Also available at http://amc.imfm.si
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 4 (2011) 363–374
Vertex transitive maps on the Klein bottle
Ondrej Šuch ∗
Institute of Mathematics and Computer Science
Ďumbierska 1, 974 01 Banská Bystrica, Slovak Republic
and
Fakulta prı́rodných vied, Univerzita Mateja Bela
Tajovského 40, 974 01 Banská Bystrica, Slovak Republic
Received 10 December 2009, accepted 4 August 2011, published online 10 October 2011
Abstract
We examine properties of groups associated with vertex transitive maps on the Klein
bottle. As an application, we prove that all 4-line vertex transitive maps on the Klein bottle
introduced by Babai admit a free vertex transitive action, but there are others that do not.
Keywords: Klein bottle, wallpaper group, vertex-transitive map.
Math. Subj. Class.: 57M60, 05C10; 20H15
1 Introduction
This paper is motivated by a question raised in [8], namely whether all vertex transitive
maps on the Klein bottle admit a free vertex-transitive action as is the case on torus. Vertex
transitive maps on the Klein bottle and groups acting on them have been considered in the
works of Babai [1] and Thomassen [9]. In this work we take a different approach, working
exclusively on the universal covering space of the Klein bottle, the plane.
Suppose a map M on the Klein bottle KB is given with a vertex-transitive action, by
a group G. The map lifts to the universal covering space [1, p. 610], where one obtains a
vertex transitive action [3, 4, 5] of a wallpaper group [12] Γ on a semiregular tiling [6, 11]
T , together with a normal subgroup K (isomorphic to pg) of transformations preserving
the canonical map R2 → KB. One can recover G as the quotient Γ/K. The action of G
is free if and only if the action of Γ is free.
In this paper we aim to provide an insight into the groups involved (Proposition 2.1,
Table 1 and 2) and resolve the question of free action (Proposition 5.1).
∗Research on this work was supported by VEGA grant 1/0722/08.
E-mail address: ondrejs@savbb.sk (Ondrej Šuch)
Copyright c© 2011 DMFA Slovenije
364 Ars Math. Contemp. 4 (2011) 363–374
plane R2

semiregular tiling T with a vertex transitive
action of a wallpaper group Γoo

Klein bottle R2/K
a map T /K with a vertex transitive action of
a finite group Goo
2 KBU groups
The group pg is central in our work. It can be introduced in several equivalent ways.
• geometrically by giving two parallel glides P , Q of equal length,
• geometrically by giving a glide g and an orthogonal translation t,
• as an abstract group 〈P,Q; P 2 = Q2〉,
• or as an abstract group 〈g, t; gtg−1 = t−1〉.
By a wallpaper or plane crystallographic group one means a group having a discrete
action on the plane, and containing two linearly independent translations. There are 17
isomorphism types of wallpaper groups [2]. Not every wallpaper group gives rise to an
action on the Klein bottle, because some have no glides, and hence no subgroup isomorphic
to pg. We shall call a KBU group a wallpaper group that contains a normal subgroup
isomorphic to pg.
Proposition 2.1. A wallpaper group Γ is a KBU group if and only if Γ is isomorphic to one
of the following groups: pm, pg, cm, pmm, pmg, pgg.
Proof. Suppose Γ is a KBU group. Clearly, Γ cannot be p1 or p2, because these groups
do not contain any glides. Suppose Γ contains a rotation r of degree > 2. If g is a glide
from the normal subgroup K isomorphic to pg, then g fixes a line L. Using the elementary
Lemma 2.2 we see that rgr−1 fixes r · L. It follows that K contains non-parallel glides,
and hence a rotation. This contradicts the fact that pg is torsion-free.
It remains to show that the group listed have normal subgroups isomorphic to pg. Such
subgroups for all of them are listed in Table 2.
Lemma 2.2. Suppose group G acts on a set X . Suppose a subgroup H leaves a subset Y
fixed. If g is any element of G then the subgroup gHg−1 leaves a subset gY fixed.
Let us recall presentations of KBU groups from [2]:
pm := 〈X,Y,R; XY = Y X,RY = Y R, (RX)2 = R2 = 1〉
pg := 〈P,Q; P 2 = Q2〉
cm := 〈R,S; (RS)2 = (SR)2, R2 = 1〉
pmm := 〈R1, R2, R3, R4; R21 = R22 = R23 = R24 = 1,
(R1R2)
2 = (R2R3)
2 = (R3R4)
2 = (R4R1)
2 = 1〉
pmg := 〈R, T1, T2; R2 = T 21 = T 22 = 1, T1RT1 = T2RT2〉
pgg := 〈P,O; (PO)2 = (P−1O)2 = 1〉
cmm := 〈T,R1, R2; T 2 = R21 = R22 = 1, (R1R2)2 = (R1TR2T )2 = 1〉
O. Šuch: Vertex transitive maps on the Klein bottle 365
Even though the groups are given in terms of presentations, their elements can be clas-
sified in geometric terms. Let g be an element of Γ. If the centralizer Z(g) of g is a free
abelian group of rank 2, then g is a translation. The translations form a free abelian sub-
group T (Γ) of rank two, of finite index (≤ 4) in Γ. Suppose next that the intersection
Z(g)∩T (g) is a free abelian group of rank 1. Then if g is an involution, then g is a mirror,
otherwise g is a glide (and g is of infinite order). In the remaining cases g is a rotation.
2.1 Translations and other elements in KBU groups
Geometric representations of KBU groups can be seen in Figure 1 with indicated gener-
ators. Each picture shows the basis of the translation subgroup (the bounding parallelo-
gram), glides (dotted lines), mirrors (thick lines) and rotations by angle π (diamonds). We
will now summarize the structure of the translation subgroups of the above seven classes
of wallpaper groups.
2.1.1 Elements of pm
The group is generated by three elements: a pair of orthogonal translations X and Y , and
a mirror in the direction of Y . The translation subgroup is generated by X and Y and is of
index 2 in pm. There is a coset decomposition pm = T (pm) ∪ R · T (pm). Elements of
form RXkY l are mirrors and glides according to whether l = 0 or not. From definining
relations it follows that
RX = X−1R RY = Y R
2.1.2 Elements of pg
The group is generated by a pair of parallel glides P and Q satisfying P 2 = Q2. The
translation subgroup is generated by P 2 and P−1Q and is of index 2 in pg. There is a coset
decomposition pg = T (pg)∪P · T (pg). Elements of form P ·P 2l(P−1Q)k are all glides.
We have
P · P 2 = P 2 · P P (P−1Q) = (P−1Q)−1P
2.1.3 Elements of cm
The group is generated by a translation S and a mirror R, which are neither parallel, nor
orthogonal. The translation subgroup is generated by S and RSR and is of index 2 in cm.
There is a coset decomposition cm = T (cm)∪R ·T (cm). Elements of form RSk(RSR)l
are mirrors and glides according to whether k = −l or not. From defining relations it
follows
R · S = RSR ·R R ·RSR = S ·R
2.1.4 Elements of pmm
This group is generated by four mirrors R1–R4 forming the sides of a square. The transla-
tion subgroup is generated by R1R3 and R2R4, and is of index 4 in pmm. There is a coset
decomposition pmm = T (pmm)∪R1 ·T (pmm)∪R2 ·T (pmm)∪R1R2 ·T (pmm). All
366 Ars Math. Contemp. 4 (2011) 363–374
VERTEX TRANSITIVE MAPS ON THE KLEIN BOTTLE 3
(a) Group pm (b) Group pg
(c) Group cm (d) Group pmm
(e) Group pmg (f) Group pgg
(g) Group cmm
Figure 1. KBU groups
.Figure 1: KBU groups.
.
O. Šuch: Vertex transitive maps on the Klein bottle 367
elements of the last coset are rotations. Elements of the form R1(R1R3)k and R2(R2R4)k
are mirrors, and the remaining non-translations are glides. From the defining relations we
have:
R1(R2R4) = (R2R4)R1 R1(R1R3) = (R1R3)
−1R1
R2(R2R4) = (R2R4)
−1R2 R2(R1R3) = (R1R3)R2
(R1R2)(R2R4) = (R2R4)
−1(R1R2) (R1R2)(R1R3) = (R1R3)
−1(R1R2)
2.1.5 Elements of pmg
This group is generated by a mirror R and two translations T1, T2 lying on a line parallel
to it. The translation subgroup is generated by (T1R)2 and T1T2, and is of index 4 in pmg.
There is a coset decomposition pmm = T (pmg) ∪ R · T (pmg) ∪ T1 · T (pmg) ∪ RT1 ·
T (pmg). We have
R(T1T2) = (T1T2)R (T1R)
2 ·R = R · (T1R)−2
T1(T1T2) = (T1T2)
−1T1 T1(T1R)
2 = (T1R)
−2T1
(RT1)(T1T2) = (T1T2)
−1(RT1) (RT1)(T1R)
2 = (T1R)
2(RT1)
2.1.6 Elements of pgg
This group is generated by a pair of orthogonal glides O and P . The translation subgroup
is generated by P 2 and O2. There is a coset decomposition pgg = T (pgg)∪O · T (pgg)∪
P · T (pgg) ∪ (OP ) · T (pgg). We have
P · P 2 = P 2 · P P ·O2 = O−2 · P
O · P 2 = P−2 ·O O ·O2 = O2 ·O
OP · P 2 = P−2 ·OP OP ·O2 = O−2 ·OP
2.1.7 Elements of cmm
This group is generated by a pair of orthogonal mirrors R1, R2 and a rotation not lying on
any of them. The translation subgroup is generated by R1R2T and R2TR1. We have a
coset decomposition cmm = T (cmm) ∪ R1 · T (cmm) ∪ R2 · T (cmm) ∪ T · T (cmm).
We have
R1(R1R2T ) = (R2TR1)R1 R1(R2TR1) = (R1R2T )R1 (2.1)
R2(R1R2T ) = (R2TR1)
−1R2 R2(R2TR1) = (R1R2T )
−1R2 (2.2)
T (R1R2T )T = (R1R2T )
−1T T (R2TR1) = (R2TR1)
−1T (2.3)
3 Ten families of glides of KBU groups
In order to analyze normal subgroups of KBU groups isomorphic with pg it is first neces-
sary to understand the set of glides in these wallpaper groups.
In Table 1 we list ten families of glides parametrized by positive integers n > 0. Us-
ing the facts from the previous section it is straightforward to verify that the elements g
368 Ars Math. Contemp. 4 (2011) 363–374
listed there are glides, that is they are not involutions, and centralize a rank 1 subgroup of
translations. The subgroup of translations is in fact generated by the corresponding parallel
translation.
Family Group G glide g orthogonal translation parallel translation
M1 pm RY n X Y
M2 pmm R1(R2R4)n R1R3 R2R4
M3 pmg R(T1T2)n (T1R)2 T1T2
M4 cm R(RS)2n RSRS−1 RSRS
M5 cmm R1(TR2)2n (TR1)2 (TR2)2
G1 pg P 2n−1 (PQ−1)2 P 2
G2 pmg (T1R)2n−1 T1T2 (T1R)2
G3 pgg O2n−1 P 2 O2
G4 cm (RS)2n−1 RSRS−1 RSRS
G5 cmm (R1T )2n−1 (R2T )2 (R1T )2
Table 1: Glides in KBU groups.
Take for instance the glide g = (T1R)2n−1 in family G2. The translation subgroup of
pmg is generated by (T1R)2 and T1T2. It is clear that g commutes with (T1R)2, and using
equations from 2.1.5 we check that it anticommutes with T1T2:
g(T1T2)g
−1 = (T1R)
2n−1(T1T2)(RT1)
2n−1
= (T1R)
2n−1(RT1)
2n−1(T1T2)
(−1)2n−1
= (T1T2)
−1
Since g2 =
(
(T1R)
2
)2n−1
we see that g is not an involution, and thus is not a mirror.
The list provided in Table 1 is in a way exhaustive.
Proposition 3.1. If γ is a glide in KBU group Γ, then there exists n ≥ 1 and an isomor-
phism Γ→ G, such that γ maps to one of the glides g listed in Table 1.
Proof. Suppose γ is a glide. Clearly, γ cannot belong to T (Γ). If γ belonged to α · T (Γ)
for some rotation α, then γ2 = 1, which contradicts the fact that all glides are of infinite
order. Let us now distinguish several cases.
Suppose next Γ admits an orthogonal basis of T (Γ). This is true for groups pm, pg,
pmm, pmg, pgg. As shown in 2.1 we need to distinguish two cases.
Firstly suppose γ = RXkY l, where X and Y is an orthogonal basis of T (Γ) such that
RX = X−1R RY = Y R
Then γ2 = Y 2l and thus γ is a glide if and only if l 6= 0. The next step is to show that
there exists an automorphism of Γ such thatRY l is mapped toRY −l. In the case of groups
pmm, pmg, pgg we can take it to be conjugation by any rotation. For groups pm and pg,
any rotation of plane of order 2 accomplishes the same, but in these cases it is an outer
automorphism. Conjugating RY l by Xs we obtain
XsRXkY lX−s = RXk−2sY l X−sRXkY lXs = RXk+2sY l
O. Šuch: Vertex transitive maps on the Klein bottle 369
Thus modulo inner automorphisms there are at most two classes of glides in the form
RXkY l. These are exchanged by conjugation with 12X , which is an outer automorphism.
In the case of group pmm we remark that the map
φ(R1) := R2, φ(R2) := R3, φ(R3) := R4, φ(R4) := R1
is an outer automorphism exchanging glides R1(R2R4)k with R2(R3R1)k.
Secondly, we need to consider the case γ = P · P 2kY l, where P is a glide and Y is an
orthogonal translation satisfying
PY = Y −1P
Then γ2 = P 4k+2, which shows that γ is a glide for all choices of k and l. Arguing as in
the previous case we obtain that modulo automorphisms of the group two glides P ·P 2kY l
and P · P 2k′Y l′ are equivalent if and only if k = ±k′.
Suppose now that Γ is isomorphic to cm or cmm. We can assume that the γ belongs
to a coset RT (Γ), where R is a mirror and T (Γ) is generated by (equal length) translations
S, RSR. Then we can explicitly write
γ = RSk(RSR)l
γ2 = Sk+l(RSR)k+l
and thus γ is a glide if and only if k + l 6= 0. In case of cmm any rotation in Γ conjugates
RSk(RSR)l with RS−k(RSR)−l. In case of cm any plane rotation does the same, but it
is now an outer automorphism. Conjugating by Ss gives
Ss(RSk(RSR)l)S−s = RSk−s(RSR)l+s.
This shows that if k+l = 2n, then the glideRSk(RSR)l is conjugate withRSn(RSR)n =
R(SR)2n = R(RS)2n. If on the other hand k + l = 2n + 1 the the glide RSk(RSR)l is
conjugate with RSSn(RSR)n = RS(RSR)nSn = (RS)2n+1.
4 Groups acting on the Klein bottle
In the previous section we listed ten families of glides in KBU groups. We now consider
the problem of determining the normal closure of a glide g in a KBU group. Conjugating
by an orthogonal translation in view of Lemma 2.2 gives a different, parallel glide g′. Thus
the normal closure always contains a subgroup isomorphic to pg.
On the other hand, any KBU group is a subgroup of finite index in pmm ([2]). This
can also be seen geometrically, or by giving explicit injections:
i1 : pm→ pmm i1(X) := R1R3, i1(Y ) := R2R4, i1(R) := R1
i2 : pg → pmm i2(P ) := R1R2R4, i2(Q) := R1R3R1R2R4
i3 : cm→ pmm i3(R) := R1, i3(S) := (R1R3)(R2R4)
i4 : pmg → pmm i4(P ) := R1R2R4, i4(Q) := R1(R1R3)2R2R4, i4(R) = R3
i5 : pgg → pmm i5(P ) := R1R2R4, i5(O) := R2R1R3
i6 : cmm→ pmm i6(R1) := R1, i6(R2) := R2, i6(T ) := R3R4
370 Ars Math. Contemp. 4 (2011) 363–374
The normal closure of a glide in a KBU group will thus be a subgroup of the normal
closure of a glide in pmm. Without loss of generality, consider the normal closure of the
glide g = R1(R2R4)n in pmm. We have
R1gR1 = g
R2gR2 = R2R1(R2R4)
nR2 = R1(R4R2)
n = g−1
R3gR3 = R3R1(R2R4)
nR3 = R3R1R3(R2R4)
n = (R3R1)
2g
R4gR4 = R4R1(R2R4)
nR4 = R1(R4R2)
n = g−1
One readily checks that R2 and R4 fix the translation (R3R1)2, while R1 and R3 invert it.
It follows that the normal closure of g in pmm is the subgroup isomorphic to pg generated
by g and the translation (R3R1)2 orthogonal to g. We have arrived at the following lemma.
Lemma 4.1. The normal closure of a glide in a KBU group is a subgroup isomorphic to
pg.
Proof. Indeed, the normal closure contains two parallel glides, and since it is a subgroup
of pg, which contains no mirrors, nor rotations, it has to be isomorphic to pg.
By geometric analysis using Lemma 2.2 one can determine the normal closure of glides
in KBU groups precisely.
Proposition 4.2. The normal closure of a glide g in a KBU group G for families M1–M5,
G1–G5 is the group 〈g, t〉 as given in Table 2. One has gtg−1 = t−1, with the indicated
quotient and a set of generators of G/〈g, t〉.
Family Group G glide g translation t F := G/〈g, t〉 Generators of F
M1 pm RY n X2 Z2 × Z2n X̄; Ȳ
M2 pmm R1(R2R4)n (R1R3)2 Z2 ×D4n R̄1R̄3; R̄2, R̄4
M3 pmg R(T1T2)n (T1R)2 D4n T̄1T̄2, T̄1
M4 cm R(RS)2n RSRS−1 Z4n R̄S̄
M5 cmm R1(TR2)2n (TR1)2 D8n T̄ R̄2, R̄2
G1 pg P 2n−1 (PQ−1)2 Z2(2n−1) Q̄
G2 pmg (T1R)2n−1 (T1T2)2 Z2 ×D2(2n−1) T̄1T̄2; T̄1R, R̄
G3 pgg O2n−1 P 2 D2(2n−1) Ō, P̄
G4 cm (RS)2n−1 RSRS−1 Z2(2n−1) S̄
G5 cmm (R1T )2n−1 (R2T )2 Z2 ×D2(2n−1) R̄2; R̄1T̄ , R̄1
Table 2: Group factors of KBU groups.
Proof. Determination of normal closure in each case proceeds analogously to the case of
M2 analyzed above.
Let us illustrate how one proves the rest in the case G5. First, since
R2TR1 ·R1R2T = (R2T )2
O. Šuch: Vertex transitive maps on the Klein bottle 371
we have from (2.1) and (2.3)
gtg−1 = (R1T )
2n−1(R2T )
2(TR1)
2n−1 = (R2T )
−2(R1T )
2n−1(TR1)
2n−1
= (R2T )
−2 = t−1
Since R1 · R1T = T , the images R̄1T̄ , R̄1, R̄2 of elements R1T,R1, R2 generate F . Ele-
ments R, T1, T2 are all involutions. From (R1R2)2 = 1 it follows that R2 commutes with
R1, and from (R̄2T̄ )2 = 1 it follows that R̄2 commutes with T̄ . We can now compute the
quotient (omitting redundant relations):
cmm/〈g, t〉 = 〈T̄ , R̄1, R̄2; T̄ 2 = R̄1 = R̄22 = 1, (R̄1T̄ )2n−1 = [R̄2, T̄ ] = [R̄2, R̄1] = 1〉
= 〈R̄2; R̄22 = 1〉 × 〈R̄1, T̄ ; R̄21 = T̄ 2 = (R̄1T̄ )2n−1 = 1〉
= Z2 ×D2(2n−1)
Corollary 4.3. (Babai) A group G acts on the Klein bottle if and only if it is a subgroup of
Z2 ×D2n for some n ≥ 1.
Proof. It is known ([1] Lemma 7.3 and [7] Theorem 6.2.4, also [10]) that if a group H acts
on a surface, then there exists a Cayley map for this group on the surface. Thus any group
action can be lifted to a Cayley action of a KBU group Γ on a semiregular tiling, such that
it has a normal subgroup K isomorphic to pg that preserves the canonical projection. One
has H = Γ/K. The group K contains a glide γ, and since it is normal, it contains the
normal closure of γ. From the previous proposition it follows that H is isomorphic to a
factor of F from Table 1, all of which are subgroups of Z2 ×D2n.
Conversely, groups Z2 ×D2n act on the Klein bottle for both n even (pmm with glide
family M2), and n odd (pmg with glide family G2 or cmm with glide family G5).
5 Vertex-transitive maps on the Klein bottle
In Babai’s work [1] one can find a list of vertex-transitive maps on the Klein bottle. Let us
explain his terminology. He represents semiregular tilings with letters as follows:
A the triangle tiling
B the square tiling
C the hexagonal tiling
D the elongated square tiling
E the snub square tiling
F the truncated square tiling
Then he shows 12 families of positions for glides acting on the tilings A1, A2, B1, B2, C1–
C3, D1, D2, E1, F1, F2. Each of them has the property that there is a glide g in the position,
and a translation t orthogonal to g such that the quotient of the plane by the group 〈g, t〉
is a vertex-transitive map on the Klein bottle. Moreover, the vertices of the semiregular
tiling fall into 4 lines modulo translations by t. He calls the last property a 4-line condition.
He notes that every position, except B1, has a natural parity condition on the glides in the
372 Ars Math. Contemp. 4 (2011) 363–374
10 ONDREJ ŠUCH
(a) pgg action on the triangu-
lar tiling
(b) cmm action on the square
tiling
(c) pmg action on the hexag-
onal tiling
(d) pgg action on the triangu-
lar tiling
(e) pgg action on the square
tiling
(f) pgg action on the hexago-
nal tiling
(g) pgg action on the elon-
gated triangular tiling
(h) pmg action on the elon-
gated triangular tiling
(i) pgg action on the snub
square tiling
(j) cmm action on the trun-
cated square tiling
(k) cm action on the triangle
tiling that is not free
Figure 2. Vertex-transitive actions of KBU groups on semiregular tilings
Figure 2: Vertex-transitive actions of KBU groups on semiregular tilings.
O. Šuch: Vertex transitive maps on the Klein bottle 373
position. This results in 13 families of 4-line vertex-transitive maps, and he lists 5 2-line
families (A10, A20, B10, B20, D10) and one 1-line family arising from the square tiling.
We can now state our final result.
Proposition 5.1. All 4-line vertex transitive maps on the Klein bottle admit a free vertex
transitive action. There are however vertex-transitive maps on the Klein bottle that do not
admit a free vertex transitive action.
Proof. The first statement is demonstrated by exhibiting free vertex transitive actions for
each of Babai’s families. This is done in Figures 2(a)-2(j). Note that for position B1, we
show a free vertex transitive action for both even and odd glides [1]. Also note that for
positions C1, D3, the resulting maps are indeed 4-line. This is because in the family G2,
the normal closure is generated by the square of the smallest translation orthogonal to the
glide.
We will prove the second statement by contradiction. Let us consider the vertex transi-
tive action of the group Γ = cm on the triangular tiling as shown in Figure 2(k). For any in-
teger n ≥ 1 one obtains a vertex transitive map on the Klein bottle by taking the quotient by
the normal closure in Γ of a glide in the direction g of length n×(length of triangle side). In
fact, the images of translations act transitively. This shows there are vertex transitive maps
arising from triangle tilings with number of vertices divisible by arbitrarily large power of
2.
Since any mirror of a triangular tiling fixes a vertex of the tiling, there are free vertex
transitive actions only by groups pg and pgg on the tiling. Any maps arising from those
actions are factors by odd multiple sized glides. Since there are only finitely many classes
of actions of pg and pgg modulo automorphisms of the triangle tiling, we conclude that the
power of 2 dividing the size of free vertex transitive maps has an upper bound.
This contradiction shows that there are maps that do not admit a free vertex transitive
action.
Finally we remark there is a 2-line family of vertex transitive maps for position C10
shown in Figure 2(f), omitted in Babai’s list.
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