Also available at http://amc.imfm.si
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 6 (2013) 25–35
Nonorientable regular maps over
linear fractional groups
Gareth A. Jones
University of Southampton, Southampton, U.K.
Martin Mačaj
Comenius University, Bratislava, Slovakia
Jozef Širáň
Open University, Milton Keynes, U.K.
Slovak University of Technology Bratislava, Slovakia
Received 28 October 2011, accepted 22 January 2012, published online 1 June 2012
Abstract
It is well known that for any given hyperbolic pair (k,m) there exist infinitely many
regular maps of valence k and face length m on an orientable surface, with automorphism
group isomorphic to a linear fractional group. A nonorientable analogue of this result was
known to be true for all pairs (k,m) as above with at least one even entry. In this paper we
establish the existence of such regular maps on nonorientable surfaces for all hyperbolic
pairs.
Keywords: Regular map, linear fractional group.
Math. Subj. Class.: 05C10, 05C25, 20G99
1 Introduction
A map on a compact, orientable surface is orientably regular if the group of all orientation
preserving automorphisms of the map is transitive, and hence regular, on darts of the map.
A map on a compact, nonorientable surface is regular if its automorphism group is transi-
tive, and hence regular, on flags of the map. In either case, such maps have all vertices of
the same degree and all faces of the same length; if these quantities are k and m we speak
of a map of type {m, k}. The type is said to be hyperbolic if 1/k+ 1/m < 1/2. Regarding
type, the following basic fact was rediscovered a number of times in the past.
E-mail addresses: g.a.jones@soton.ac.uk (Gareth A. Jones), macaj@fmph.uniba.sk (Martin Mačaj),
j.siran@open.ac.uk (Jozef Širáň)
Copyright c© 2013 DMFA Slovenije
26 Ars Math. Contemp. 6 (2013) 25–35
Theorem 1.1. For every hyperbolic pair (k,m) there exist infinitely many orientably reg-
ular maps of type {m, k}.
A brief history of the development around this result together with a list of various
proofs can be found in [9]. A particularly important way of proving Theorem 1.1 follows
from [8] and implies that all such maps can be chosen to have the orientation preserving
automorphism group isomorphic to a linear fractional group over a finite field.
It is quite surprising that a nonorientable analogue of Theorem 1.1 has not been con-
sidered. A proof might follow from the study of generation of symmetric and alternating
groups by pairs of permutations of given order in [6], but this work does not appear to
have a corresponding follow-up and it is not our intention to do so. Instead, motivated by
the result of [8] mentioned above, we will be interested in possibilities to prove a stronger
form of a nonorientable analogue of Theorem 1.1 for maps with automorphism group iso-
morphic to a linear fractional group over a finite field. In fact, this has already been done
for three quarters of the cases by the third author in [11] where it is shown that for any
hyperbolic pair (k,m) with at least one even entry there are infinitely many nonorientable
regular maps of type {m, k} with automorphism group isomorphic to PGL(2, F ) for suit-
able finite fields F , but the case when both k and m are odd was left untouched.
In this note we fill in this gap, establishing thus the existence of an infinite number of
nonorientable regular maps of type {m, k}with automorphism group isomorphic to a linear
fractional group for any given hyperbolic type {m, k}. The main results are presented in
Section 4, preceded by background information summed up in Section 2 and auxiliary
number theoretic results in Section 3.
2 Preliminaries
Foundations of the theory of regular maps have been laid in [4] and [2] and in what follows
we just briefly review a few basic facts; for surveys we recommend [7] and [10].
Orientably regular maps of type {m, k} can be identified with their orientation pre-
serving automorphism groups and these are in a one-to-one correspondence with the finite
groups G presented in the form
G = 〈r, s; rk = sm = (rs)2 = . . . = 1〉 (2.1)
where r and s represent a k-fold rotation about a fixed vertex of the map and an m-fold
rotation about the centre of a face incident with the vertex. In particular, we require that k,
m and 2 are the true orders of r, s, and rs, respectively. Vertices, faces and edges of the
orientably regular map Mor(G) corresponding to a presentation of a group G as in (2.1)
can be identified with left cosets of the cyclic subgroups 〈r〉, 〈s〉 and 〈rs〉, with incidence
determined by non-empty intersection; the group G then acts as the orientation preserving
automorphism group of Mor(G) by left multiplication.
Regular maps on nonorientable surfaces are also in a one-to-one correspondence with
presentations of finite groups as in (2.1) but satisfying the extra condition that G contains
an involution t such that trt = r−1 and tst = s−1. This time, the nonorientable regular
map Mnor(G) corresponding to such a group G has vertices, faces and edges identified
with the left cosets of the dihedral subgroups 〈r, t〉, 〈s, t〉 and 〈rt, ts〉. Incidence is again
determined by non-empty intersection and G acts as the automorphism group of Mnor(G)
by left multiplication.
G. A. Jones et al.: Nonorientable regular maps over linear fractional groups 27
Thus, if a group G with presentation (2.1) admits an inner automorphism induced by
an involution and inverting r and s, the above correspondences allow one to associate
two maps with G, namely, Mor(G) and Mnor(G). To remove this ambiguity in what
follows, for a group G as in (2.1) we define the map M(G) by letting M(G) = Mnor(G)
if G has an inner automorphism induced by an involution, inverting both r and s, and
M(G) = Mor(G) if G has no such inner automorphism.
As stated in the Introduction we will be interested in regular maps of a given type with
automorphism group isomorphic to a linear fractional group. We begin by recalling the
characterisation of such automorphism groups from [8]; for a much more detailed proof
we refer to [3].
Proposition 2.1. Let (k,m) be a hyperbolic pair and letK be an algebraically closed field
of a prime characteristic p coprime to km. Let ξ and η be primitive (δk)th and (δm)th roots
of unity in K, where δ = 1 if p = 2 and δ = 2 if p > 2. Let D = −(ξ2 + ξ−2 + η2 + η−2)
and let
R = ±
[
ξ 0
0 ξ−1
]
and S = ±(ξ − ξ−1)−1
[
−(η + η−1)ξ−1 D
1 (η + η−1)ξ
]
be elements of PSL(2,K). Then,
(a) the orders of R, S, and RS in PSL(2,K) are k, m, and 2, respectively, and
(b) if G is a subgroup of PSL(2,K) with presentation (2.1), then there exist primitive
(δk)th and (δm)th roots of unity such that G is conjugate to the subgroup generated
by the matrices R and S as above.
It is therefore sufficient to study the groups G(ξ, η) = 〈R,S〉 with R and S as above.
Necessary and sufficient conditions for G(ξ, η) to give rise to a nonorientable regular map
were given in [3]. Here we present an excerpt sufficient for our purposes.
Theorem 2.2. Let (k,m) be a hyperbolic pair and letK be an algebraically closed field of
a prime characteristic p relatively prime to both k and m. Let ξ and η be primitive (δk)th
and (δm)th roots of unity in K, where δ = 1 if p = 2 and δ = 2 if p > 2. Let e = e(k,m)
be the smallest positive integer j such that n | (pj − εn)/δ for each n ∈ {k,m} and some
εn ∈ {+1,−1}. Then:
(1) if e is even, e = 2f , then G(ξ, η) is isomorphic to PGL(2, pf ) if and only if the
quantity D = −(ξ2 + ξ−2 + η2 + η−2) is not equal to zero, and either (a) there is
an even entry n ∈ {k,m} and an ε ∈ {+1,−1} such that n divides (pf − ε) but 2n
does not, while the other entry divides (pf − ε′)/2 for some ε′ ∈ {+1,−1}, or (b)
both k and m are even and for any n ∈ {k,m} there exists an ε′n such that n is a
divisor of (pf − ε′n) but 2n is not;
(2) G(ξ, η) is isomorphic to PSL(2, pe) if and only if D 6= 0 and either e is odd, or the
pair (k,m) together with an even e do not satisfy any of the above conditions (a) and
(b); and
(3) if D 6= 0, the map M(G(ξ, η)) is nonorientable if and only if either e = 2f and
G(ξ, η) ∼= PGL(2, pf ), or if G(ξ, η) ∼= PSL(2, pe) and D is a square in GF (pe);
in particular, in the last case M(G(ξ, η)) is always nonorientable if p = 2 and both
k and m are odd.
28 Ars Math. Contemp. 6 (2013) 25–35
From the last part of this result one sees that to obtain nonorientable regular maps it is,
for example, sufficient to make sure that e = 2f , D 6= 0 and G(ξ, η) ∼= PGL(2, pf ). In
[11] it is shown that this can be guaranteed whenever at least one of k and m is even:
Theorem 2.3. Let (k,m) be a hyperbolic pair with at least one even entry. Then, there is
an infinite number of finite, nonorientable, regular maps of type {m, k} with automorphism
group isomorphic to PGL(2, F ) for suitable finite fields F .
To be able to extend this result to the case when both k and m are odd, by Theorem
2.2 one can only hope to establish the existence of infinitely many regular maps of type
{m, k} with automorphism group isomorphic to PSL(2, F ) for suitable finite fields F . In
particular, by part (3) of Theorem 2.2, to achieve this we need to make sure that for an
infinite number of primes p one can select primitive 2k-th and 2m-th roots of unity ξ and
η in GF (pe) in such a way that the quantity D = −(ξ2 + ξ−2 + η2 + η−2) is a square
in GF (pe), where e = e(k,m); note that e depends on p as well but this dependence is
not shown in our notation. Observe that if k and m are odd, ξ2 and η2 are primitive k-th
and m-th roots of unity, respectively. Thus, in this case D has the form D = −(ωk + ωm)
where ωn denotes the sum of an n-th primitive root of unity and its reciprocal in GF (pe).
In what follows we will investigate such quantities in general, first over the field of complex
numbers and subsequently over finite fields by considering factor fields of rings of algebraic
integers.
3 Auxiliary results involving complex roots of unity
Let ζn denote any primitive n-th root of unity, but this time taken in the field C of complex
numbers unless stated otherwise. It is known that all the primitive n-th roots of unity are
conjugate over the rationals Q and their common minimal polynomial is the n-th cyclo-
tomic polynomial Φn of degree ϕ(n), the value of the Euler totient function at n. By ωn
we denote any number of the form ζn + ζ−1n ; these quantities are again conjugate over Q
and their common minimal polynomial will be denoted by Ψn(x). It is well known that if
n > 2, then
xϕ(n)/2Ψn(x+ x
−1) = Φn(x). (3.1)
Finally, let Un denote the set of all primitive n-th roots of unity in C and let Un stand for
the set of all the corresponding quantities ωn.
We continue with some observations. From the fact that Φ1(x) = x − 1, Φp(x) =
1 +x+ · · ·+xp−1 and Φpn(x) = Φn(xp) if p|n and Φpn(x) = Φn(xp)/Φn(x) otherwise,
we obtain the following auxiliary result by easy calculations.
Lemma 3.1. Let Φn(x) be the n-th cyclotomic polynomial. Then, Φ1(1) = 0, Φpk(1) = p
for p prime and k > 0, and Φn(1) = 1 otherwise. Also, Φ1(−1) = −2, Φ2(−1) = 0,
Φ2pk(−1) = p for p prime and k > 0, and Φn(−1) = 1 otherwise.
With the help of these facts we obtain our basic result on products of the quantities
−(ωk + ωm) for any ωk ∈ Uk and ωm ∈ Um.
Proposition 3.2. Let k,m be odd positive integers and let
P (k,m) =
∏
ωk∈Uk
∏
ωm∈Um
−(ωk + ωm) .
Then, P (1, 1) = −4, P (k, k)2 = (−2)ϕ(k) for k ≥ 3, and P (k,m)2 = 1 otherwise.
G. A. Jones et al.: Nonorientable regular maps over linear fractional groups 29
Proof. Obviously P (k,m) = P (m, k) and we will therefore assume that k ≥ m in what
follows. The values of P (k,m) for k,m ≤ 2 are trivial. If k ≥ 3 and m = 1, then
P (k, 1) =
∏
ωk∈Uk −(2 + ωk) = Ψk(−2) = (−1)
−ϕ(k)/2Φk(−1) = (−1)ϕ(k)/2 and so
P (k, 1)2 = 1. For the remaining part of the proof we assume that k ≥ m > 1.
By the properties of the polynomials
Ψm(x) =
∏
ωm∈Um
(x− ωm)
we obtain, for any ωk = ζk + ζ−1k ∈ Uk, the equality∏
ωm∈Um
−(ωk + ωm) = Ψm(−ωk) = (−ζk)−ϕ(m)/2Φm(−ζk) .
Let U ′k be a subset of Uk of cardinality ϕ(k)/2 such that Uk = {ζk + ζ
−1
k ; ζk ∈ U ′k}. The
previous computation then implies that
P (k,m) =
∏
ζk∈U ′k
(−ζk)−ϕ(m)/2Φm(−ζk) .
Extending the product above from U ′k to Uk means squaring the last equation; combining
this with the fact that the product of all the (even number of) k-th primitive roots of unity
is equal to 1 we obtain
P (k,m)2 =
∏
ζk∈Uk
(−ζk)−ϕ(m)/2Φm(−ζk) =
∏
ζk∈Uk
Φm(−ζk).
Invoking the well known identity Φm(x) =
∏
d|m(x
d − 1)µ(m/d), where µ is the Moebius
function, we have
Φm(−ζk) =
∏
d|m
(−ζdk − 1)µ(m/d) .
This product is non-zero since both k and m, and hence all the divisors d, are odd and so
(−ζk)d 6= 1; note also that the divisors satisfy d ≤ k because of the assumption m ≤ k.
Let us analyze the system of powers U = (ζdk ; ζk ∈ Uk) appearing in the last equality.
For any positive divisor d of m let n(d) = k/(d, k) and r(d) = ϕ(k)/ϕ(n(d)); of course,
both quantities depend on k as well. It can now be seen that the system U is a collection,
for any d dividing m, of primitive n(d)-th roots of unity, each repeated r(d) times. With
the help of all these facts together with Φt(x) =
∏
ζt∈Ut(x− ζt) evaluated at x = −1 we
successively obtain
P (k,m)2 =
∏
ζk∈Uk
∏
d|m
(−ζdk − 1)µ(m/d) =
∏
d|m
∏
ζk∈Uk
(−1− ζdk)µ(m/d)
=
∏
d|m
∏
ζn(d)∈Un(d)
(−1− ζn(d))r(d)µ(m/d) =
∏
d|m
(Φn(d)(−1))r(d)µ(m/d) .
As all the values of n(d) are odd here, we have Φn(d)(−1) = 1 if d < k and Φn(d)(−1) =
−2 if d = k, where the second possibility occurs if and only ifm = k and then r(d) = ϕ(k)
and µ(m/d) = 1. We conclude that for k ≥ 3 we have P (k, k)2 = (−2)ϕ(k), and
P (k,m)2 = 1 if 1 < m < k. This completes the proof.
30 Ars Math. Contemp. 6 (2013) 25–35
A different and more powerful approach to the investigation of the quantity D from
Theorem 2.2 relies on some known facts on algebraic integers in algebraic number fields.
We refer to [1] as a suitable introductory reference and recall here just a few basic concepts
and results.
Let K be an algebraic number field, that is, an extension of Q of a finite degree. Let
O = OK be the ring of algebraic integers in K. The ring O is known to be a Dedekind
domain, but apart from a few facts the theory of such domains will not be needed. A basic
property of O is that every non-zero ideal J ⊂ O has a finite index [O : J ]. Without
going into too much detail we recall that the the index [O : J ] is the norm N(J) of J .
Another important property of O is that any prime ideal J ⊂ O is maximal. Thus, for
any such J the quotient ring O/J is a finite field and so there exists a unique rational
prime p such that N(J) = pj for some j ∈ {1, 2, . . . , d}, where d = [K : Q] is the
degree of the extension. Further, it is known that K admits exactly d distinct injective
homomorphisms σ1, . . . , σd into C. The norm N(z) of any element z ∈ K is defined as
the product N(z) = σ1(z) . . . σd(z); the elements σt(z), 1 ≤ t ≤ d, are the conjugates of
z over K. The norm is multiplicative, that is, N(z1z2) = N(z1)N(z2) for any z1, z2 ∈ K.
Norms of elements of O and ideals in O are known to be related by the fact that, for every
non-zero algebraic integer z ∈ O, the absolute value of N(z) is equal to the norm of the
ideal (z) ⊂ O generated by z. In particular, the norm of every non-zero element z ∈ O is
a non-zero integer, and it is well known that |N(z)| = 1 if and only if z is a unit, that is,
an invertible element in the ring O. We will also repeatedly use the fact that if an element
z ∈ O belongs to an ideal I of O, then N(I) divides N(z).
For illustration we present some of the consequences of Proposition 3.2 in the language
of algebraic number theory. Let α = ζ2k and β = ζ2m be complex primitive 2k-th and 2m-
th roots of unity, respectively, and letA = −(α2+α−2+β2+β−2). In what follows, letK
denote the algebraic number field Q[α, β]. Since the generators α and β of K are roots of
unity in C, every injective homomorphism σ : K → C is uniquely determined by positive
integers i and j, relatively prime to k andm, such that σ(α) = αi and σ(β) = βj . Observe,
however, that whether particular i and j give rise to such an injective homomorphism may
also depend on α and β and not just on k and m. As before, let O = OK be the ring of
algebraic integers of K. Since α and β themselves are algebraic integers in K, we have
A ∈ O; in particular, the norm N(A) in O is an integer.
Lemma 3.3. If α 6= β, then A is a unit in O, and if α = β, then |N(A)| is a power of 2.
Proof. Observe that all factors in the product P (k,m) in Proposition 3.2 are algebraic
integers, with all conjugates of A being among the factors. By the same Proposition we
have P (k,m) = ±1 if k 6= m, while P (k, k)2 = (−2)ϕ(k) for k ≥ 3. Since algebraic
integers have integral norm, it follows that A is a unit in O if k 6= m. In the case when
k = m and α = β, the absolute value of the norm of−2(α2 +α−2) is equal to (−2)ϕ(k)/2,
and therefore for α 6= β the absolute value of the norm of A must be 1.
Returning to our main theme, until the end of this section we will assume that (k,m) is
a fixed hyperbolic pair with no restriction on the parity of the two entries. We begin by an
elementary observation that will turn out to be crucial later.
Lemma 3.4. The quantity A− n2 is never a unit in O for any integer n > 2.
Proof. We recall the known fact that K is isomorphic to the cyclotomic field Q[γ], where
γ = cos( 2π` ) + i sin(
2π
` ) is a primitive `-th root of unity for ` = lcm{2, k,m}. The
G. A. Jones et al.: Nonorientable regular maps over linear fractional groups 31
conjugates of γ over K have the form cos( 2π` j) + i sin(
2π
` j), where 1 ≤ j < ` and
(j, `) = 1. All the ϕ(`) distinct injective homomorphisms σt : K → C preserve the
rationals pointwise. Since the explicit form of the conjugates of γ over K implies that
|σt(A)| < 4, we have |σt(A − n2)| = |σt(A) − n2| ≥ n2 − |σt(A)| > n2 − 4 for
any t such that 1 ≤ t ≤ ϕ(`). Thus, by the definition of the norm, for n > 2 we have
|N(A− n2)| > 1, which means that A− n2 is not invertible in O.
It is useful to realise that our considerations before Lemma 3.1 did not depend on the
parity of k and m and hence we may use them in what follows. Observe that in the general
case we want to deal with, the value of A could be equal to zero in K, which happens
precisely if iβ ∈ {±α,±α−1}. If, however, {k,m} is a hyperbolic pair, it is easy to see
that we can choose α and β avoiding this condition. Keeping to the notation introduced
above, for any n ≥ 3 let I = In be a maximal ideal in O containing the element A − n2
and let p = pn be the characteristic of the field F = O/I . Letting ξ = α + I , η = β + I ,
and D = A+ I , we have:
Lemma 3.5. If n is relatively prime toN(A), then the elementD = −(ξ2+ξ−2+η2+η−2)
is a non-zero square in F and p does not divide n. Moreover, if p is not a divisor of 2km,
then ξ and η are primitive 2k-th and 2m-th root of unity in F .
Proof. Since A− n2 ∈ I , that is, A+ I = n2 + I , the element D = A+ I is a square in
F . As p ∈ I and I is a prime ideal, by our earlier remarks on norms of elements and ideals
of the Dedekind ring O the condition A ∈ I is equivalent to each of the conditions n2 ∈ I ,
n ∈ I , and p|n. Hence p divides both n and N(A), contrary to our assumption on their
relative primeness.
It is obvious that ξ is a 2k-th root of unity in F . Assume that αu− 1 ∈ I , where αu is a
primitive c-th root of unity in C for a proper divisor c of 2k. As the ideal generated by the
algebraic integer αu− 1 is contained in I , the norm of I divides the norm of αu− 1, which
implies that the norm of αu − 1 is divisible by p. On the other hand, all conjugates of αu
are c-th primitive roots of unity in C. Arguments analogous to those used in the proof of
Proposition 3.2 imply that, up to sign, the norm of αu − 1 is a power of Φc(1). Thus, by
Lemma 3.1, c is a power of p, contrary to the assumption that p - 2k. It follows that ξ is a
primitive 2k-th root of unity in F . By the same token, η is a primitive 2m-th root of unity
in F .
By suitably varying the parameter n one obtains an infinite sequence of primes as in
Lemma 3.5.
Lemma 3.6. If A 6= 0, then there exists an infinite set of values n and an infinite sequence
of prime ideals In of O containing the element A − n2 such that the fields O/In have
pairwise distinct prime characteristic pn.
Proof. Referring to the way the primes pn have been introduced for any n > 2, let us
define an infinite sequence nj of integers by letting n1 = 2|N(A)|+ 1 and nj =
∏j−1
i=1 pni
for j > 1. Applying Lemma 3.5 inductively we deduce that pnj does not divide nj for
any j ≥ 1. By our construction, for any j ≥ 2 the prime pnj differs from all the previous
primes pni for i < j.
32 Ars Math. Contemp. 6 (2013) 25–35
4 Results
Two immediate consequences in the direction of our interest can be obtained by exploring
earlier results. Firstly, there is a much more general version of Theorem 2.2 in which the
prime p is not necessarily coprime to k and m, in particular, covering the case when both
k and m are equal to p and G ∼= PSL(2, p); see Propositions 3.1, 3.2, 4.6 and 6.1 of [3].
In order to avoid a rather long re-statement of these facts we invite the reader to check that
part (2) of Proposition 6.1 combined with Proposition 3.1 of [3] imply:
Theorem 4.1. If p is a prime congruent to 1 mod 4, then there exists a nonorientable
regular map of type (p, p) with automorphism group isomorphic to PSL(2, p). 2
Secondly, if both k and m are odd and p = 2, part (3) of Theorem 2.2 directly yields
the following result, where e = e(k,m) is as introduced in the statement of Theorem 2.2.
Theorem 4.2. Let (k,m) be a hyperbolic pair consisting of odd entries. Then there
is a nonorientable regular map of type {m, k} with automorphism group isomorphic to
PSL(2, 2e) for e = e(k,m). 2
Together with the earlier findings this gives at least an existence result of the sought
kind on regular maps over linear fractional groups.
Corollary 4.3. For any hyperbolic pair (k,m) there exists a nonorientable regular map of
type {m, k} with automorphism group isomorphic to a linear fractional group over a finite
field. 2
In the light of Theorem 2.3, the question of existence of an infinite number of such
maps of any given type hyperbolic type is settled by the following result. Although we are
interested mainly in the case when k and m are odd, we give a more general formulation
which yields an alternative proof of Theorem 2 of [11].
Theorem 4.4. For every hyperbolic pair (k,m) there is an infinite number of finite, nonori-
entable, regular maps of type {m, k} with automorphism group isomorphic to PSL(2, F )
or PGL(2, F ) for suitable finite fields F .
Proof. We will refer to the notation introduced in Section 3. For a fixed hyperbolic pair
(k,m) and a non-zero A = −(α2 + α−2 + β2 + β−2) with let p = pn be any prime from
Lemma 3.6 relatively prime to 2km, and let G = G(ξ, η) be the corresponding group.
By Theorem 2.2, G is isomorphic to PSL(2, F ) or PGL(2, F ′). As D is a square, the
corresponding regular map M(G) is nonorientable in both cases.
We also present two more results based on residue techniques which, although appli-
cable only to a very restricted infinite set of types with both entries odd, may be useful in
future investigations.
Theorem 4.5. Let k and m be prime powers congruent to 3 mod 4. Then, there exist
infinitely many nonorientable regular maps of type {m, k} with automorphism group iso-
morphic to PSL(2, F ) for suitable finite fields F .
Proof. Let p be a prime congruent to 1 mod 8 and let e = min{n; k|pn ± 1 and m|pn ±
1}. Then, p does not divide 2km and the equality from Proposition 3.2 holds also in
GF (pe), with appropriate interpretation of the primitive roots. Since p ≡ 1 mod 8, the four
G. A. Jones et al.: Nonorientable regular maps over linear fractional groups 33
elements ±1 and ±2 are all quadratic residues in GF (p) and hence also in GF (pe). By
Proposition 3.2, the element P (k,m) is a quadratic residue inGF (pe); note that in the case
k 6= m it would have been sufficient to assume p ≡ 1 mod 4 to obtain the same conclusion.
By our assumptions, both ϕ(k)/2 and ϕ(m)/2 are odd. The product P (k,m) has therefore
an odd number of factors and so at least one of them must be a quadratic residue inGF (pe).
That is, there exist ωk ∈ Uk and ωm ∈ Um such that the valueD = −(ωk+ωm) is a square
in GF (pe). This gives, by Theorem 2.2 and the remark after Theorem 2.3, a nonorientable
regular map of type {m, k}. Since there are infinitely many primes p as above, our result
follows.
Theorem 4.6. Let k andm be odd integers forming a hyperbolic pair such that the number
ϕ(k)ϕ(m)/4 is even, that is, at least one of k, m is not a prime power congruent to 3 mod
4. If P (k,m) < 0, then there are infinitely many finite, nonorientable, regular maps of type
{m, k} with automorphism group isomorphic to PGL(2, F ) for suitable finite fields F .
Proof. Let p ≡ 3 mod 4 be a prime such that e is odd (e.g., any p ≡ ±1 mod km). If
k 6= m, then P (k,m) = −1 not only in C but also in GF (pe). Similarly if k = m, then
we have P (k,m) = (−2)ϕ(k)/2 both in C and inGF (pe). Note that if k = m, then ϕ(k)/2
is even and 2ϕ(k)/2 is a quadratic residue in GF (p). As p ≡ 3 mod 4 and e is odd, the
product P (k,m) is a quadratic nonresidue in both GF (p) and GF (pe). On the other hand,
P (k,m) has an even number of factors, and therefore at least one of them is a quadratic
residue in GF (pe).
5 Remarks
By Theorem 4.4, for any given hyperbolic pair (k,m) there exists an infinite number of
nonorientable regular maps of type {m, k} with automorphism group isomorphic to linear
fractional groups over finite fields. Our approach was based on developing some results
obtained in [3] in the course of analysing regular maps over linear fractional groups. The
scope of [3] is, however, broader and covers regular hypermaps. For a general theory of
hypermaps and their surface representations we recommend [5]. Here we just recall that
a finite regular hypermap of type (k,m, l) can be identified with a finite quotient group of
the triangle group T (k,m, l) = 〈r, s, t; rk = sm = (rs)l = 1〉. Thus, regular hyper-
maps are a natural generalisation of regular maps (corresponding to the case when l = 2).
Facts collected in [8, 3] imply that for any hyperbolic triple (k,m, l), that is, such that
1/k + 1/m + 1/l < 1, there exist infinitely many regular hypermaps of type (k,m, l) on
orientable surfaces, with automorphism group isomorphic to a linear fractional group over
a finite field. By the theory developed in [3] combined with the findings in this paper, to
establish a nonorientable analogue of this result requires analysing conditions under which
the quantityA′ = 4+(α+α−1)(β+β−1)(γ+γ−1)−(α+α−1)2−(β+β−1)2−(γ+γ−1)2,
where α, β and γ are primitive 2k-th, 2m-th, and 2l-th roots of unity in C, projects onto a
non-zero square in a quotient field of the ring of algebraic integers of Q[α, β, γ] generated
by a suitable prime ideal; note that for l = 2 we have γ2 = 1 and γ+ γ−1 = 0 and then A′
reduces to the quantity A introduced earlier. In fact, methods of Section 3 can be adapted
in an obvious way to construct, for any hyperbolic triple (k,m, l) and for suitable triples
(α, β, γ) of primitive roots of unity as above, an infinite number of nonorientable regular
hypermaps of type (k, l,m) over linear fractional groups.
A comparison of Theorems 4.4, 4.5 and 4.6 reveals their different nature. Theorem 4.4
is more universal since it applies to all hyperbolic pairs and it is, in essence, constructive,
34 Ars Math. Contemp. 6 (2013) 25–35
but it yields no information on the corresponding set of primes. On the other hand, The-
orems 4.5 and 4.6 apply to a very restricted set of hyperbolic pairs and are, in essence,
existential, but the sets of primes for which they guarantee the existence of nonorientable
regular maps have positive density in the set of all primes. This leaves the intriguing ques-
tion of whether it is possible, for any given hyperbolic pair (k,m), to determine all primes
p such that there exists a nonorientable regular map of type {m, k} with its automorphism
group isomorphic to a linear fractional group over a field of characteristic p.
For possible further interest we present a table of values of the product P (k,m) for
odd k,m such that 3 ≤ k,m ≤ 41. Observe that for k 6= m the table shows negative
entries only if both k and m are powers of primes congruent to 3 mod 4. If this observation
carries through to all odd k and m, Theorem 4.6 would be applicable only in the case when
k = m, and the values 5, 13, 25, 29 and 37 show that this Theorem is not void.
3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
2 1 −1 −1 −1 1 1 1 −1 1 −1 1 −1 1 −1 1 1 1 1 1
1 −22 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
−1 1 −23 −1 −1 1 1 1 −1 1 −1 1 −1 1 −1 1 1 1 1 1
−1 1 −1 23 −1 1 1 1 −1 1 −1 1 −1 1 −1 1 1 1 1 1
−1 1 −1 −1 25 1 1 1 −1 1 −1 1 −1 1 −1 1 1 1 1 1
1 1 1 1 1 −26 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 24 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 28 1 1 1 1 1 1 1 1 1 1 1 1
−1 1 −1 −1 −1 1 1 1 29 1 −1 1 −1 1 −1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 26 1 1 1 1 1 1 1 1 1 1
−1 1 −1 −1 −1 1 1 1 −1 1 −211 1 −1 1 −1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 −210 1 1 1 1 1 1 1 1
−1 1 −1 −1 −1 1 1 1 −1 1 −1 1 29 1 −1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 −214 1 1 1 1 1 1
−1 1 −1 −1 −1 1 1 1 −1 1 −1 1 −1 1 −215 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 210 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 212 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −218 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 212 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 220
Acknowledgement
Research of the second author was supported by the APVV Research Grants 0111-07 and
0223-10, and the VEGA Research Grants 1/0588/09 and 1/0406/09. The third author ac-
knowledges support by the APVV Research Grants 0104-07 and 0223-10, and the VEGA
Research Grants 1/0280/10 and 1/0781/11. Both the second and the third authors also
acknowledge the APVV support as part of the EUROCORES Programme EUROGIGA
(project GREGAS, ESF-EC-0009-10) of the European Science Foundation.
G. A. Jones et al.: Nonorientable regular maps over linear fractional groups 35
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