Also available at http://amc.imfm.si
ISSN 1855-3966 (printed ed.), ISSN 1855-3974 (electronic ed.)
ARS MATHEMATICA CONTEMPORANEA 1 (2008) 223–241
Injectivity Radius of Representations of Triangle
Groups and Planar Width of Regular Hypermaps
Martin Mačaj
Comenius University, Bratislava, Slovakia
Jozef Širáň
Open University, UK, and STU, Bratislava, Slovakia
Mária Ipolyiová
Matej Bel University, Banská Bystrica, Slovakia
Received 3 January 2008, accepted 19 December 2008, published online 31 December 2008
Abstract
We develop a rigorous algebraic background for representations of triangle groups in
linear groups over algebras arising from factor rings of multivariate polynomial rings. This
is then used to substantially improve the existing upper bounds on the order of epimorphic
images of triangle groups with a given injectivity radius and, analogously, the size of the
associated hypermaps of a given type with a given planar width.
Keywords: Triangle group, epimorphism, injectivity radius, regular hypermap, planar width.
Math. Subj. Class.: 05C10, 05C25, 05C65, 05E99, 20C99
1 Introduction
The theory of regular maps and hypermaps has become increasingly important due to connec-
tions with hyperbolic geometry, group theory, Riemann surfaces, and Galois theory. These
and many other links between regular (hyper)maps and other parts of mathematics are sur-
veyed, for example, in [8]. Research in this area has, for the most part, focused on classifi-
cation problems. Three most prominent (but by far not the only) examples are classification
of regular embeddings of complete graphs [6], classification of regular hypermaps over 2-
dimensional special linear groups over finite fields [13], and classification of regular maps on
surfaces with negative prime Euler characteristic [2].
E-mail addresses: martin.macaj@fmph.uniba.sk (Martin Mačaj), jozef.siran@stuba.sk (Jozef Širáň),
maria.ipolyiova@umb.sk (Mária Ipolyiová)
Copyright c© 2008 DMFA – založništvo
224 Ars Mathematica Contemporanea 1 (2008) 223–241
In this paper we will study a phenomenon of a different nature. Informally, we will be
interested in the problem of estimating the number of vertices of regular hypermaps con-
taining ‘parts’ of prescribed ‘size’ which look ‘locally planar’. We will explain the terms in
quotation marks in what follows. For now we just note that there are exciting connection of
this problem with finite representations of triangle groups, theory of matrices over algebraic
number fields, arc-transitive graphs of given girth, and vertex-transitive non-Cayley graphs.
We begin with a description of certain hyperbolic tessellations. A triple (l,m, n) of posi-
tive integers is said to be hyperbolic if 1/l+1/m+1/n < 1. With any such hyperbolic triple
we associate a trivalent tessellation U(l,m, n) of the hyperbolic plane H with a proper 3-
colouring of faces as follows. All faces of colour 1, 2, and 3 are congruent 2l-gons, 2m-gons,
and 2n-gons, respectively, such that every vertex is incident to precisely one face of each
colour. The underlying trivalent 1-complex of U(l,m, n) is a bipartite graph, the vertices of
which split into two classes according to whether the clockwise cyclic order of colours of the
incident faces is 1, 2, 3 or 1, 3, 2.
It is well known that the group of orientation and face colour preserving automorphisms
of U(l,m, n) is the triangle group T (l,m, n) with full presentation
T (l,m, n) = 〈r, s, t| rm = sn = tl = rst = 1〉 . (1)
Let ϕ : T (l,m, n) → H be an epimorphism onto a finite group H and let N be the kernel
of ϕ. We say that ϕ is torsion-free if N is a torsion-free group. Any such torsion-free
epimorphismϕ : T (l,m, n)→ H determines the face-3-coloured quotient hypermapMϕ =
U(l,m, n)/N of type (l,m, n) in the compact, orientable surface S = H/N . Orbits of the
cyclic subgroups generated by ϕ-images of r, s, t are vertices, faces, and (hyper)edges ofM.
The group H acts onM as colour and orientation preserving automorphism group. Since H
acts regularly on vertices ofM that belong to the same bipartition class, the hypermapM is
also called regular. The fact that all finite, regular hypermaps on compact, orientable surfaces
arise this way is well known and we recommend [8] for details. If l = 2, we may contract all
faces coloured 2 to points and, at the same time, collapse the quadrangles coloured 1 to edges;
the resulting embedded graph forms a regular map. The regular action of H on vertices of
one colour class in the corresponding hypermap then manifests as a regular action on arcs.
Intuitively, with an increasing index of N in T (l,m, n) the quotient mapM will contain
‘locally planar’ parts of increasing size. To make this observation more precise, let K be the
1-skeleton ofM. Since (l,m, n) is a hyperbolic triple, the supporting surface S ofM has
genus at least two. For any closed, non-contractible curve C on S, the number of intersections
|C ∩ K| of C with K is always non-zero, since M induces a cellular decomposition of S.
Following [12] we define the planar width ofM as the minimum of |C ∩K|, where C ranges
over all non-contractible closed curves on S. An interesting question is then to determine, or
at least estimate, the smallest number of vertices of a finite, regular hypermap of a hyperbolic
type (l,m, n) with planar width larger than d.
As we shall see, the same idea can also be formulated in the language of group epimor-
phisms, without invoking quotient hypermaps. Let T (l,m, n) be presented as in (1). For
any non-identity element w ∈ T (l,m, n) let `(w) be the smallest length of a word over the
alphabet {r, r−1, s, s−1, t, t−1} that represents w in T (l,m, n). The injectivity radius of an
epimorphism ϕ : T (l,m, n) → H is the largest k such that ϕ(w) 6= 1 for all non-identity
w ∈ T (l,m, n) such that `(w) ≤ k. We emphasise that the injectivity radius is a function of
both the epimorphism ϕ as well as the generating set {r, s, t}. Now, the interesting problem
M. Mačaj et al.: Injectivity Radius of Representations of Triangle Groups . . . 225
is to estimate the smallest order of a finite group H for which there exists an epimorphism
T (l,m, n)→ H of injectivity radius at least δ.
Let us note that the two problems have solid foundations, since for every hyperbolic
triple (l,m, n) there exist finite, regular hypermaps of type (l,m, n) with arbitrarily large
planar width, as well as epimorphisms from T (l,m, n) into finite groups with arbitrarily
large injectivity radius. This is a consequence of residual finiteness of hyperbolic triangle
groups, which follows from a general result of [9] and also from the specific approach in [10]
that we will refer to later.
Although planar width was mostly considered in the context of maps, in our brief over-
view of the state-of-the-art we describe the results in the equivalent language of hypermaps in
which l = 2. A covering space approach to proving existence of regular hypermaps of hyper-
bolic type (2,m, n) and arbitrary planar width was proposed in [12], furnishing no explicit
upper bound on the number of vertices. Using faithful representations of triangle groups in
3-dimensional orthogonal groups over algebraic extensions of the integers, an explicit up-
per bound of the form C̃(2,m, n)d for the smallest number of vertices of a regular map of
hyperbolic type (2,m, n) with planar width larger than d was obtained in [14]. Exploring
the same method in more detail, improvements towards a smaller order of magnitude of the
functions C̃(2,m, n) and, in general, C̃(l,m, n), were later found in [5] for regular maps
and [1] for regular hypermaps. In some sense, this is the best possible type of a bound, since
it follows from [11] that the number of vertices of a regular hypermap of type (l,m, n) and
planar width larger than d is bounded below by c̃(l,m, n)d; we note that both C̃(l,m, n)
and c̃(l,m, n) are independent of d. However, the current gap between the two quantities is
enormous. From [11] we know that
√
lmn < c̃(l,m, n), while, say, the best result of [1] only
gives log2 C̃(l,m, n) < 9(l2 +m2 + n2)(l+m+ n)lmn. This motivates a search for better
bounds on the two parameters.
In this contribution we develop methods for improvements of the upper bound. Varying
the methods and their ingredients we actually prove a number of estimates on C̃(l,m, n) that
are asymptotically much better that the above bound; for instance, one of our estimates has
the form log2 C̃(l,m, n) < 64lmn log2(4µ), where µ = max{l,m, n}. We will also discuss
the interaction between the orders of magnitude of the new estimates. Existence of regular
maps of ’small’ planar width implies the existence of relatively small arc-transitive graphs
with interesting additional properties, such as large girth, or non-Cayleyness; see [14]. Our
results transfer into new upper bounds on the size of such graphs; we omit the details that can
be obtained by direct substitution.
Our paper is organised as follows. The relationship between injectivity radius of epimor-
phisms of triangle groups and planar width of hypermaps is discussed in section 2 in some
detail. Section 3 is the first encounter with orthogonal representations of triangle groups over
polynomial rings factored by suitable ideals generated by reciprocal polynomials. The alge-
braic machinery for working with such representation is introduced in section 4. The results
obtained are then applied in section 5 to obtain the first large-scale improvements over the
above upper bounds. Further refinements of our methods are considered in section 6, where
we not only improve the previous results but also derive estimates of a slightly different kind.
The last section 7 contains a discussion on planar width of regular maps.
We will arrive at our main result on planar width through a series of step-by-step improve-
ments. Formally, it would have been possible to present a direct proof of our best bound.
Nevertheless, such an approach would result in a long and technical proof and the ideas lead-
226 Ars Mathematica Contemporanea 1 (2008) 223–241
ing to partial improvements in various directions would be overwhelmed by the large number
of computational details. Our presentation leaves sufficient room to develop these ideas sep-
arately. This way the reader will be able to compare the contribution of individual ideas to
the final result.
2 Hypermaps, planar width and injectivity radius
Consider a finite, trivalent, face-3-coloured regular hypermapM of a hyperbolic type (l,m,
n) on a compact, clockwise oriented surface S, induced by a torsion-free epimorphism ϕ :
T (l,m, n) → H . Denoting the ϕ-images of r, s, and t by ρ, σ, and τ , the automorphism
group H ofM has a partial presentation of the form
H = 〈ρ, σ, τ | ρm = σn = τ l = ρστ = . . . = 1〉 . (2)
BecauseM is of hyperbolic type and so S has genus at least two, there will be at least one
additional relator in the presentation of H , as indicated by the dots. Suppose that vertices of
the (bipartite) 1-skeleton ofM have been properly two-coloured, say, black and white. Fix
a black vertex, say, u. Since H acts regularly on the black vertices, for every black vertex v
there is a unique ω ∈ H such that v = ω(u). This unique ω will be called the label of v and
denoted λ(v).
Let v1, v2, and v3 be the clockwise next black vertices on the face incident with v and
coloured 1, 2, and 3, respectively. We have assumed that the colours correspond to faces of
length 2l, 2m, and 2n, in that order, and hence also to the generators τ , ρ, and σ, in the same
order. Then, the labels of v2, v3, and v1 are ωρ, ωσ, and ωτ , as indicated in Fig. 1. Indeed,
it is easy to see that, say, ωρω−1 is the automorphism that takes v = ω(u) onto v2, and then
v2 = ωρω−1(v) = ωρ(u), showing that λ(v2) = ωρ as claimed; the argument for v3 and v1
is similar. In other terms, for any given black vertex, the label of the clockwise next black
vertex in the face coloured 2, 3, and 1 is obtained by right multiplication of the label of the
given vertex by the generators ρ, σ, and τ ofH , respectively. The reader familiar with Cayley
graphs and maps will immediately notice the analogy between the adjacency defined by right
multiplication by the generators in the Cayley setting and the situation in our case.
ωσ
ωτωρ
ω
2
3
1
3
1
2
Figure 1: A fragment of a regular hypermap with group labelling of black vertices.
M. Mačaj et al.: Injectivity Radius of Representations of Triangle Groups . . . 227
The relation between elements of H as words over the alphabet {ρ±1, σ±1, τ±1} and
non-contractible closed curves on the supporting surface S of our hypermap M with the
1-skeleton K can now be explained conveniently. Let C be a non-contractible closed curve
on S. Since we are interested in the minimum of |C ∩ K|, we may assume without loss
of generality that the only points in C ∩ K are black vertices. Let |C ∩ K| = k and let
C ∩K = {v0, v1, . . . , vk−1}. Select a direction on C and choose v0 to be the initial (and, at
the same time, the terminal) point of C. We may assume that the notation has been chosen
so that vi and vi+1 are consecutive for any i (mod k) in the direction of traversal of C. The
subpath of C from vi to vi+1 lies entirely within an open face of our hypermap; for the sake of
the argument, assume that the face has colour 2. Then, by the rule of the right multiplication
by generators, we have λ(vi+1) = λ(vi)ρa for some a such that 0 < |a| ≤ m/2; the
inequality on the right is strict if m is odd. Repeating this successively for i = 0, 1, . . . , k−1
for every pair of consecutive black vertices on C in its complete traversal from the initial point
back, we arrive at the equation λ(v0) = λ(v0)γ where γ is a product of k factors, each of
the form ρa with 0 < |a| ≤ m/2, or σb with 0 < |b| ≤ n/2, or τ c with 0 < |c| ≤ l/2.
Of course, γ is equal to the identity element of H , but for us the importance lies in having
γ expressed as the product of the above factors “along” the curve C as described. Note that
a different choice of the initial point results in a conjugate of γ, and choosing the opposite
direction of C inverts γ. This way, every such curve C determines a word w that cannot
be derived from the four relators in the partial presentation (2) for H . (Indeed, if w was a
consequence of the relators in (2), then the expression of w in terms of the (conjugates of the)
relators would give a way to contract C to a point, and vice versa.) Conversely, consider a
word w = w1w2 . . . wj where wi = αai for 1 ≤ i ≤ j, α is equal to ρ, σ, or τ , and the ai are
nonzero integers the absolute value of which do not exceed l/2, m/2 and n/2, respectively.
If w represents the identity in H and is independent on the four relators given in (2), then w
defines a non-contractible closed curve on S meetingK at the j black vertices labelled 1, w1,
w1w2, . . ., w1w2 . . . wj−1. We sum up our observations as follows.
Lemma 1. Let (l,m, n) be a hyperbolic triple, let µ = max{l,m, n}, and let d be a positive
integer. If ϕ : T (l,m, n) → H is an epimorphism onto a finite group H with injectivity
radius at least dµ/2, then the planar width of the hypermapM is larger than d. Conversely,
if the planar width ofM is larger than d, then the injectivity radius of ϕ is at least d. 
3 Orthogonal representation of triangle groups
So far we have not referred to geometric properties of universal tessellations U(l,m, n) of
hyperbolic type. As it is generally known [8], the triangle group T (l,m, n) actually acts
on U(l,m, n) as the group of orientation-preserving isometries of H, leaving the universal
tessellation U(l,m, n) invariant and preserving the face colours. It was shown in [10] that,
in a coordinate system derived from the model of H as the upper-half of a two-sheeted hy-
perboloid x2 + y2 − z2 = −1 in R3, the generators r, s, and t of T (l,m, n) in (1) can be
represented by the 3× 3 matrices R, S, and T defined by
R =
 η2 − 1 0 ηζ + ξη 1 ξ
−η 0 −1
 , S =
 −1 −ζ 0ζ ζ2 − 1 0
η ξ + ηζ 1
 , T =
 1 ζ η + ζξ0 −1 −ξ
0 ξ ξ2 − 1

(3)
228 Ars Mathematica Contemporanea 1 (2008) 223–241
where ξ = 2 cos(π/l), η = 2 cos(π/m), and ζ = 2 cos(π/n) are algebraic integers over
Z. The group 〈R,S, T 〉 = 〈R,S〉 is a subgroup of the orthogonal group of transformations
preserving the inner product defined by the quadratic form given by the matrix 1− ξ2 ζ + ξη η + ζξζ + ξη 1− η2 ξ + ηζ
η + ζξ ξ + ηζ 1− ζ2
 .
Combined with the general knowledge about hyperbolic geometry, it follows from [10] that
the assignment r 7→ R, s 7→ S, and t 7→ T extends to a faithful representation, that is, to an
injective homomorphism, from T (l,m, n) to O(3,R). This important linear representation
will be invoked throughout. Two other proofs of faithfulness of this representation, both
based on the much more general idea of associating canonical bilinear forms to reflection
groups, can be found in [3] presenting the original approach of Tits [15], or in [4] using a
more direct method.
Our goal is to construct “small” finite groups onto which there is an epimorphism from
a triangle group with a given injectivity radius. The orthogonal representation introduced
above seems to be an ideal candidate for this task. The main idea has roots in [9] and consists
in looking at congruence subgroups, that is, normal subgroups determined by certain congru-
ences, and at the corresponding factor groups. A way to achieve this is to regard the image
of the above representation of T (l,m, n) as a subgroup of O(3,Z[ξ, η, ζ]) and factorise the
ring Z[ξ, η, ζ] by an ideal generated by a suitably chosen integer.
In order to investigate injectivity radius of epimorphisms of triangle groups into finite
groups arising as indicated, it turned out to be useful to adopt a more algebraic approach. Let
M = l.c.m.(l,m, n) = l0l = m0m = n0n, let θ = 2 cos(π/M), and let fθ(x) ∈ Z[x] be
the minimal polynomial for θ. Then, Z[ξ, η, ζ] ⊆ Z[θ] ∼= Z[x]/(fθ(x)), and for ξ, η and ζ
we then have ξ = Pl0(θ), η = Pm0(θ), and ζ = Pn0(θ), where Pk(x) is the k-th modified
Tchebyshev polynomial; see [14] for details. An improvement towards smaller orders of
the target groups was obtained by replacing the above ring with Z[x, y, z]/J where the ideal
J = (f(x), g(y), h(z)) is chosen in such a way that, substituting x+ J , y + J and z + J for
ξ, η and ζ in (3), the matrices arising from R, S and T give rise to a faithful representation of
T (l,m, n), cf. [1, 5].
To open up a way for further improvements, let us have a look at spectral properties of the
matrices R, S, and T . Without loss of generality, we will confine our analysis to the matrix
S. First, observe that the block structure of the matrix shows that S is, in fact, the matrix
of an affine transformation ū 7→ ūL + ā where ū ∈ R2 is an arbitrary two-dimensional row
vector, L is the submatrix of S obtained by deleting the third row and the third column, and
ā = (η, ξ+ηζ). Let us write ζ in the form ζ = ζ∗+ζ
−1
∗ where ζ∗ = cos(2π/n)+i sin(2π/n)
and i2 = −1. An easy calculation shows that the eigenvalues of L are ζ2∗ and ζ−2∗ and hence
the spectrum of S is {1, ζ2∗ , ζ−2∗ }; in particular, the order of L is equal to the order of S.
The observation about the spectrum is the cornerstone of this work, the essence of which
is fusing the outlined algebraic methods with representing ξ, η and ζ as a sum of an element
and its inverse. Although our results will be valid in a more general setting, we will confine
ourselves mostly to considerations in the rings Z[x]/(Φ2M (x)) where M = l.c.m.(l,m, n)
and Φ2M (x) is the 2M -th cyclotomic polynomial, and in Z[x, y, z]/(f(x), g(y), h(z)) where
f(x) = (x2l − 1)/(x2 − 1), g(y) = (y2m − 1)/(y2 − 1) and h(z) = (z2n − 1)/(z2 − 1).
The reason for choosing the two rings will become clear soon; at this stage we just show
that they lead to faithful representations of type (3). From now on, we will assume that R,
M. Mačaj et al.: Injectivity Radius of Representations of Triangle Groups . . . 229
S, and T are matrices as in (3), but the parameters ξ, η and ζ appearing there will belong to
more general structures (rings or fields) that will be specified in the course of our exposition.
We derive simple sufficient conditions for the matrices R, S, and T to still generate a faith-
ful representation of T (l,m, n). For convenience, we will sometimes identify the (faithful)
representations of T (l,m, n) with the corresponding images 〈R,S, T 〉. We begin with an
auxiliary result about orders of such matrices and without loss of generality we focus on S.
Lemma 2. Let n ≥ 2 be an integer, let F be a field of characteristic not dividing n, and let
S be the matrix in (3) such that ξ, η, ζ ∈ F . Let ζ = ζ∗ + ζ−1∗ where ζ∗ is either an element
of F or an element of an extension of F of degree two. Then, the order of S divides n if and
only if ζ2n∗ = 1 and ζ
2
∗ 6= 1. If the characteristic of F divides n, then the condition ζ2∗ 6= 1 is
no longer necessary.
Proof. Our earlier observations about the orders extend to matrices over any field. Specif-
ically, the order of S is equal to the order of the submatrix L of S formed by deleting the
third row and the third column. The eigenvalues of L are ζ2∗ and ζ
−2
∗ . It can be checked that
L is conjugate to
(
1 1
0 1
)
if and only if ζ2∗ = 1, in which case the order of L is equal to the
characteristic of F . If ζ2∗ 6= 1, then L is conjugate to the matrix Diag(ζ2∗ , ζ−2∗ ), making the
conclusion about the order of S obvious. 
Now we are ready to state and prove the indicated results on representations of triangle
groups.
Proposition 3. Let (l,m, n) be a hyperbolic triple and let M = l.c.m.(l,m, n) = l0l =
m0m = n0n. Let f(x) ∈ Z[x] be a monic reciprocal polynomial of even degree. Let R, S,
and T be matrices as in (3) defined over the ring Q[x]/(f(x)), where ξ = xl0 +x−l0 +(f(x)),
η = xm0+x−m0+(f(x)), and ζ = xn0+x−n0+(f(x)). If the 2M -th cyclotomic polynomial
divides f(x) and if f(x) divides (x2M − 1)/l.c.m.(x2l0 − 1, x2m0 − 1, x2n0 − 1), then the
assignment r 7→ R, s 7→ S and t 7→ T extends to a faithful representation of the triangle
group T (l,m, n) in SL(3,Q[x]/(f(x))).
Proof. Let us denote by Gx the group generated by R, S, and T . For any root b of f(x) let
Gb denote the group generated by the three matrices obtained from R, S, and T by replacing
ξ, η, and ζ with bl0 + b−l0 , bm0 + b−m0 , and bn0 + b−n0 , respectively. By the substitution
theorem we have an obvious epimorphism Gx → Gb. Since f(x) does not have repeated
roots, there is a natural embedding Gx ↪→
∏
Gb where the product ranges over conjugacy
classes of roots of f(x). Therefore,Gx is a faithful representation of T (l,m, n) if everyGb is
a representation of T (l,m, n) and at least one of them is faithful. Faithfulness of at least one
ofGb follows from the condition Φ2M (x)|f(x), that is, from Mennicke’s representation [10].
Since f(x) divides (x2M − 1)/l.c.m.(x2l0 − 1, x2m0 − 1, x2n0 − 1), Lemma 2 guarantees
that every Gb is a representation of T (l,m, n). 
In a similar way one can prove the following result; we leave the details to the reader.
Proposition 4. Let (l,m, n) be a hyperbolic triple. Let J = (f(x), g(y), h(z)) be an ideal in
Q[x, y, z] where f(x) ∈ Z[x], g(y) ∈ Z[y], and h(z) ∈ Z[z] are monic reciprocal polynomi-
als of even degree. Let R, S, and T be matrices as in (3) defined over the ring Q[x, y, z]/J ,
where ξ = x+ x−1 + J , η = y + y−1 + J , and ζ = z + z−1 + J . Assume that Φ2l(x)|f(x)
230 Ars Mathematica Contemporanea 1 (2008) 223–241
and f(x)|(x2l − 1)/(x2 − 1), Φ2m(y)|g(y) and g(y)|(y2m − 1)/(y2 − 1), and Φ2n(z)|h(z)
and h(z)|(z2n − 1)/(z2 − 1). Then, the assignment r 7→ R, s 7→ S and t 7→ T extends to a
faithful representation of the triangle group T (l,m, n) in SL(3,Q[x, y, z]/J).
4 Injectivity radius and norms in algebras
From the methods and results of [11] used to investigate the growth rate of maps, the follow-
ing fact can be extracted.
Proposition 5. Let (l,m, n) be a hyperbolic triple and let δ be a positive integer. Then, for
every epimorphism ϕ : T (l,m, n)→ H of injectivity radius at least δ onto a finite group H
we have |H| > cδ where c depends on l,m, n but not on δ. 
The natural question of constructing groups H as above to which a similar upper bound
applies was answered in the affirmative in [14] and improvements of the construction were
found later in [1, 5]. We briefly sum up the essence of the approach in the cited papers, using
a more general framework.
Let A be a u-dimensional abelian Z-algebra, that is, an abelian ring with unity such that
the additive group of A is a u-dimensional free Z-module and multiplication is bilinear. Let
β1, . . . , βu be an (additive) basis ofA. Then, every element α ∈ A can be uniquely expressed
in the form α = a1β1 + . . . + auβu, where ai ∈ Z for 1 ≤ i ≤ u. The integral norm of α
with respect to the basis β1, . . . , βu is the integer
〈α〉 = maxi|ai| .
This integral norm extends in a natural way to vectors and matrices over A (by simply taking
the maximum norm over all entries of a vector or a matrix) and we will reserve the same
notation for such extensions.
The previous work on injectivity radius [1, 5, 14] was based on variations of the following
general result.
Theorem 6. LetA be a u-dimensional abelian Z-algebra. Let ϑ : T (l,m, n)→ SL(q, A) be
a faithful representation of the triangle group T (l,m, n) = 〈r, s, t| rm = sn = tl = rst = 1〉
and let Γ = {ϑ(r)±1, ϑ(s)±1, ϑ(t)±1}. Further, let λ = maxX∈Γ〈X〉 where the norm is with
respect to a fixed basis of A containing 1 and let κ = max{〈Xα〉/〈α〉; α ∈ Aq \ {0}, X ∈
Γ}. Assume that there is a positive integer δ and a prime p such that κδ−1λ + 1 < p < κδ .
If ψp is the natural projection SL(q, A) → SL(q, A/pA), then the representation ψpϑ has
injectivity radius at least δ and |Im(ψpϑ)| < Cδ where C = κ(q
2−1)u.
Proof. It is an easy exercise to show that, for an integral norm onAwith respect to an arbitrary
basis, the parameter κ is well defined and κ ≥ λ ≥ 1. To prove the theorem, it is sufficient
to show that for any j ≤ δ and any X1, X2, . . . , Xj ∈ Γ such that Xj . . . X2X1 6= I in
SL(q, A) we have 〈Xj . . . X2X1 − I〉 < p; this will guarantee that the product modulo
pA cannot be equal to the identity matrix. Let α be any column of X1. By induction we
obtain 〈Xj . . . X2α〉 ≤ κj−1〈α〉 ≤ κδ−1λ, which implies that 〈Xj . . . X2X1〉 ≤ κδ−1λ. If
p is a prime such that κδ−1λ + 1 < p < κδ , then all integral coefficients in all entries of
the matrix Xj . . . X2X1 − I are, in absolute value, smaller than p. Since the basis of A is
assumed to contain 1, no nonzero integral coefficient in any entry of the above matrix mod
pA collapses. Therefore Xj . . . X2X1 6= I in SL(q, A/pA); for the order of this group we
M. Mačaj et al.: Injectivity Radius of Representations of Triangle Groups . . . 231
have |SL(q, A/pA)| ≤ |A/pA|q2−1 = p(q2−1)u < κ(q2−1)uδ . 
In section 3 we indicated that earlier improvements on the bounds of injectivity radius of
triangle groups given in [14] focused on factor rings associated with minimal polynomials of
algebraic integers of type 2 cosπ/k. This way the best construction in [1] gives, after a rec-
tification in the theoretical part of the argument, a finite group H admitting an epimorphism
ϕ : T (l,m, n)→ H of injectivity radius at least δ such that
|H| < Cδ where log2 C < 9(l2 +m2 + n2)lmn . (4)
Restricting to l = 2 and denoting by φ the Euler’s function, the analogous best result of [5]
yields
|H| < Cδ where log2 C < 2(φ(2m)2 + φ(2n)2)φ(2m)φ(2n) . (5)
Since the original bound in [14] for l = 2 was log2(C) < 72(mn)3, the contribution of (5)
consists, roughly speaking, in replacing the factor (mn)2 by (m+ n)2. The actual improve-
ment therefore depends on the order of magnitude of the ratio between m and n.
With the help of the two faithful representations of hyperbolic triangle groups introduced
in section 3, in the next section we will derive a substantial improvement of these construc-
tions. We will do so by applying Theorem 6 to the Z-algebras used in Propositions 3 and 4.
For this we need to prove estimates on integral norms with respect to suitable bases. In view
of the general philosophy outlined in Section 3 it should be no surprise that the bases will be
formed by sums of certain elements and their inverses.
Consider a monic, reciprocal polynomial f(x) ∈ Z[x] of even degree 2u, given by
f(x) = x2u + c1x2u−1 + · · ·+ cu−1xu+1 + cuxu + cu−1xu−1 + · · ·+ c1x+ 1 .
Reciprocality allows us to work with elements of the form x + x−1 + (f(x)) in the ring
Z[x]/(f(x)). Let
θ = x+ x−1 + (f(x)) and θi = xi + x−i + (f(x))
for any i ∈ Z; in particular, θ0 = 2 and θ1 = θ. The set B = {1, θ1, θ2, . . . , θu−1} forms a
basis of the Z-algebra Z[θ]. The following obvious relations between the elements of B will
help us simplify expressions involving multiplication:
θiθj = θi+j + θi−j , (6)
θu = −cu −
u−1∑
i=1
cu−iθi, (7)
θi = θ−i, (8)
θi+1 = θ1θi − θi−1. (9)
The norm 〈 〉 we will be considering below will always be relative to the basis B. To simplify
the notation, we will write 〈f〉 instead of 〈f(x)〉 = maxi|ci|, and 〈g〉 instead of 〈g(θ)〉 for
elements g(θ) ∈ Z[θ].
232 Ars Mathematica Contemporanea 1 (2008) 223–241
Lemma 7. Let i be a positive integer and let g = g(θ) = a0 +
∑u−1
j=1 ajθj where a0 and the
aj are integers. Then
〈θ1g(θ)〉 ≤ (2 + 〈f〉)〈g〉, (10)
〈θu+i〉 ≤ (3 + 〈f〉)i〈f〉, (11)
〈θig(θ)〉 ≤
(
2 + 〈f〉 (3 + 〈f〉)
i − 1
2 + 〈f〉
)
〈g〉 < (3 + 〈f〉)i〈g〉. (12)
Proof. With the help of (6) and (8) we obtain
θ1g(θ) = 2a1 +
u−2∑
i=1
(ai−1 + ai+1)θi + au−2θu−1 + au−1θu
and applying (7) yields (10). To prove (11) it suffices to repeatedly apply (9) and (10), starting
with 〈θu〉 = 〈f〉. For the last inequality, using (6) and (8) we arrive at
〈θig(θ)〉 ≤ 2〈g〉+ (〈θu〉+ · · ·+ 〈θu+i−1〉)〈g〉 . (13)
Combined with (11), the above transforms into
〈θig(θ)〉 ≤
(
2 + 〈f〉 (3 + 〈f〉)
i − 1
2 + 〈f〉
)
〈g〉 ,
which is the first part of (12). The last inequality in (12) is obvious. 
We conclude with another note regarding the earlier work on bounding the injectivity
radius. In [1, 5, 14], modifications of Theorem 6 were used relative to the basis {1, θ, θ2, . . . ,
θu−1}. In contrast with this, here we will be using the basis {1, θ1, θ2, . . . , θu−1}. As we
shall see, and as one may have anticipated from the outlined relation of the new basis with
spectra, the effect of the change of basis will be considerable.
5 New bounds on injectivity radius
We are now prepared to improve the bounds (4) on the injectivity radius and the associated
target group. Our point of departure will be Propositions 3 and 4 where we introduced two
representations of triangle groups; from now on we will call them accordingly Representation
1 and 2.
We begin with Representation 1. Let (l,m, n) be a hyperbolic triple and let M =
l.c.m.(l,m, n) = l0l = m0m = n0n. Let f(x) = Φ2M (x) ∈ Z[x] be the 2M -th
cyclotomic polynomial of degree φ(2M) = 2u. As in the previous section, we define
θ = x + x−1 + (f(x)) and θi = xi + x−i + (f(x)) for any i ∈ Z and consider the ba-
sis B = {1, θ1, θ2, . . . , θu−1} of the Z-algebra Z[θ] together with the associated integral
norm 〈 〉. Further, let R, S, and T be matrices as in (3) over Z[θ], where ξ = θl0 , η = θm0 ,
and ζ = θn0 . At last, let µ0 = max{l0,m0, n0}. In what follows we derive estimates on the
parameters κ, λ, and C introduced in Theorem 6.
We first show that κ ≤ (3 + 〈f〉)2µ0 . To this end, it is sufficient to show that 〈Xα〉 <
(3 + 〈f〉)2µ0〈α〉 for every X ∈ {R±1, S±1, T±1} and every 3-dimensional column vector
α with entries gj = gj(θ) ∈ Z[θ] for j = 1, 2, 3. We outline the calculations for the matrix
M. Mačaj et al.: Injectivity Radius of Representations of Triangle Groups . . . 233
X = S only, as the particulars for the remaining five matrices are analogous. To simplify
the matters, in estimating 〈Sα〉 we will restrict ourselves to the contribution coming from the
third row of S. Thus, we will replace 〈Sα〉 with the scalar product 〈S(3)α〉 where S(3) is the
third row of S; from our computations it will be obvious that the resulting upper bound on
〈S(3)α〉will also be an upper bound on 〈Sα〉. Also, if g ∈ {g1, g2, g3} is such that 〈α〉 = 〈g〉,
then in our estimates we may for simplification assume that all entries of α are equal to g.
We then have
〈S(3)α〉 ≤ 〈(η + (ξ + ηζ) + 1)g(θ)〉 ≤ 〈g〉+ 〈ηg(θ)〉+ 〈ξg(θ)〉+ 〈ηζg(θ)〉 . (14)
For any positive integer i let χi = 2 + (3 + 〈f〉)i − 1)〈f〉/(2 + 〈f〉). By (12), we
have 〈ηg(θ)〉 = 〈θm0g(θ)〉 ≤ χm0〈g〉, 〈ξg(θ)〉 = 〈θl0g(θ)〉 ≤ χl0〈g〉, and 〈ηζg(θ)〉 =
〈θm0θn0g(θ)〉 ≤ χm0χn0〈g〉. This yields
〈S(3)α〉 ≤ (1 + χl0 + χm0 + χm0χn0)〈α〉 .
A routine calculation shows that 1 + χl0 + χm0 + χm0χn0 ≤ (3 + 〈f〉)2µ0 , and hence
〈S(3)α〉 ≤ (3 + 〈f〉)2µ0〈α〉. Similarly, 〈S(1)α〉 ≤ (1 + χn0)〈α〉 ≤ (3 + 〈f〉)2µ0 and
〈S(2)α〉 ≤ (1 + χn0 + χ2n0)〈α〉 ≤ (3 + 〈f〉)2µ0 . It is now clear that the right hand side of
the last inequality is an upper bound on 〈Sα〉 as well, which proves that κ ≤ (3 + 〈f〉)2µ0 .
We continue our analysis by estimating λ = max〈X〉 for X ∈ {R±1, S±1, T±1}. If
2µ0 < u = φ(2M)/2, then applying (6) to entries of X we obtain λ ≤ 2. If 2µ0 ≥ u,
then the inequality (11) of Lemma 7 in combination with the observation that u ≥ 2 for all
hyperbolic triples shows that λ ≤ 3+(3+〈f〉)µ0−u〈f〉+(3+〈f〉)2µ0−u〈f〉 ≤ (3+〈f〉)2µ0−1.
Summing up our discussion, we conclude that λ ≤ (3 + 〈f〉)2µ0−1 for all hyperbolic triples.
Set κ0 = (3 + 〈f〉)2µ0 and λ0 = (3 + 〈f〉)2µ0−1. By Bertrand’s Postulate, for any positive
integer δ ≥ 1 there exists a prime p between κδ−10 λ0 + 1 and κδ0. We may, of course, apply
Theorem 6 with the upper bounds κ0 and λ0 in place of κ and λ, arriving thus at our first
specific result.
Theorem 8. Let (l,m, n) be a hyperbolic triple, let M = l.c.m.(l,m, n) = l0l = m0m =
n0n, and let µ0 = max{l0,m0, n0}. Let f(x) be the 2M -th cyclotomic polynomial and let
〈f〉 be the largest absolute value of its coefficients. Then, for every positive integer δ there
exists an epimorphism ϕ : T (l,m, n)→ H of injectivity radius at least δ such that |H| ≤ Cδ
where C ≤ (3 + 〈f〉)8µ0φ(2M). 
The corresponding result for the existence of relatively “small” regular hypermaps of pla-
nar width larger than a preassigned integer follows immediately from the preceding theorem
and Lemma 1.
Corollary 9. Let (l,m, n) be a hyperbolic triple, let µ = max{l,m, n}, letM = l.c.m.(l,m,
n) = l0l = m0m = n0n, and let µ0 = max{l0,m0, n0}. Let f(x) be the 2M -th cyclotomic
polynomial and let 〈f〉 be the largest absolute value of its coefficients. Then, for every positive
integer d there exists a regular hypermap of type (l,m, n) of planar width larger than d, the
number of vertices of which does not exceed 2C̃d where C̃ ≤ (3 + 〈f〉)4µµ0φ(2M). 
The estimates in Theorem 8 and Corollary 9 clearly supersede the improvements (4) and
(5) of the original bound given in [14]. The exact order of magnitude of the bounds on C
is, however, not easy to extract without a more detailed information about coefficients in
234 Ars Mathematica Contemporanea 1 (2008) 223–241
cyclotomic polynomials, and we will address this issue later. Nevertheless, this provides
motivation for finding good bounds the order of magnitude of which would be more obvious.
We do so by focusing on Representation 2.
Again, let (l,m, n) be a hyperbolic triple. Let f(x) ∈ Z[x], g(y) ∈ Z[y], and h(z) ∈
Z[z] be defined by f(x) = (x2l − 1)/(x2 − 1), g(y) = (y2m − 1)/(y2 − 1), h(z) =
(z2n − 1)/(z2 − 1). All the three polynomials are monic, reciprocal, and of even degree;
let uf = deg(f)/2, ug = deg(g)/2, uf = deg(h)/2, and u = ufuguh < lmn. Consider
the ideal J = (f(x), g(y), h(z)) ∈ Q[x, y, z] and let ξ = x + x−1 + J , η = y + y−1 + J ,
and ζ = z + z−1 + J . Similarly, for any integers i, j, k we define ξi = xi + x−i + J ,
ηj = yj + y−j + J , and ζk = zk + z−k + J . We will now be working with the Z-algebra
A = Z[ξ, η, ζ]. The set B of all products ξiηjζk/2ε where 0 ≤ i < uf , 0 ≤ j < ug ,
0 ≤ k < uh, with ε equal to the number of subscripts i, j, k in ξiηjζk equal to 0, forms a
basis of A containing 1; the dimension of A is equal to u.
Results of Lemma 7 carry over to bounds on the norm of elements of A = Z[ξ, η, ζ]
with a minor modification of the argument. We will therefore invoke it in our subsequent
considerations regarding Representation 2.
By (6) we have ξ2 − 1 = ξ2 + 1, η2 − 1 = η2 + 1, and ζ2 − 1 = ζ2 + 1. If l ≥ 4, then
ξ2 ∈ B. If l = 2 or 3, a calculation shows that ξ2 + 1 = 1 or 0, respectively. The similar
holds for m and n, and therefore λ = 1 for any hyperbolic triple.
An application of the inequalities (10) and (12) of Lemma 7 shows that for every g ∈ A
and for every ε ∈ {ξ, η, ζ} we have 〈ε1g〉 ≤ 3〈g〉 and 〈ε2g〉 ≤ 7〈g〉. Combining these
inequalities with the obvious analogue of the estimate (14) yields 〈S(3)α〉 ≤ (3 + 3 + 9 +
1)〈α〉 = 16〈α〉. In a similar way, 〈S(2)α〉 ≤ 11〈α〉 and 〈S(1)α〉 ≤ 4〈α〉. This extends to all
X ∈ {R±1, S±1, T±1}, and hence κ ≤ 16. Using these facts as ingredients in Theorem 6
we obtain:
Theorem 10. Let (l,m, n) be a hyperbolic triple. Then, for every positive integer δ there
exists an epimorphism ϕ : T (l,m, n)→ H of injectivity radius at least δ such that |H| ≤ Cδ
where C ≤ 232lmn. 
A further improvement on the exponent at C can be obtained by observing that the above
calculations remain valid if, for k ∈ {l,m, n}, the polynomial (x2k−1)/(x2−1) is replaced
with xk + 1 or (xk + 1)/(x + 1), depending on whether k is even or odd. This way the
dimension of A decreases by a factor of 8, giving the bound C ≤ 24lmn.
The planar width version of the above result reads as follows.
Corollary 11. Let (l,m, n) be a hyperbolic triple and let µ = max{l,m, n}. Then, for every
positive integer d there exists a regular hypermap of type (l,m, n) of planar width larger than
d, the number of vertices of which does not exceed 2C̃d where C̃ ≤ 22µlmn. 
We bypass the problem of comparing the bounds from Theorems 8 and 10 by establishing
yet another bound that is asymptotically better. At this place we just note that a naive experi-
ence with cyclotomic polynomials might suggest the false impression that coefficients of the
k-th cyclotomic polynomial are always small compared to its degree. However, in [16] it was
proved that the natural logarithm of the norm of the k-th cyclotomic polynomial is larger than
exp(ln 2 ln k/ ln ln k) for infinitely many k.
M. Mačaj et al.: Injectivity Radius of Representations of Triangle Groups . . . 235
6 Refining the calculations
The bounds on planar width obtained so far were extracted from the fact that 〈Xα〉 ≤ κ〈α〉
implies 〈Xkα〉 ≤ κk〈α〉. In this section we use a more direct approach. We begin with
determining the powers of the matrix S.
Proposition 12. Let A be a Z-algebra, let ζ∗, η1, ξ1 ∈ A, and let ζi = ζi∗+ ζ−i∗ . Assume that
ζ2∗ − 1 is invertible. Then, for the matrix S from (3) and any positive integer k we have
Sk =
 −1−∑k−1i=1 ξ2i −∑ki=1 ξ2i−1 0∑k
i=1 ξ2i−1 1 +
∑k
i=1 ξ2i 0
η1Σ1 + ζ1Σ4 η1Σ2 + ζ1Σ3 1

and
S−k =
 1 +∑ki=1 ξ2i ∑ki=1 ξ2i−1 0−∑ki=1 ξ2i−1 −1−∑k−1i=1 ξ2i 0
η1Σ3 + ζ1Σ2 η1Σ4 + ζ1Σ1 1
 ,
where Σ1 = k+
∑k−1
i=1 (k− i)ξ2i, Σ2 =
∑k
i=1(k+ 1− i)ξ2i−1, Σ3 = k+
∑k−1
i=1 (k− i)ξ2i,
and Σ4 =
∑k−1
i=1 (k − i)ξ2i−1.
Proof. Although it is possible to use induction, we sketch a more elegant way inspired by the
spectral properties of S. As we saw in Section 3, S is the matrix of an affine transformation
given by ū 7→ ūL + ā. Consequently, Sk represents the mapping ū 7→ ūLk + ā(I + L +
. . .+ Lk−1). Realising that
L = P−1DP =
1
1− ζ2∗
(
1 −ζ∗
−ζ∗ 1
)(
ζ2∗ 0
0 ζ−2∗
)(
1 ζ∗
ζ∗ 1
)
we obtain
Lk = P−1DkP =
1
1− ζ2∗
(
1 −ζ∗
−ζ∗ 1
)(
ζ2k∗ 0
0 ζ−2k∗
)(
1 ζ∗
ζ∗ 1
)
which, after evaluation of the product, leads to the expression for the linear part of S.
A fast way to deal with the affine part of S is based on the following four relations
involving eigenvectors:
ā = η1(1, ζ1) + ξ1(1, 0),
(1, ζ1) = (1, ζ∗) + (1, ζ−1∗ )− (1, 0),
(1, ζ∗)L = ζ2∗(1, ζ∗), and
(1, ζ−1∗ )L = ζ
−2
∗ (1, ζ
−1
∗ ).
Details of the computations are left to the reader. 
In order to effectively utilise the above result in Representation 2, we have to improve
the bound (11). We continue with a technical lemma, presented in a more general form than
actually needed here.
236 Ars Mathematica Contemporanea 1 (2008) 223–241
Lemma 13. Let f(x) ∈ Z[x] be a monic, reciprocal polynomial of even degree, dividing
x2k − 1. Let θi = xi + x−i + (f(x)) ∈ Z[x]/(f(x)). Then, for every j such that 1 ≤ j < k
and every a0, a1, . . . , ak ∈ Z,
θj(a0 +
k∑
i=1
aiθi) =
aj + aj−1θ1 + . . .+ a1θj−1 + a0θj + a1θj+1 + . . .+ ak−j−2θk−2+
ak−j−1θk−1 + ak−jθk + aj + aj+1θ1 + . . .+ ak−1θk−j−1 + akθk−j+
ak−1θk−j+1 + . . .+ ak−j+2θk−2 + ak−j+1θk−1 + akθk−j .
Proof. First, let us note that f(x)|x2k − 1 implies that xi + (f(x)) = x2k−i + (f(x)), and
consequently θi = θ2k−i. We evaluate the product on the left-hand side of the equation in the
lemma by first applying (6) and then rearranging:
θj(a0 +
k∑
i=1
aiθi) = a0θj +
k∑
i=1
aiθj−i +
k∑
i=1
aiθj+i =
[aj+(
j−1∑
i=1
aiθj−i)+a0θj+(
k−j∑
i=1
aiθj+i)]+[aj+(
k∑
i=j+1
aiθj−i)+(
k−1∑
i=k−j+1
aiθj+i)+akθj+k].
The two lines of the formula in the lemma can now be easily obtained by applying the
relations θi = θ−i = θ2k−i to the two expressions that are enclosed in square brackets. 
We now use Lemma 13 to the situation in Representation 2 with the Z-algebra A =
Z[ξ, η, ζ], that is, we will consider θ ∈ {ξ, η, ζ} and k ∈ {l,m, n}. We will show details of
the computations only for θ = ζ, k = n, and for elements of the form g(ζ) = a0+
∑n−2
i=1 aiζi.
Since in our basis for Representation 2 the elements ζi appear only with subscripts i ≤ n−2,
the term anζn−j is not present at all in the formula of Lemma 13. Taking the norm of
both sides of the formula and using (11) we obtain 〈ζjg〉 ≤ (2 + 2〈ζn−1〉 + 〈ζn〉)〈g〉 ≤
(2 + 2 + 4)〈g〉 = 8〈g〉 for every g ∈ A and 1 ≤ k ≤ n− 1.
Using the standard approach combined with the above results we obtain
〈Skα〉 ≤ (3[2k + 8(k(k + 1)
2
+ 3
k(k − 1)
2
)] + 1)〈α〉 = (48k2 − 18k + 1)〈α〉
Recalling that because of the order of S we are only interested in the values k ≤ n/2, we
conclude that for any power X of S we have
〈Xα〉
〈α〉
≤ 12n2 − 9n+ 1 ≤ 16n2
Combining this with the arguments used in the proof of Theorem 6 we finally arrive at:
Theorem 14. Let (l,m, n) be a hyperbolic triple and let µ = max{l,m, n}. Then, for every
positive integer d there exists a regular hypermap of type (l,m, n) of planar width larger
than d, the number of vertices of which does not exceed 2C̃d where C̃ ≤ (4µ)64lmn. 
M. Mačaj et al.: Injectivity Radius of Representations of Triangle Groups . . . 237
This is in general much better than the bound from Corollary 11, since in the exponent
the term 2µ has been replaced with 64(2 + log2 µ).
Let us sum up the advantages and possible disadvantages of the two representations we
have been using throughout. A strong point for Representation 1 is a smaller dimension of
the algebra, especially when g.c.d.(l,m, n) is relatively large compared to l,m, n. More-
over, Representation 1 enables one to work over the ring Z[2 cos(π/M)], which is a ring
of algebraic integers. We have used the fact that this ring embeds in a field, and there are
other important properties of such rings that could be useful in future research in this area. A
weaker point of Representation 1 is the potentially large upper bound on κ. On the contrary,
the corresponding bound on κ in Representation 2 is small, even constant. This advantage
is, however, negatively outweighed by large dimension of the algebra. The effect of this be-
comes significant if g.c.d.(l,m, n) is relatively large compared to l,m, n. Also, the algebra
is bound to have divisors of zero.
In what follows we will show that the drawback of potentially large exponents κ in Repre-
sentation 1 can be overcome. This will be achieved by applying Lemma 12 to Representation
1 with the goal to obtain bounds on C comparable to the ones appearing in Representation 2.
First, we extend the definition of a norm to an arbitrary generating set {βi; i ∈ I} of a
Z-algebra A by setting
〈g〉 = min{max
i∈I
|ai|; g =
∑
i∈I
aiβi, ai ∈ Z}
where the minimum is taken over all possible ways of expressing g as an integral linear
combination of generators. It is easy to see that for any two norms 〈 〉1 and 〈 〉2 with respect
to two finite generating sets of a Z-algebra A there exists a constant c such that for every
g ∈ A we have
〈g〉1 ≤ c〈g〉2.
Let (l,m, n) be a hyperbolic triple, with M = l.c.m(l,m, n) = l0l = m0m = n0n
and µ = max{l,m, n}. Consider the 2M -th cyclotomic polynomial f(x) ∈ Z[x]. As in
Representation 1 before, in the ring Z[x]/(f(x)) we let θ = x + x−1 + (f(x)) and θi =
xi + x−i + (f(x)). Further, we define ξ = θl0 , η = θm0 , and ζ = θn0 . Our underlying
Z-algebra will be A = Z[θ]. Let R, S, and T be matrices as in (3) where ξ, η, and ζ are as
defined above. The new idea is to work with two norms simultaneously. The first norm 〈 〉
wlll be the previously used integral norm with respect to the basis {1, θ1, θ2, . . . , θu−1} of
A where u = φ(2M)/2. The second norm [ ] will be the integral norm with respect to the
generating set {1, θ1, θ2, . . . , θM}.
Yet another application of the methods permeating this paper gives the following result
the proof of which is left to the reader.
Lemma 15. Let g(θ) ∈ A and X ∈ {R±1, S±1, T±1}. Then, for any integer k,
(1) [g] ≤ 〈g〉 ≤ (3 + 〈f〉)M−u+1[g],
(2) [θig] ≤ 3[g] for 1 ≤ i < M ,
(3) [Xα] ≤ 16[α],
(4) [Xkα] ≤ (9/2)µ2[α].
238 Ars Mathematica Contemporanea 1 (2008) 223–241
These inequalities imply that for an arbitrary product X ′ = X1X2 . . . Xδ , where Xi ∈
{R±1, S±1, T±1}, we have [X ′] ≤ 16δ−1 and 〈X ′〉 ≤ (3 + 〈f〉)M−u+116δ−1. Leaving
the reiterating details aside, as the result we obtain the following conclusion.
Theorem 16. Let (l,m, n) be a hyperbolic triple and let M = l.c.m.(l,m, n). Let f(x) be
the 2M -th cyclotomic polynomial. Then, for every positive integer δ there exists an epimor-
phism ϕ : T (l,m, n) → H of injectivity radius at least δ such that |H| ≤ D · Cδ where
D ≤ (3 + 〈f〉)4Mφ(2M) and C ≤ 216φ(2M). 
The key point, however, on our way to further refinements is to consider more general
exponents in the product in conjunction with Proposition 12. If X ′′ = Xj11 X
j2
2 . . . X
jd
d ,
then [X ′′] ≤ (9µ2/2)d−1µ/2, and 〈X ′′〉 ≤ (3 + 〈f〉)M−u+1(9µ2/2)d−1µ/2. Again, for
every δ we may choose a prime p such that (3 + 〈f〉)M−u+1(4µ)2d−1 + 1 < p < (3 +
〈f〉)M−u+1(4µ)2d, making the norm 〈X ′′ − I〉 smaller than p. Invoking Theorem 6 and its
proof we eventually obtain:
Theorem 17. Let (l,m, n) be a hyperbolic triple, let M = l.c.m.(l,m, n), and let µ =
max{l,m, n}. Let f(x) be the 2M -th cyclotomic polynomial and let 〈f〉 be the largest abso-
lute value of its coefficients. Then, for every positive integer d there exists a regular hypermap
of type (l,m, n) of planar width larger than d with number of vertices not exceeding 2D · C̃d
where D ≤ (3 + 〈f〉)2Mφ(2M) and C̃ ≤ (4µ)8φ(2M). 
We note that even though the constant D looks enormous, its effect with increasing δ be-
comes insignificant. Therefore the last two bounds are asymptotically better than the bounds
of Theorem 10 and Theorem 14, respectively.
7 Maps, Hurwitz groups, and conclusion
We have seen in the Introduction that the research into injectivity radius of epimorphisms of
triangle groups was actually motivated by questions about planar width of regular maps rather
than hypermaps. Recall that a regular map of type (m,n) can be identified with a regular hy-
permap of type (2,m, n). If the triple (2,m, n) is hyperbolic, that is, if 1/m + 1/n < 1/2,
the pair (m,n) is said to be hyperbolic as well. The usual topological representation of
a regular map is obtained from the face-3-coloured embedding of the corresponding hy-
permap by contracting the the faces coloured 2 to points and collapsing the quadrangles
coloured 1 to line segments, giving rise to vertices and edges of the map, respectively. Ac-
cordingly, the group T (2,m, n) is usually presented as a two-generator group in the form
T (2,m, n) = 〈r, s| rm = sn = (rs)2 = 1〉. This change of viewpoint affects the definition
of injectivity radius and planar width. In particular, the injectivity radius of an epimorphism
φ : T (2,m, n) → H = 〈ρ, σ| ρm = σn = (ρσ)2 = . . . = 1〉 is defined with respect to the
set {r, r−1, s, s−1}. The topological definition of planar width of regular maps carries over in
the obvious way, that is, it is again the smallest number of intersections of a non-contractible
closed curve with the 1-skeleton of the map. Since the 1-skeleton of the map is different from
the 1-skeleton of the corresponding hypermap, the algebraic characterisation of planar width
is slightly different as well and reads as follows. The planar width of the map associated with
an epimorphism ϕ : T (2,m, n)→ H is the smallest positive integer d such that there exists
w = (ri1sj1)(ri2sj2) . . . (ridsjd) such that w 6= 1 and ϕ(w) = 1.
Estimates on injectivity radius of epimorphisms of T (2,m, n) and on planar width of
regular maps can be obtained as consequences of our general approach. One may, of course,
M. Mačaj et al.: Injectivity Radius of Representations of Triangle Groups . . . 239
substitute the value l = 2 in all our previous results. A better way, however, is to observe that
the fact that 2 cos(π/2) = 0 means that in every representation of T (2,m, n) based on the
matrices R, S, and T given in (3), we can use ξ = 0 and, of course, ignore T . Carrying the
preceding computation through with suchR and S one obtains the following specific versions
of our earlier results.
Theorem 18. Let (m,n) be a hyperbolic pair and let M = l.c.m(m,n). Then, for any
positive integer δ there exists a representation of T (2,m, n) of injectivity radius at least δ
into a finite group of order at most D · Cδ , where C ≤ 134φ(2M) and D depends only on m
and n, and not on δ. 
Theorem 19. Let (m,n) be a hyperbolic pair and let M = l.c.m(m,n). Then, for any
positive integer d there exists a regular map of type (m,n) of planar width larger than d with
automorphism group of order at most D · C̃δ , where C̃ ≤ (5mn)8φ(2M) and D depends only
on m and n, and not on d. 
The reader may have realised that there is still room for further improvement of the pre-
sented methods and results. Our intention, however, was to focus on the main ideas and not to
get stuck in details. Nevertheless, we at least indicate an improvement for the hyperbolic pair
(3, 7), which plays an important role in the theory of maps. The finite quotients of T (2, 3, 7)
are known as the Hurwitz groups.
From the last part of the proof of Theorem 6 we see that the estimate on the order of the
target group in a representation of T (2, 3, 7) depends on three parameters: The type of the
groups in which the representation takes place, dimension of the algebra A, and the prime p.
We now show a way to reduce all three parameters in the case of T (2, 3, 7). First, since the
representation is, in fact, in the orthogonal group, the number 8 = 32 − 1 can be reduced
to 3. Second, realising that 2 cos(π/2) = 0 and 2 cos(π/3) = 1, we can work in the ring
Z[2 cos(π/7)] instead of Z[2 cos(π/42)], decreasing thereby the dimension of the algebra
from 12 to 3. Third, a detailed evaluation of the products RiSj instead of using just the
powers Ri and Sj enables us to show that for the injectivity radius at least a given positive
integer δ ≥ 2 it suffices to choose a prime p such that 2·5δ−1+1 < p < 4·5δ−1. Similarly, for
planar width at least d ≥ 1 is is sufficient to choose p satisfying 6·21d−1+1 < p < 12·21d−1.
Putting the pieces together gives the following results, leaving the technical details to the
reader.
Theorem 20. For any positive integer δ ≥ 2 there exists a representation of T (2, 3, 7) of
injectivity radius at least δ into a finite group of order at most 59δ 
Theorem 21. For any positive integer d there exists a regular map of type (3, 7) of planar
width larger than d with automorphism group of order at most 219d. 
Even though the constants 59 and 219 are still rather large, they are considerably smaller
than the constants 1348 and 10596, obtained by substituting the values m = 3 and n = 7 in
the bounds of Theorems 18 and 19.
We conclude with a few remarks. It may be instructive to sum up the individual stages
leading from the original upper bound in [14] through the intermediate steps in [1, 5] to the
improvements we presented in the last three section. The innovation of [1, 5] was in con-
sidering representations over factor rings of multivariate polynomials and a more accurate
240 Ars Mathematica Contemporanea 1 (2008) 223–241
evaluation of the bounds at the expense of compactness of the obtained formulae. The con-
tribution of the present paper is threefold. First, we derived a rigorous algebraic background
for the ideas used in the previous papers. Second, utilisation of variables of the form of a sum
of an element and its inverse enabled us to abandon Tchebyshev’s polynomials and, instead,
work with polynomials (x2k−1)/(x2−1) of norm 1. The associated change of basis opened
the room to working with powers of the generating matrices and led to direct computation of
planar width. Third, extending the concept of norm to generating sets enabled us to return to
representations over complex numbers which, at the same time, have a smaller dimension.
Our improvements on the estimates on C and C̃, although significant, are still (roughly)
exponential in lmn, while the lower bound extracted from [11] is polynomial in lmn. To im-
prove the lower bound one would have to use some knowledge about fundamental regions of
normal, torsion-free, finite-index subgroups of triangle groups; such regions are, in general,
far from disk-neighbourhoods considered in [11]. As regards the upper bound, additional re-
finements of our methods combined by a use of different bases may result in further progress.
A possibly fruitful direction could be to incorporate the classification of 2-dimensional linear
representations of triangle groups over finite fields presented in [13]. Working with linear
representations over u-dimensional algebras, however, implies that one will not be able to
get away with the exponent u appearing in the bound on C, due to Minkowski’s theorem on
lattice points in convex subsets of multidimensional spaces (see e.g. [7]).
Acknowledgement. The authors acknowledge partial support from the VEGA Grants No.
1/3020/06, 1/3022/06, 1/0489/08, and APVV Research Grants 20-000704, 0040-06, 0104-
07, and LPP-0203-06.
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