on
dessins can be found in [27].)
G
H
Figure 5: Joining G and H to form G(1)H.
Using (1)-handles in G and H , we first form an infinite diagram
H(1)G(1)G(1)G(1)G · · · .
corresponding to an infinite planar map H(1)G(1)G(1)G(1)G · · · of type {7, 3}: the (1)-
handle at the top of each map H or G is joined, as in Figure 5, to that at the bottom of
the next map G. In this chain, each copy of G has an unused (1)-handle; we join these
18 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
arbitrarily in pairs, using (1)-compositions. Each such join adds a bridge to the underlying
surface, increasing the genus by 1, so the result is an oriented trivalent map N of type {7, 3}
and of infinite genus. This gives an infinite transitive permutation representation X 7→ x,
Y 7→ y, T 7→ t of Γ on the directed edges of N , with Γ+ again acting as its monodromy
group, and T acting as a reflection.
We need to prove that Γ and Γ+ act primitively. As shown by Conder [7] the permu-
tation w = yxt (= xyt in his notation) induced by Y XT has cycle structures 13133 and
1131101111171 in G and H . In each of the (1)-compositions we have used, two fixed
points of w are paired to form a cycle of length 2 of w, and a cycle of w of length 13 in G is
merged with one of length 13 or 10 in G or H to form a cycle of length 26 or 23. All other
cycles of w are unchanged, so in particular its cycle C of length 17 in H remains a cycle in
the final diagram. Since all other cycles of w have finite length coprime to 17, some power
of w acts on C as w and fixes all other points. Since 17 is prime, it follows that if Γ+ acts
imprimitively, then all points in C must lie in the same equivalence class E. Now C is
what Conder calls a ‘useful cycle’, since it contains a fixed point of y not in a (1)-handle
(the right-hand free edge β in the central circle in H in Figure 4) and a pair of points from
a 3-cycle of x (namely β and βx = βw8). It follows that X and Y leave E invariant,
which is impossible since they generate the transitive group Γ+. Thus Γ+ acts primitively
(as therefore does Γ), so the point-stabilisers N = Γα and N+ = Γ+α of a directed edge
α are maximal subgroups of Γ and Γ+. By the Reidemeister–Schreier algorithm, N+ is a
free product of four cyclic groups of order 2 (arising from fixed points of y in H not in the
(1)-handle), and infinitely many of infinite order (two arising from each bridge between a
pair of copies of G). Thus N+ admits an epimorphism onto F∞, so Proposition 3.3 shows
that Γ+ is countably abundant.
We can choose α to be fixed by t, so that T ∈ N , and hence N is a semidirect product
of N+ by ⟨T ⟩. The action of t is to reflect H and all the copies of G in the diagram,
together with the bridges added between pairs of them. Acting by conjugation on N+,
T therefore induces two transpositions on the elliptic generators of order 2. Each bridge
contributes a pair of free generators to N+, one of them (represented by a loop crossing
the bridge and returning ‘at ground level’), centralised by T , the other (represented by
a loop transverse to the first, following a cross-section of the bridge) inverted by T ; by
sending T , together with the inverted generators and the four elliptic generators of N+, to
the identity, we can map N onto the free group of countably infinite rank generated by the
centralised generators, so Proposition 3.3 shows that Γ is countably abundant. In fact, there
are 2ℵ0 ways of pairing the copies of G to produce bridges, giving mutually inequivalent
permutation representations and hence mutually non-conjugate subgroups N and N+ of Γ
and Γ+, so this argument establishes countable superabundance.
The extension to the case r ≥ 7 is essentially the same, but based on the coset dia-
grams in Conder’s later paper [8] on alternating and symmetric quotients of ∆(3, 2, r) and
∆[3, 2, r]. In this case his coset diagrams S(h, d) and U(h, d) play the roles of G and H ,
where r = h+ 6d with d ∈ N and h = 7, . . . , 12.
Proposition 8.1 accounts for a proportion 121/216 of all hyperbolic triples. It seems
plausible that coset diagrams of Everitt [13] and others, constructed to extend Conder’s
results on alternating group quotients to all finitely generated non-elementary Fuchsian
groups, could be used to prove that all cocompact hyperbolic triangle groups ∆(p, q, r)
and ∆[p, q, r], together with their associated categories, are countably superabundant, thus
proving Conjecture 1.4.
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 19
9 Realisation of other groups
Finally, although this paper is mainly about triangle groups and their associated categories
of maps and hypermaps, we can deduce realisation properties for many other groups and
categories.
Theorem 9.1. If a group Γ has an epimorphism onto a non-abelian free group then it is
finitely and countably superabundant.
Proof. By Theorem 1.3 the free group F2 = ∆(∞,∞,∞) has these properties. Since
F2 is an epimorphic image of every other non-abelian free group, the result follows from
Lemma 3.1.
For example, Theorem 9.1 applies to the fundamental groups Γ of many topological
spaces, so their categories of coverings are finitely and countably superabundant.. Exam-
ples include compact orientable surfaces of genus g with k punctures, where 2g + k ≥ 3.
Taking g = 0, k = 3 shows that the category D of all dessins is finitely superabundant (see
Theorem 1.3(a) for a more specific result); in fact, Cori and Machı̀ [9] proved that every
finite group is the automorphism group of a finite oriented hypermap, two years before
Grothendieck introduced dessins in [20].
Hidalgo [21] has proved the stronger result that every action of a finite group A by
orientation-preserving self-homeomorphisms of a compact oriented surface S is topologi-
cally equivalent to the automorphism group of a dessin. One way to see this is to triangulate
S/A, with all critical values of the projection π : S → S/A among the vertices, none of
which has valency 1, and then to add an edge to an additional 1-valent vertex v in the in-
terior of a face. This gives a map (that is, a dessin with q = 2) on S/A which lifts via
π to a dessin D on S with A ≤ Aut(D). The only 1-valent vertices in D are the |A|
vertices in π−1(v). These are permuted by Aut(D), with A and hence Aut(D) acting
transitively; however, the stabiliser of a 1-valent vertex (in any dessin) must be the identity,
so Aut(D) = A, as required. By starting with inequivalent triangulations of S/A one can
obtain infinitely many non-isomorphic dessins realising this action of A.
ORCID iD
Gareth A. Jones https://orcid.org/0000-0002-7082-7025
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