Corollary 8.2.7. Every vertex uof Γ is contained in exactly two partst, t0 ∈ V(T). In addition,NΓ(u)⊆Vt∪Vt0 and every part is the disjoint union of its adhesion sets.
Moreover, let{x, y}be an adhesion set. Observe thatxy−1{x, y}is again an adhesion set containingx, so xy−1{x, y}={x, y}withxy−1x=y. We obtain:
Lemma 8.2.8. For every adhesion set{x, y}, we have (xy−1)2= 1.
Lemma8.2.8implies the following Corollary for the edge stabilizers ofT. Corollary 8.2.9. Let tt0∈E(T). ThenG(Vt∩Vt0)∼=Z2.
Lastly, we will invoke the following folklore Lemma from the well-known theory of tree decompositions into 3-connected components (see [78,97] as an example) when we argue about the planarity of Γ andGin each case that arises.
Lemma 8.2.10. Let (T,V) be a tight tree-decomposition of a (locally finite) connected graph H with finite parts and adhesion at most 2. Then Γ is planar if and only if the torso of every part of(T,V)is planar.
Proof. The forward implication follows from the fact that the torso of a part in (T,V) is a topological minor ofH: for every virtual edge of the part realized by an adhesion set of size exactly two, there is always a path outside of the part that connects the two vertices of the adhesion set.
For the backward implication, embedTon the plane. It is straightforward to combine the planar embeddings of every torso along the adhesion sets according toT following its embedding.
Our goal in the following sections is to determine the structure of the parts of the type 0 tree-decomposition of Γ in order to compute their stabilizers and apply Lemma 2.4.8or 2.4.9.
Lemma 8.3.1. Gacts with inversion on(T,V)if and only if b inverts Vt and Vt0.
The following general lemma for tree-decompositions is folklore.
Lemma 8.3.2. Let (T,V) be a tree-decomposition of a connected graphΓ and t∈V(T)such that every adhesion set oftinduces a connected subgraph. Then Γ[Vt]is connected. In particular, the torso of every part of (T,V)is connected.
Moreover, the following Lemma holds regardless of the number of generators in S.
Lemma 8.3.3. Every part ofV induces a finite cycle.
Proof. Let t∈ V(T). Since every adhesion set induces a connected subgraph, we conclude by Lemma8.3.2that Γ[Vt] is connected. Moreover, Corollary8.2.7 implies that Γ[Vt] is 2-regular. It follows that Γ[Vt] is either a finite cycle or a double ray. Recall that by Lemma 8.2.6 all adhesion sets are disjoint. The conclusion follows by observing that every vertex of Vt is a cut vertex whenVt
induces a double ray and hence, the graph Γ is not 2-connected.
It will be clear by Lemma8.2.10that we will obtain in all subcases planar Cayley graphs.
8.3.1 Two generators
Assume thatG=ha,bi, wherebis an involution. We distinguish the following cases depending on the colors of the edges incident to the adhesion sets, depicted as in the following Figure.
Vt
Vt0
Vt
Vt0
Case I Case II
Figure 8.2: Cases ofType Iwith two generators
Case I
Suppose that the edges incident to each adhesion set in N are as in Case I of Figure 8.2. Observe that{a−1, ba} ⊆Vt and{a, ba−1} ⊆Vt0 are the neighbors
of 1 and bin Vtand Vt0, respectively. Sinceb{a−1, ba}={a, ba−1}, it must be that bVt =Vt0 and bVt0 =Vt. Lemma 8.3.1implies that Gacts on E(T) with inversion (and hence transitively on V(T)).
By Lemma8.3.3, there is ann∈Nsuch that that (ba)n= 1 and Vt={1, b, ba, . . . ,(ba)n−1b=a−1}.
This gives a partition hbai t hbaib of Vt. We next conclude that G(Vt)⊆Vt by noting that 1∈Vt. Clearly, we have hbai ⊆G(Vt). Moreover, for the element ba∈Vt, we observe that
(ba)ib(ba) = (ba)ia6∈Vt.
Since Vt =hbai t hbaib , we conclude thatGVt =hbai ∼=Zn. Moreover, GVt ∩ GVt∩Vt0) =hbai ∩ hbi= 1.
We apply Lemma2.4.9and obtain that G∼=Zn∗Z2. Case II
By the structure of the neighbourhood of{1, b}and Lemma8.3.1we see thatb cannot invertVtandVt0, henceGacts on (T,V) without inversion.
Now, consider the adhesion seta−1{1, b}= (a−1Vt)∩(a−1Vt0). Froma{1, b} ⊆ Vtwe deduce that{Vt, Vt0}={a−1Vt, a−1Vt0}. Since the adhesion set{1, b}has ingoing a-edges but a{1, b} has outgoing a-edges in Vt, we cannot have that a−1Vt = Vt. Consequently, it must be that a−1Vt0 = Vt. The fact that two adjacent parts lie in the same orbit under the action ofG implies thatGacts transitively on V (andV(T)).
By Lemma8.3.3, there is in this case ann∈Nsuch that (ba−1ba)n= 1 and Vt={1, b, ba−1, ba−1b, . . . ,(ba−1ba)n−1ba−1b=a−1}.
In other words, hba−1bai t hba−1baibt hba−1baiba−1t hba−1baiba−1b forms a partition ofVt. Notice thathba−1baiis the trivial group whenba−1ba= 1. As before, since 1∈Vt we infer thatGVt ⊆Vt. Clearly, we havehba−1bai ⊆GVt. Moreover, we see thathba−1baiba−16⊆GVtbecause we have (ba−1ba)iba−1(ba−1ba)6∈
Vtand that hba−1baiba−1a6⊆GVt because (ba−1ba)iba−1b(a−1ba)6∈Vt.
Lastly, observe that sinceb is an involution and all adhesion sets induce a b-edge, we have that the action of b on Γ fixes every adhesion set. Hence, we have that b ∈GVt. It follows thathba−1ba, bi ⊆ GVt. Therefore, we conclude
that
GVt =hba−1ba, b|b2,(ba−1ba)n,(a−1ba)2i ∼=D2n. By Lemma2.4.8, we have that
G∼= D2n ∗
Z2
(t).
We collect both cases in the following theorem.
Theorem 8.3.4. If (T,V) is of Type I with two generators, then G satisfies one of the following cases:
(i) G∼=Zn∗Z2. (ii) G∼=D2n ∗
Z2
(t).
The definitions of a free product with amalgamation, an HNN-extention and the proof of Theorem8.3.4immediately imply:
Corollary 8.3.5. [37, Theorem 1.1]If(T,V)is ofType Iwith two generators, thenG has one of the following presentations:
(i) ha, b|b2,(ba)ni. (ii) ha, b|b2,(ba−1ba)ni.
8.3.2 Three generators
Let G = ha,b,ci, where a, b and c are involutions. Suppose that the edges induced by the adhesion sets inN are colored withb. Up to rearranginga, b, c, there are two cases for the local structure of the separators in N, as in the following figure:
Vt
Vt0
Vt
Vt0
Case I Case II
Figure 8.3: Cases ofType Iwith three generators
Case I
First, we observe by Lemma8.3.1thatGacts onT without inversion, since by the structure of the neighbourhood of {1, b} we see thatb must stabilize both Vt and Vt0. Consequently, G must act with two orbitsO1, O2 on Γ[V], where the parts inO1 contain thea-edges and the parts inO2contain thec-edges. By Lemma 8.3.3we deduce that (ba)n= 1 and (bc)m= 1 and soVt=hbai t hbaib andVt0 =hbci t hbcib
To compute the stabilizers of the parts, observe that we can escape a part inO1only withc-edges. Hence, we have GVt =Vt=hba, b|b2= (ba)n=a2i ∼= D2nand similarlyG(Vt0)=Vt0=hbc, b|b2= (bc)m=c2i ∼=D2m. Therefore, by Lemma 2.4.8we obtain
G∼=D2n ∗
Z2
D2m. Case II
In this case, we see thatb invertsVtandVt0, soGacts onT with inversion by Lemma 8.3.1. Hence, Galso acts transitively onV(T).
Letx:=bcba. By Lemma8.3.3we see that (bcba)n= 1 and thathxi t hxibt hxibct hxibcbis a partition ofVt. Clearly, we have thathbcbai ⊆GVt. We show that we actually have equality:
• xib·bc=xic6∈Vt, hencehxib6∈GVt,
• xibc·a6∈Vt, hencehxibc6∈GVt,
• xibcb·c6∈Vt, hencehxibcb6∈GVt.
We conclude that Gt = hbcbai ∼= Zn and consequently we also have that GVt∩G(Vt∩Vt0)=hbcbai ∩ hbi= 1. It follows from Lemma2.4.9that
G∼=Zn∗Z2. In conclusion, we have proved:
Theorem 8.3.6. If(T,V)is of Type I with three generators, thenGsatisfies one of the following cases:
(i) G∼=D2n ∗
Z2
D2m. (ii) G∼=Zn∗Z2.
Corollary 8.3.7. [37, Theorem 1.1]If(T,V)is ofType Iwith three generators, thenG has one of the following presentations:
(i) ha, b, c|a2, b2, c2,(ba)n,(bc)mi. (ii) ha, b, c|a2, b2, c2,(bcba)ni.