Splitting groups with cubic Cayley graphs of connectivity two

A group $G$ splits over a subgroup $C$ if $G$ is either a free product with amalgamation $A \underset{C}{\ast} B$ or an HNN-extension $G=A \underset{C}{\ast} (t)$. We invoke Bass-Serre theory and classify all infinite groups which admit cubic Cayley graphs of connectivity two in terms of splittings over a subgroup.


Introduction
The study of the structure of groups in terms of the connectivity of their Cayley graphs was started by Jung and Watkins.They characterized infinite transitive graphs of connectivity one whose automorphism groups act on their vertex sets as primitive and regular permutation groups [12].Later, Watkins [17] characterized all Cayley graphs of connectivity one.
A topic that has been already paired with the connectivity of Cayley graphs in order to study them is the planarity of infinite Cayley graphs.A finitely generated group G is called planar if it admits a generating set S such that the Cayley graph Cay(G, S) is planar.In that case, S is called a planar generating set.For the first time, in 1896, Maschke [13] characterized all finite groups admitting planar Cayley graphs.Infinite planar groups attracted more attention, as some of them are related to surface and Fuchsian groups [18,Section 4.10] which play a substantial role in complex analysis, see survey [18].Hamann [10] uses a combinatorial method in order to show that planar groups are finitely presented.His method is based on tree-decompositions, a crucial tool of graph minor theory which we also utilize extensively in this paper.
In [7], Droms, B. Servatius, and H. Servatius characterized planar groups with low connectivity in terms of the fundamental group of the graph of groups.Indeed, they showed: Theorem.[7,Theorem 4.4] If a group G has planar connectivity 1 2, then either G is a finite cyclic or dihedral group, or it is the fundamental group of a graph of groups whose edge groups all have order two or less and whose vertex groups all have planar connectivity at least three.In the latter case, the vertex groups have planar generating sets which include the nontrivial elements of the incident edge groups.
Later, Georgakopoulos in [9] determines all presentations of groups which admit planar cubic Cayley graphs with connectivity two.Furthermore, he provides partial information about the presentations of non-planar ones.Our result provides full information not only for the planar, but also the non-planar groups with cubic Cayley graphs of connectivity two.
More specifically, Georgakopoulos' method does not assert anything regarding (and is, in a sense, independent of) splitting the group over subgroups to obtain its structure.By combining tree-decompositions and Bass-Serre theory, we give a simple proof for the full characterization of all groups with cubic Cayley graphs of connectivity two via the following theorem: Theorem 1.1.Let G = S be a group such that Γ = Cay(G, S) is a cubic graph of connectivity two.Then G is isomorphic to one of the following groups: Theorem 1.1 is a direct consequence of Theorems 4.3, 4.5, 5.4 and 5.8, where we also discuss in detail the planarity of the corresponding Cayley graphs in each case, as well as their presentations.This allows us to obtain as a corollary the results of [9], as well as full presentations for the non-planar groups with cubic Cayley graphs of connectivity two.
Compared to the methods in [9], we believe that our graph theoretical arguments are simplified, while we inevitably spend more time to recover the full algebraic structure of the group in terms of splitting this time around.Moreover, even though the planar part of our result can be relatively quickly recovered from [9] by applying Tietze transformations accordingly, such an approach usually works only provided that one knows or guesses beforehand the new desired presentation (in our case, the one that expresses the splitting of the group) in order to apply the correct Tietze transformations.By applying Bass-Serre theory, we naturally determine the structure of the group in terms of splitting avoiding the nuisance above, which was also the way we originally obtained it.

Graphs
Throughout this paper, Γ always denotes a connected locally finite graph with vertex set V (Γ) and edge set E(Γ).A ray is a one-way infinite path and a tail of a ray is an infinite subpath of the ray.Two rays R 1 and R 2 are equivalent if there is no finite set S of vertices such that R 1 and R 2 have tails in different components of G \ S. The equivalence classes of rays are called ends.We refer the reader to surveys [4,5] for a detailed study of the end structure of graphs.
For a subset U ⊆ V (Γ), we denote by Γ[U ] the subgraph induced by the vertices of U .A separation of Γ is an ordered pair (A, B), where Otherwise, we say that the two separations cross.We say that a vertex set X separates vertex sets U and W if there exists a separation (A, B) such that U ⊆ A, W ⊆ B and X = A ∩ B.
Let S be a set of vertices of Γ.The set of neighbors of S is denoted by Two ends ω 1 and ω 2 are k-distinguishable if there is a separation of order at most k distinguishing ω 1 and ω 2 efficiently.
A separation is splitting if it distinguishes at least two ends efficiently.We note that if (A, B) is splitting, then (A, B) is a tight separation.Let (A, B) be a splitting k-separation.The crossing number cn(A, B) of (A, B) is the cardinality of the set containing all crossing tight ℓ-separations distinguishing at least two ends, where ℓ ≤k (which can be seen to be finite [3]).
Let Γ be an arbitrary connected graph.A tree-decomposition of Γ is a pair (T, V) of a tree T and a family V = (V t ) t∈V (T ) of vertex sets V t ⊆ V (Γ), which are called parts, such that: (T1) V (Γ) = t∈T V t , (T2) for every edge e ∈ E(Γ), there exists a t ∈ V (T ) such that both ends of e lie in V t , (T3) V t1 ∩ V t3 ⊆ V t2 whenever t 2 lies on the (t 1 , t 3 )-path in T .
In order to distinguish them from the vertices of the graph Γ, we will usually refer to the vertices of the underlying tree T of a tree decomposition of Γ as nodes.
An adhesion set of (T, V) is a set of the form V t ∩ V t ′ , where tt ′ ∈ E(T ).The adhesion of (T, V) is the maximum over the sizes of its adhesion sets.It is not hard to see that each adhesion set leads to a separation of Γ.More precisely, assume that T t and T ′ t are the components of T − tt ′ containing t and t ′ respectively.Then the adhesion set V t ∩ V t ′ induces the separation (W t\t ′ , W t ′ \t ) of Γ, where W t\t ′ = s∈Tt V s and W t ′ \t = s∈T t ′ V s .When every such separation is tight, we call the tree-decomposition tight as well.Finally, a tree decomposition is reduced if no part is contained in another one.
The following folklore fact about tree decompositions and nested set of separations is well-known, see [2].
Remark 2.1.Every nested set N of separations gives rise to a reduced treedecomposition whose adhesion sets are exactly the elements of N .On the other hand, each adhesion set of a tree-decomposition induces a separation and the set of all induced separations of adhesion sets of a tree-decomposition is a nested set of separations.
Let Γ be a locally finite graph with a tree-decomposition (T, V).We call the torso of a part V t the supergraph of Γ[V t ] obtained by adding to it all possible edges in the adhesion sets incident to V t .The following general lemma for tree-decompositions is folklore.Lemma 2.2.Let (T, V) be a tree-decomposition of a connected graph Γ and t ∈ V (T ) such that every adhesion set of t induces a connected subgraph.Then Γ[V t ] is connected.In particular, the torso of every part of (T, V) is connected.
In this paper, we are studying groups admitting cubic Cayley graphs of connectivity two.

Groups
Let G be a group acting on a set X. Then the setwise stabilizer of a subset Y of X is the set of all elements g ∈ G stabilizing Y setwise, i.e Let G be a group acting on a graph Γ.Then this action induces an action on E(Γ).We say that G acts without inversion on Γ if g(uv) = vu for all uv ∈ E(Γ) and g ∈ G.In the case that g(uv) = vu, we say that g inverts u, v. Notice that when G acts transitively with inversion on the set E(T ) of edges of a tree T without leaves, it must also act transitively on the set V (T ) of its vertices. Let The crux of Bass-Serre theory is captured in the next lemma which determines the structure of groups acting on trees.
Lemma 2.3.[15] Let G act without inversion on a tree which has no vertices of degree one and let G act transitively on the set of (undirected) edges.If G acts transitively on the vertices of the tree, then G is an HNN-extension of the stabilizer of a vertex over the stabilizer of an edge.If there are two orbits on the vertices of the tree, then G is the free product of the stabilizers of two adjacent vertices with amalgamation over the stabilizer of an edge.
There is a standard way to deal with the case where we cannot apply Lemma 2.3 directly when G acts with inversion on the tree.Lemma 2.4.Let G act transitively with inversion on the edges of a tree T without leaves.Then G is the free product of the stabilizer of a vertex and the stabilizer of an edge with amalgamation over their intersection.
Proof.Subdivide every edge tt ′ of T to obtain a tree T ′ and let v tt ′ be the corresponding new node.Notice that G now acts transitively on E(T ′ ) without inversion and with two orbits on V (T ′ ).Each old node t of T has the same pointwise stabilizer in T ′ .Observe that for each new node

General structure of the tree-decomposition
Our key tool is the canonical tree decomposition (T, V) of Lemma 3.3, which will allow us to translate the action of G on Γ to an action of G on T and apply Bass-Serre theory.The lemma follows easily by the following results of [3], which we slightly reformulate for our needs.
Theorem 3.1.[3, Corollary 1.2.]Let Γ be a locally finite graph with more than one end.For each ℓ ∈ N, there is a canonical tree-decomposition distinguishing all ℓ-distinguishable ends efficiently.Moreover, its adhesion sets induce splitting separations of minimum crossing number.
As a last preparatory step before we state and prove Lemma 3.3, we need the following result which ensures that our Cayley graph is multi-ended.For the rest of the paper, we assume that G = S is an infinite finitely generated group such that Γ = Cay(G, S) is cubic with connectivity two.
The proof idea of the following lemma is similar to [11,Corollary 4.3].
Lemma 3.3.Let G = S be an infinite finitely generated group such that Γ = Cay(G, S) is cubic with connectivity two.Then there exists a reduced treedecomposition (T, V) with the following properties: (i) The adhesion sets of (T, V) have size exactly 2 and induce splitting separations of minimum crossing number.
(ii) The action of G on Γ induces an action on V (T ) and a transitive action on the set of separations corresponding to the adhesion sets.
Proof.Since Γ is an infinite graph of connectivity exactly 2, Lemma 3.2 implies that Γ has at least two ends.Let (T ′ , V ′ ) be a canonical tree-decomposition obtained by Theorem 3.1 which distinguishes all 2-distinguishable ends and since Γ is 2-connected, its adhesion sets have size 2. We consider the set S of all induced separations by (T, V) and choose a separation (A, B) of S. It follows from [3, Lemma 2.1] that the orbit of (A, B) under the action of G is a nested set N of separations of order 2. Now by Remark 2.1, the set N gives rise to a tree-decomposition (T, V) which has the desired properties.
Notice that the transitive action on the set of separations in Lemma 3.3 (ii) implies at most two orbits for Γ[V] := {Γ[V t ] | t ∈ V (T )} under the action of G.Moreover, we can translate the action of item (ii) to an action of G on T in the natural way (and G will clearly act transitively on E(T )): Let N be a nested set of separations of order two in such a way that N gives a tree-decomposition as in Lemma 3.3.It is easy to see that every 2-separation of Γ such that A ∩ B is a proper subset of A and B distinguishes at least two ends, see [8,Lemma 3.4].For an arbitrary element (A, B) ∈ N , there are three cases in terms of the degrees of the vertices of the separator in each side of the separation:

Type I
Type II Type III First, we dismiss the case of Type III separations.This follows as an easy corollary of the following lemma.Lemma 3.4.There is always a splitting 2-separation of crossing number 0 in Γ.
Proof.Any Type I separation has crossing number 0, as the two vertices of the separator are connected with an edge and thus they are inseparable.Hence we can assume that every splitting 2-separation is not of Type I. We show that there is always a splitting 2-separation (A, B) on A ∩ B = {x, y} such that there are at least two internally disjoint (x, y)-paths in Γ[A].This will directly imply that there is no 2-separation (C, D) crossing such an (A, B) as in that case the single vertex in C ∩ D ∩ A would separate x and y in Γ[A], so cn(A, B) = 0.In what follows, (T, V) will always be as in Lemma 3.3, either of Type I or Type II if not explicitly stated otherwise.For a node t ∈ V (T ), we define Recall that every adhesion set V t ∩ V t ′ of (T, V) induces the separation (W t\t ′ , W t ′ \t ) of Γ. Assume that (T, V) and the separations (W t\t ′ , W t ′ \t ) it induces are of Type II.We call such a separation (W t\t ′ , W t ′ \t ) small if the vertices of the separator V t ∩ V t ′ have degree 1 in W t ′ \t and big if they have degree 2 in W t ′ \t .
One of our main goals towards the general structure of the tree-decomposition of Γ is to eventually prove in Lemma 3.8 that all adhesion sets of (T, V) are disjoint.As a preparatory step for that, we need the following Lemma.Lemma 3.6.Every vertex u belongs in at least one and at most two different adhesion sets of (T, V) (as subsets of V (Γ) and not as intersections of different pairs of parts).Moreover, for every node t of T and every t 1 , t 2 ∈ N T (t), we have Proof.The lower bound for the first assertion of the lemma follows directly from the transitivity of the actions of G on Γ and E(T ).For the upper bound, let {x, u} and {y, u} be two adhesion sets of the tree-decomposition meeting on u.Since G acts transitively on E(T ), there is a 1 = g ∈ G such that g{x, u} = {y, u}.Observe that since g = 1, we must have gx = u and gu = y, from which we obtain ux −1 u = y.Since {x, u} and {y, u} were arbitrary adhesion sets containing u, the upper bound follows.
For the second assertion, we clearly have that It follows from the definition of a tree-decomposition that V t1 ∩V t2 ⊆ V t and so V t1 ∩V t2 is a subset of both V t ∩V t1 and V t ∩V t2 .Therefore, we have We observe that the 2-connectivity of Γ implies that all components of Γ \ S are tight (and in particular, the one containing V t \ S).
Let T S be the subtree of T whose corresponding parts contain S. Then |V (T S )| ≥ 3 and assume that |V (T S )| ≥ 4. Observe that Γ \ S then has at least four tight components, which contradicts the fact that Γ is cubic.Hence, |V (T S )| = 3 and so V (T S ) = {t, t 1 , t 2 }.Consequently, we see that This means that both vertices of S must have degree one in each of C 1 , C 2 , C 3 , respectively, and that S induces an independent set.Since G acts transitively on Γ and E(T ) and t was an arbitrary node of t, it follows that every vertex has degree at most one in every part it belongs in.We conclude that every part of V induces at most a matching where every pair of vertices in the same adhesion set is unmatched.It easily follows that Γ is the disjoint union of two infinite cubic trees, contradicting the fact that Γ is connected.
Let H be an arbitrary graph with a set U ⊆ V (H) and a subgraph H ′ of H.The set U is called connected in H ′ if for every pair of vertices u, u ′ ∈ U there is a (u, u ′ )-path in H ′ .Lemma 3.7.Let t be an arbitrary vertex of T .Then for every t ′ ∈ N T (t), the following holds: Proof.(i) Let V t ∩ V t ′ = {u, u ′ } and P be a path between u and u ′ .Suppose that P is contained in W t\t ′ and consider the tree decomposition (T ′ , V ′ ) of P by restricting (T, V) on the parts that contain at least two vertices of P .Notice that every adhesion set of (T ′ , V ′ ) has size exactly 2 and root T ′ on t.Since T ′ ⊆ T , the second assertion of Lemma 3.6 holds for (T ′ , V ′ ) as well, therefore we have that every part of (T ′ , V ′ ) contains at least one new vertex of P compared to its predecessor in the tree-order of T ′ when rooted on t.Since P is finite, it follows that T ′ is finite as well.We eventually find a part V s of (T, V) -in particular a leaf of T ′such that P ′ = V (P ) ∩ V s is a subpath of P whose end vertices constitute exactly one of the adhesion sets S of V s .Recall that G acts transitively on the set of adhesion sets of (T, V).Hence, we can map S to V t ∩ V t ′ , say gS = V t ∩ V t ′ .Then gs ∈ {t, t ′ }.Thus, gP ′ is a (u, u ′ )-path that either lies in V t or V ′ t .(ii) Since Γ is connected, the torso of V t is a connected graph.The result follows by replacing the virtual edges of a path within the torso of V t by paths obtained by (i).
The next crucial lemma implies that all adhesion sets in N are disjoint.Recall that (T, V) is a tree-decomposition as in Lemma 3.3.Lemma 3.8.Let t be a node of T .Then for every t 1 , t 2 ∈ N T (t), we have Proof.By Lemma 3.6, we have that z}.By Lemma 3.6, these are the only adhesion sets of V t containing x.We can assume that (T, V) is of Type II: indeed, assume that (T, V) is of Type I.By the tightness of all separations in N , we have that x has at least one neighbor in each of V t1 \ V t and V t2 \ V t in addition to y and z, a contradiction to Γ being cubic.Hence, (T, V) is of Type II.
Now, by the transitive action on E(T ) we have that (W t\t1 , W t1\t ) is either isomorphic to (W t\t2 , W t2\t ) or its inverse (as an ordered separation).In the latter case, assume there are g ∈ G, t ∈ V (T ), such that (W t\t1 , W t1\t ) = (gW t2\t , gW t\t2 ). ( Recall the definition of small and big separations from Page 7. We can assume w.l.o.g. that the above separations are small (a similar situation arises in case they are big separations).We observe that this implies that deg W t 1 \t (x) = 1, deg W t 2 \t (x) = 2 and the degree of x in the component of Γ \ {x, y, z} containing V t \ {x, y, z} is 0. By Lemma 3.7, there is an (x, y)-path P lying completely within V t1 , but by (1) we have that the (x, y)-path gP lies within V t , which yields a contradiction to the degree of x.
Otherwise, for every t ∈ V (T ) there is a g ∈ G such that Then we directly observe that both separations (W t\t1 , W t1\t ), (W t\t2 , W t2\t ) must be small.The only way for this to happen is when x has degree one in each of C 1 , C 2 , C 3 and therefore degree at most one in V t1 , V t2 , V t (in other words, every part that contains it).By the transitivity of Γ and the fact that t was arbitrary, we conclude that every vertex of Γ has degree at most one in every part that contains it, to obtain a contradiction to Lemma 3.7 exactly as before.
Lemma 3.8 has some important consequences.Combined with Lemma 3.6, we immediately obtain the following.Corollary 3.9.Every vertex u of Γ is contained in exactly two parts t, t ′ ∈ V (T ).In addition, N Γ (u) ⊆ V t ∪ V t ′ and every part is the disjoint union of its adhesion sets.
Moreover, let {x, y} be an adhesion set.Observe that xy −1 {x, y} is again an adhesion set containing x, so xy −1 {x, y} = {x, y} with xy −1 x = y.We obtain: Lemma 3.10.For every adhesion set {x, y}, we have (xy −1 ) 2 = 1.Lemma 3.10 implies the following Corollary for the edge stabilizers of T .
Lastly, we will invoke the following folklore Lemma from the well-known theory of tree decompositions into 3-connected components (see [14,16] as an example) when we argue about the planarity of Γ and G in each case that arises.Lemma 3.12.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 of H: 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, we will inductively embed the planar torsos of the parts of (T, V) on the plane by gluing (the torso of) each new bag at the appropriate adhesion set.For the inductive step, we consider the adhesion set V t ∩ V t ′ = {x, y} of the new bag V t along which it is going to be amalgamated with the already embedded torso of V t ′ .We simply replace the edge xy with a planar embedding of the torso of V t (restricted inside a thin disk around the embedded xy), keeping the edge xy in the so far embedded part of Γ depending on whether it is an actual edge of Γ or a virtual edge of the torsos of V t , V t ′ , accordingly.It is straightforward to check that the above inductive strategy to combine the planar embeddings of the torsos along the adhesion sets produces an embedding of Γ on the plane, since each virtual edge of the torsos is replaced with a new bag at some inductive step.
Our goal in the following sections is to determine the structure of the parts of the tree-decomposition of Γ obtained by Corollary 3.3 in order to compute their stabilizers and apply Lemma 2.3 or 2.4.
Lastly, observe that since b 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 ∈ St G (V t ).It follows that ba −1 ba, b ⊆ St G (V t ).Therefore, we conclude that By Lemma 2.3, we have that We collect both cases in the following theorem.Theorem 4.3.If (T, V) is of Type I with two generators, then G satisfies one of the following cases: The definitions of a free product with amalgamation, an HNN-extension and the proof of Theorem 4.3 immediately imply:

Three Generators
Let G = a, b, c , where a, b and c are involutions.Suppose that the edges induced by the adhesion sets with corresponding separations in N are colored with b.Up to rearranging a, b, c, there are two cases for the local structure of the adhesion sets with separations in N , as in the following figure:

Case II
In this case, we see that b inverts V t and V t ′ , so G acts on T with inversion by Lemma 4.1.Hence, G also acts transitively on V (T ).
Let x := bcba.By Lemma 4.2 we see that (bcba) n = 1 and that x ⊔ x b ⊔ x bc ⊔ x bcb is a partition of V t .Clearly, we have that bcba ⊆ St G (V t ).We show that we actually have equality: We conclude that St G (t) = bcba ∼ = Z n and consequently we also have that In conclusion, we have proved: Theorem 4.5.If (T, V) is of Type I with three generators, then G satisfies one of the following cases: Corollary 4.6.[9, Theorem 1.1]If (T, V) is of Type I with three generators, then G has one of the following presentations: 5 Tree-decomposition of Type II Even though at first glance there can be several cases for Type II separations, we will in fact be able to quickly exclude most of them using appropriately the following lemma.
Lemma 5.1.Let G = a, b, c (with possibly c = a −1 ), where b is an involution and let {x, y} be the adhesion set of a Type II separation in (T, V) of Γ as in Lemma 3.3.Let v 1 , v 2 , v 3 be any consecutive vertices in a shortest (x, y)-path P with at least two edges and suppose there is g ∈ G such that gv 2 ∈ {x, y}.Then gv 1 and gv 3 lie in the same component of Γ \ {x, y}.
Proof.Suppose not.We observe that gx, gy must then lie in different components of Γ \ {x, y} as well: if not, then gx, gy lie in the same component, say, C. The fact that gv 1 , gv 3 lie in different components implies that gP leaves C, therefore we have that both x, y ∈ V (gP ).Since gv 2 ∈ {x, y} is an inner vertex of gP , the subpath of gP from x to y contradicts the choice of P .
Hence, g{x, y} is a separator where gx, gy lie in different components of Γ \ {x, y}.It easily follows that {x, y} and {gx, gy} are not nested, a contradiction to Lemma 3.1.Now, let W 2n+1,2k , where n ≥ 1, k ≥ 3, n ≤ k, denote the cubic graph obtained by the 2k-cycle with vertices {0,1,. . .,2k-1} along with "chord" edges of the form {2i, 2i + 2n + 1} (mod 2k) forming a matching.Moreover, we define V 2n , n ≥ 2 as the cubic graph obtained by the 2n-cycle along with the "diagonal" edges {i, i + n} (mod 2n).We note that for k = 2n + 1 we have W 2n+1,4n+2 = V 4n+2 (Fig. 4).Lastly, let R 2m+1 be the cubic graph obtained by a double ray with vertex set Z (defined in the natural way) and by adding the edges of the form {2i, 2i + 2m + 1} (Fig. 5).We note that we will see in the next subsections that the tree-decomposition of Γ obtained by Lemma 3.3 will have two orbits of parts and that the torsos of the parts of one of the two orbits will always be isomorphic to either W 2n+1,2k , V 2n or R 2n+1 .The fact that W 2n+1,2k is never planar, whereas V 2n and R 2m+1 are planar if and only if n = 2 and m = 1, respectively, will allow us by Lemma 3.12 to determine exactly when Γ will be planar.

Two generators
Let G = a, b , where b is an involution.Let (T, V) the corresponding treedecomposition obtained by Lemma 3.3 and N the set of separations obtained from its adhesion sets.Then we have the following cases for the neighborhood of such an adhesion set {x, y}: Case II Case III Figure 6: Cases of Type II with two generators.
Proof.Let {x, y} be an adhesion set.First, observe that no path in Γ contains two consecutive b-edges, hence every path of length two contains at least one a-edge.Let P be a shortest (x, y)-path2 , necessarily of length at least two.Assume that either Case I or Case II happen.Notice that -in both casesfor every possible edge-coloring of a path of length two there exists a path Q of length two whose middle vertex belongs in {x, y} and its two endpoints lie in different components of Γ \ {x, y} that realizes the same edge-coloring.Consider an arbitrary subpath P ′ = v 1 v 2 v 3 of P of length two and an appropriate Q as above that realizes the edge-coloring of P ′ .Let w be the middle vertex of Q and g = wv −1 2 .Then gP = Q and gv 1 , gv 3 lie in different components of Γ \ {x, y}, contradicting Lemma 5.1.
Consequently, we can assume for the rest of this subsection that only Case III happens.It follows that no part of (T, V) contains edges of all colors: otherwise, by Corollary 3.9 and the fact that no adhesion set contains both a vertex incident In particular, G is planar if and only if n = 2.

Three generators
Let G = a, b, c , where a, b and c are involutions.Then -up to rearranging a, b, c-we have the following cases for the separations in N : Lemma 5.6.The adhesion sets of (T, V) satisfy Case II.
Since the torso of every part of (T, V) is a connected graph, we deduce that the tree-decomposition has two orbits of parts: parts in O 1 contain only b-and c-edges and parts in O 2 induce perfect a-matchings.Clearly, G then acts on (T, V) without inversion.Let us quickly obtain the analogue of Lemma 5.3.Proof.Let V t ∈ O 1 and {x, y} = V t ∩ V t ′ be an adhesion set of t.Since all neighbours of t induce an a-matching, it follows by Lemma 3.7(ii) that Γ[V t ] is connected.
Hence, there exists an (x, y)-path P of length i within V t , necessarily alternating with b-and c-edges.Then, either x = y(bc) n or x = y(bc) n b, up to swapping b and c.To obtain the structure of the 2-regular, connected graph V t we distinguish the following cases.
(i) If x = y(bc) n , then the (x, y)-path xy −1 P intersects P only in x, y and by Lemma 3.10, we obtain (bc) 2n = 1.In this case, V t induces an alternating (b, c)-cycle of length 4n.
If x = y(bc) n b, then xy −1 P = P and: (ii) either V t induces an alternating (b, c)-cycle, (iii) or V t induces an alternating double (b, c)-ray.
By the 2-connectivity of Γ, the connected, 2-regular torso of a part V s ∈ O 2 must be a finite cycle.Depending on which of the cases of Lemma 5.7 we have, we can label its edges with (bc) n or (bc) n b (corresponding to the virtual edges of the torso) and a in an alternating fashion.Therefore, there is an m ≥ 2 such that (a(bc) n ) m = 1 or (a(bc) n b) m = 1.It remains to infer the structure of G in each case.Then St G (V t1 ) = V t1 ∼ = D 4n , as V t1 forms a group.Next, let V t2 ∈ O 2 with 1 ∈ V t2 .Notice that (a(bc) n ) m = a 2 = 1 and (a(a(bc) n )) 2 = (bc) 2n = 1.We can deduce that V t2 is a group (and hence St G (V t2 ) = V t2 ), along with its presentation: By Lemma 2.3, we have In this case, the torso of V t1 is isomorphic to V 4n , which is planar if and only if n = 1.
and there is no edge between A \ B and B \ A. The order of (A, B) is the size of A ∩ B and we denote it by |(A, B)|.If |(A, B)| = k, we say that (A, B) is a k-separation.The set of separations of Γ can be equipped with the following partial order: ( , where tt ′ = e ∈ E(T ).The result follows from Lemma 2.3.Finally, Z n denotes the cyclic group of order n.A finite dihedral group is defined by the presentation a, b | b 2 , a n , (ba) 2 and denoted by D 2n .Moreover, the infinite dihedral group D ∞ is defined by a, b | b 2 , (ba) 2 .

Figure 1 :
Figure 1: The three types of splitting 2-separations in cubic Cayley graphs of connectivity 2.

Lemma 3 . 5 .
First, we note that for every 2-separation (A, B) both A \ B, B \ A are tight components of G \ (A ∩ B): otherwise, by the 2-connectivity of Γ there are two tight connected components in either A \ B or B \ A, hence there are two internally disjoint (x, y)-paths in Γ[A] or Γ[B].Then (A, B) or (B, A) is the desired separation.Now, consider a splitting separation (A, B) on A ∩ B = {x, y} such that there exist single vertices separating x, y in Γ[A] and let S A be the (non-empty) set of these cut vertices in Γ[A].Let P be a shortest (x, y)-path in Γ[A].Then S A ⊆ V (P ).It is easily verified that any two consecutive vertices of S A ∪ {x, y} in the ordering inherited by P constitute the separator of a 2-separation (C, D) in Γ nested with (A, B), and suppose w.l.o.g. that B ⊆ D. Since (A, B) is a splitting separation, say distinguishing ω 1 , ω 2 , there is a separation (C, D) as above that is also splitting, distinguishing ω 1 , ω 2 as well.Then (C, D) is the desired separation as any vertex separating the vertices of C ∩ D in Γ[C] must also separate x, y in Γ[A], contradicting the fact that S A ∩ (C \ D) = ∅.Any tree decomposition of Γ as in Lemma 3.3 is either of Type I or Type II.Proof.Let (A, B) be a Type III separation with A ∩ B = {x, y}.We can assume that |N (x) ∩ A| = 1 and |N (x) ∩ B| = 2. Let x ′ be the unique neighbor of x in A and y ′ be the unique neighbor of y in B. Then (A ′ , B ′ ) := (A ∪ {y ′ } \ {x}, B ∪ {x ′ } \ {x}) is a tight Type III separation on {x ′ , y ′ }, clearly crossing (A, B) and distinguishing efficiently the same ends.The lemma is now a direct consequence of Lemma 3.3 and Lemma 3.4.

Corollary 4 . 4 .
[9, Theorem 1.1]If (T, V) is of Type I with two generators, then G has one of the following presentations:

Figure 3 :
Figure 3: Cases of Type I with three generators

Figure 7 :
Figure 7: Type II cases with three generators

Lemma 5 . 7 .
Every part in O 1 induces an alternating (b, c)-cycle of even length or an alternating double (b, c)-ray.