Asphericity of positive free product length 4 relative group presentations

Abstract Excluding some exceptional cases, we determine the asphericity of the relative presentation 𝒫 = 〈 G , x ∣ a ⁢ x m ⁢ b ⁢ x n 〉 {\mathcal{P}=\langle G,x\mid ax^{m}bx^{n}\rangle} , where a , b ∈ G ∖ { 1 } {a,b\in G\setminus\{1\}} and 1 ≤ m ≤ n {1\leq m\leq n} . If H = 〈 a , b 〉 ≤ G {H=\langle a,b\rangle\leq G} , the exceptional cases occur when a = b 2 {a=b^{2}} or when H is isomorphic to C 6 {C_{6}} .

Clearly, the element x is not conjugate in G(P) to any element of G and |x| ≥ m + n > 1 in G(P). The final statement in Theorem 1.1 then follows immediately from, for example, statement (0.4) in the introduction of [6] and the fact that P is orientable.
An example is given by the following: Let F be the free group with basis u 0 , . . . , u l−1 , and let θ be the automorphism of F such that u i θ = u i+1 (mod l). For ω ∈ F, recall that the cyclically presented group G l (ω) is given by the group presentation G l (ω) = ⟨u 0 , . . . , u l−1 | ω, ωθ, . . . , ωθ l−1 ⟩.
Then G l (ω) belongs to the class of groups of type F, as defined in [8]. Now the automorphism θ of F induces an automorphism of G l (ω) and the resulting split extension E l (ω) of G l (ω) by the cyclic group of order l has presentation E l (ω) = ⟨u, t | t l , w(u, t)⟩, where w(u, t) is obtained from ω by the rewrite u i → t −i ut i [16]. In our case, ω rewrites to (ut −f ) r t rf −A (ut −f ) s t sf +A , so letting x = ut −f , we obtain E l (ω) = ⟨G, x | ax r bx s ⟩, where G = ⟨t | t l ⟩, a = t rf −A and b = t sf +A . More generally, it follows from [8, Lemma 6] and [4,Theorem 4.1] that Theorem 1.1 can be applied to obtain asphericity classifications for group of type F.
In Section 2, we give the method of proof and introduce the concepts of pictures and curvature distribution. In Section 3, some preliminary results are proved. The proof of Theorem 1.1 is completed in Section 4.

Method of proof 2.1 Pictures
The definitions in this subsection are taken from [6]. The reader is referred to [6] and [2] for more details.
A picture ℙ is a finite collection of pairwise disjoint discs {D 1 , . . . , D m } in the interior of a disc D 2 , together with a finite collection of pairwise disjoint simple arcs {α 1 , . . . , α n } embedded in the closure of D 2 − ⋃ m i=1 D i in such a way that each arc meets ∂D 2 ∪ ⋃ m i=1 D i transversely at its end points. The boundary of ℙ is the circle ∂D 2 , denoted ∂ℙ. For 1 ≤ i ≤ m, the corners of D i are the closures of the connected components of ∂D i − ⋃ n j=1 α j , where ∂D i is the boundary of D i . The regions of ℙ are the closures of the connected components of D 2 − (⋃ m i=1 D i ∪ ⋃ n j=1 α j ). An inner region of ℙ is a simply connected region of ℙ that does not meet ∂ℙ. The picture ℙ is non-trivial if m ≥ 1, is connected if ⋃ m i=1 D i ∪ ⋃ n j=1 α j is connected, and is spherical if it is nontrivial and if none of the arcs meets the boundary of D 2 . Thus the set of regions of a connected spherical picture ℙ consists of the simply connected inner regions together with a single annular region that meets ∂ℙ. The number of edges in ∂∆ is called the degree of the region ∆ and is denoted by d(∆). A region of degree n will be called an n-gon. If ℙ is a spherical picture, the number of different discs to which a disc D i is connected is called the degree of D i , denoted by d(D i ). The discs of a spherical picture ℙ are also called vertices of ℙ.
Suppose that the picture ℙ is labeled in the following sense: Each arc α j is equipped with a normal orientation, indicated by a short arrow meeting the arc transversely, and labeled with an element of x ∪ x −1 . Each corner of ℙ is oriented clockwise (with respect to D 2 ) and labeled with an element of G. If κ is a corner of a disc D i of ℙ, then W(κ) will be the word obtained by reading in a clockwise order the labels on the arcs and corners meeting ∂D i beginning with the label on the first arc we meet as we read the clockwise corner κ.
If we cross an arc labeled x in the direction of its normal orientation, we read x, else we read x −1 .
A picture over the relative presentation P = ⟨G, x | r⟩ is a picture ℙ labeled in such a way the following are satisfied: (1) For each corner κ of ℙ, W(κ) ∈ r * , the set of all cyclic permutations of r ∪ r −1 which begin with a member of x. (2) If g 1 , . . . , g l is the sequence of corner labels encountered in anticlockwise traversal of the boundary of an inner region ∆ of ℙ, then the product g 1 g 2 . . . g l = 1 in G. We say that g 1 g 2 . . . g l is the label of ∆, denoted by l(∆) = g 1 g 2 . . . g l . A dipole in a labeled picture ℙ over P consists of corners κ and κ of ℙ together with an arc joining the two corners such that κ and κ belong to the same region and such that if W(κ) = Sg where g ∈ G and S begins and ends with a member of x ∪ x −1 , then W(κ ) = S −1 h −1 . The picture ℙ is reduced if it is non-empty and does not contain a dipole. A relative presentation P is called aspherical if every connected spherical picture over P contains a dipole. If P is not aspherical, then there is a reduced spherical picture over P.
A connected spherical picture ℙ over P is defined to be strictly spherical if the product of the corner labels in the annular region taken in anticlockwise order defines the identity in G. The relative presentation P is weakly aspherical if each strictly spherical connected picture over ℙ contains a dipole. Let G(P) denote the group defined by P. It is shown in [6] that if P is weakly aspherical and if the natural map of G into G(P) is injective, then P is aspherical. Now let P = ⟨G, x | ax m bx n ⟩. Then the natural map G → G(P) is injective [17], and so it suffices to show that P is weakly aspherical. Let ℙ be a reduced connected strictly spherical picture over P. Then the vertices (discs) of ℙ are given by Figure 2.1 (i), (ii). It is clear from the orientation of the edges that a positive vertex can only be connected to a negative vertex, in particular, the degree of each region of ℙ is even.

The star graph Γ
The star graph Γ of a relative presentation P is a graph whose vertex set is x ∪ x −1 and edge set is r * . For R ∈ r * , write R = Sg, where g ∈ G and S begins and ends with a member of x ∪ x −1 . The initial and terminal functions are given as follows: ι(R) is the first symbol of S, and τ(R) is the inverse of the last symbol of S. The labeling function on the edges is defined by λ(R) = g −1 and is extended to paths in the usual way. A non-empty cyclically reduced cycle (closed path) in Γ will be called admissible if it has trivial label in G.
In general, we have that only each inner region of a reduced spherical picture ℙ over P supports an admissible cycle in Γ. However, since we are only considering strictly connected spherical ℙ, the same holds for the annular region as well.
The star graph Γ of P = ⟨G, x | ax m bx n ⟩ is given by Figure 2.1 (iii). In particular, a word obtained from a cyclically reduced closed path in Γ does not contain aa −1 , a −1 a, bb −1 , b −1 b up to cyclic permutation although it can contain the subwords 11 −1 , 1 −1 1 provided that different edges of Γ labeled by 1 are used. (Note that the structure of Γ also implies that the degree of a region must be even.) Using Γ, we see that the possible labels or regions of degree 2 or 4 are (up to cyclic permutation and inversion) as follows:

Curvature
Our aim is to show that, given certain conditions on m, n, a and b, P is aspherical. To this end, assume by way of contradiction that ℙ is a reduced connected strictly spherical picture over P. Our method is curvature distribution (see, for example, [1]). Proceed as follows: Contract the boundary ∂ℙ to a point which is then deleted. This way all regions ∆ of the amended picture, also called ℙ, are simply connected and form a tessellation of the 2-sphere. An angle function on ℙ is a real-valued function on the set of corners of ℙ. Given this, the curvature of a vertex of ℙ is 2π less the sum of all the angles at that vertex; and then the curvature c(∆) of a region ∆ of ℙ is (2 − d(∆))π plus the sum of all angles of the corners of ∆. The angle functions we will define results in each vertex having zero curvature, and so the total curvature c(ℙ) of ℙ is given by c(ℙ) = ∑ ∆∈ℙ c(∆). Given this, it is a consequence of Euler's formula, for example, that c(ℙ) = 4π, and so ℙ must contain regions of positive curvature. Our strategy will be to show that the positive curvature that exists in ℙ can be sufficiently compensated by the negative curvature. To this end, we locate each ∆ satisfying c(∆) > 0 and distribute c(∆) to near regions∆ of ∆. For such regions∆ , define c * (∆ ) to equal c(∆ ) plus all the positive curvature∆ receives minus all the curvature∆ distributes during this distribution procedure. We prove that c * (∆ ) ≤ 0 and, since the total curvature of ℙ is at most ∑ c * (∆ ), this yields the desired contradiction.
The standard angle function assigns the angle 0 to each corner of v which forms part of a region of degree 2 and assigns 2π d (v) to the remaining corners. Therefore if ∆ is a region of degree k > 2 with vertices v 1 , . . . , v k such that d (v 3) = 0 this shows, for example, that ℙ must contain a region of degree 4 (since d(∆) is even); and since c(4, 4, 4, 4) = 0, ℙ must contain a vertex of degree < 4.

Preliminary results
Recall that P = ⟨G, x | ax m bx n ⟩ (1 ≤ m ≤ n) and ℙ denotes a reduced connected strictly spherical picture over P amended as described in Section 2.3. We can make the following assumptions without any loss of generality: (P1) ℙ is minimal with respect to number of vertices. (P2) Subject to (P1), ℙ is maximal with respect to number of 2-gons. Proof. Consider the 4-gon ∆ of Figure 3.1 (i) having label (11 −1 ) 2 . Observe that there are k i 2-gons between v i and v i+1 (1 ≤ i ≤ 4, subscripts mod 4). Apply r = min(k 2 + 1, k 4 + 1) bridge moves [9] of the type shown in  Then each of the first r − 1 bridge moves will create and destroy two 2-gons leaving the total number unchanged. The r-th bridge move however will create two 2-gons but destroy at most one. Since bridge moves do not alter the number of vertices, we obtain a contradiction to (P2). The proof now proceeds by induction on k. Indeed if l(∆) = (11 −1 ) k where k > 2, then a sequence of bridge moves can produce a new picture with at least the same number of 2-gons but with a region having label (11 −1 ) k−1 .

Lemma 3.2. The following conditions hold:
Proof. (i) The result follows immediately from [5,Lemma 3.8]. (ii) If ⟨a, b⟩ is infinite cyclic, the result follows by (i); otherwise the proof of [2, Theorem 3] which uses a weight test [6] on the star graph shows that P is aspherical. Proof. If |ab −1 | = ∞, the result follows from Lemma 3.2. Let |ab −1 | = 1, and so the relator is ax m ax m . If m = 1, it is clear that every picture contains a dipole, so let m > 1. Then the degree of each vertex of any given picture ℙ is at least 4, and it follows from the last paragraph in Section 2.3 that P is aspherical. Finally, if 1 < |ab −1 | < ∞, then since (ax m ) 2 = ab −1 , it follows that |ax m | < ∞ and non-asphericity follows, for example, from [6, statement (0.4)].

Lemma 3.4.
Suppose that m ̸ = n. Then the following hold: (i) If a = b ±1 and |a| < ∞, then P is not aspherical.
Proof. (i) If a = b −1 , then |x| < ∞ and so P is not aspherical; or if a = b, then, for example, a spherical picture can easily be constructed ( it follows that |x m b| < ∞ and P is not aspherical. (iv) Let be a connected spherical picture over P. If contains a vertex of degree 2, then, since n > 2m, contains a dipole and the result follows, so assume otherwise. Now fill in the regions of using, if necessary, b-vertices with label b ±k to obtain a connected spherical picture over the ordinary presentation ⟨b, x; b k , b 2 x m bx n ⟩. The vertices of are given (up to inversion) in Figure 3.2 (iii). Clearly, each region of not of degree 2 has degree ≥ 4; and, since k ≥ 7, each b-vertex has degree ≥ 4. Curvature considerations now tells us that there must be a non-b-vertex v such that d(v) < 4, and so v must form a dipole in . But d(v) ≥ 3 in , so it follows that d(v) = 3 in both and which forces the dipole v forms in to be a dipole in , as required.  It will be assumed from now on that none of the following exceptional cases holds: It follows from Lemmas 3.2-3.4, the exceptional cases and [1] that from now on the following assumptions can be made: Given this, we have the following lemmas: Lemma 3.5. The following conditions hold: Any reduced closed path in the star graph Γ of length at most 4 involving a or b yields (see Section 2.2) one of the relators ab ±1 , a 2 b −1 , ab −2 , (ab −1 ) 2 , a 2 or b 2 , and so (A4) and (A5) together with assuming that none of (ab −1 ) 2 , a 2 , b 2 are trivial forces the degree of each region (not of degree 2) to be at least 6, and so P is aspherical (see Section 2.3).
and hence 1 2 is distributed from ∆ to each of∆ 1 and∆ 2 as shown. Note that positive curvature is distributed across a (b −1 , 1)-edge as shown in Figure 3.5 (i)-(iii) or the inverse (1 −1 , b)-edge as in Figure 3.5 (iv), where the maximum amount of π 2 is indicated. Let∆ be a region that receives positive curvature. Then l(∆ ) involves b and, since |b| = ∞ and a ∉ ⟨b⟩, must also involve a at least twice. Therefore where the r. π 2 is the contribution from the r vertices with corner label a ±1 , and α is the total curvature contributed to c * (∆ ) by the r segments a ε i w i a ε i+1 (subscripts mod r). First observe that it can be assumed without any loss that 1 −1 1 is not a sublabel of l(∆) since the contribution to c * (∆ ) made by the three edges and two intermediate vertices of Figure 3.6 (i) is at most −3π + 3π 2 = − 3π 2 , whereas the contribution made by the single edge shown is −π. Given this, it can be further assumed that 11 −1 is not a sublabel since the possibilities a −1 11 −1 b and b −1 11 −1 a each contribute at most −3π + 3π 2 + π 2 = −π which equals the contribution made by the edge of a −1 b or b −1 a (Figure 3.6 (i)). Given these two assumptions, it follows that, up to inversion, there are four types of segment a ε i w i a ε i+1 , and these are These are shown in Figure 3.6 (ii)-(v). The contribution from each segment is made up from the edges e i (1 ≤ i ≤ l), the vertices v j (1 ≤ j ≤ l − 1) and any positive curvature∆ receives across the e i . It follows that • the segment of Figure 3.6 (ii) contributes at most • the segment of Figure 3.6 (iii) contributes at most • the segment of Figure 3.6 (iv) contributes at most • the segment of Figure 3.6 (v) contributes at most Examples of when the maximum can be obtained are given by Figure 3.6 (vi)-(ix). Note that d(v 1 ) = 3 in Figure 3.6 (vii) and d(v l−1 ) = 3 in Figure 3.6 (viii) for otherwise the maximum would be − 3π 2 . For this reason we have the additional rule that π 2 is distributed from∆ to∆ 1 in Figure 3.6 (iii) when d(v 1 ) = 3 and in Figure 3.6 (iv) when d(v l−1 ) = 3. Note that the curvature is distributed as before across a (b −1 , 1)-edge or (1 −1 , b)-edge and so the above calculations remain unchanged. Therefore the net contribution to∆ of the segments of Figure 3.6 (ii)-(v) are each at most − 3π 2 and so c * (∆ ) ≤ 2π + r. π 2 − r. 3π 2 ≤ 0 for r ≥ 2, a contradiction that completes the proof.

Proof of Theorem 1.1
The statements in Lemma 3.5 imply that we need only consider 1 |a| + 1 |b| + 1 |ab −1 | ≤ 1, where at least one of |a|, |b| or |ab −1 | equals 2. Working modulo a ↔ b, it can be assumed that |a| ≤ |b|, and so there are seven possibilities, namely |a| = 2, |b| = 2 and |ab −1 | = ∞ which is not allowed by (A3), conditions (i)-(iii) of Lemma 3.6, and the following three cases that remain to be considered:  As before suppose by way of contradiction that ℙ is a reduced connected strictly spherical amended picture over P. For the remaining cases, we have found it easier to work with the dual of ℙ. This yields a so-called strictly spherical relative diagram, D say, which is connected and simply connected. The regions of D are given (up to inversion) by the region ∆ of If ab = ba, then P is not aspherical by [7, Theorem A] and noting that, although more general, nonasphericity in [7] implies the existence of a reduced spherical picture, that is, implies non-asphericity in the sense used here, so assume otherwise. Checking Γ (and using the assumption ab ̸ = ba together with the hypothesis on |a|, |b|, |ab −1 |) shows that if d(v) ≤ 8, then In particular, as shown in Figure 4.1 (iv), the vertices of two adjacent corners each with label 1 cannot both have vertex label a1 −1 a1 −1 , that is, be of degree 4.
If c(∆) > 0, then ∆ is given by    to c(∆ i ) as in . We note that in order to obtain the two possible l(v 2 ) for d(v 2 ) = 10 and l(v 2 ) = b −1 ab −1 aw, our method, here and elsewhere, is to enumerate all closed paths in Γ modulo cyclic permutation and inversion that do not contain the subwords aa −1 , a −1 a, bb −1 or b −1 b. We then, often with the use of GAP [19], delete all those words in our list that contradict either ab ̸ = ba or our hypothesis on |a|, |b| and |ab −1 |. This is a routine but lengthy procedure and we omit the details for reasons of space.
The description of curvature distribution is now complete so let d(∆ ) ≥ 5. Once more it can be seen from Figure 4.1 (iv) that the vertices of two adjacent corners each with label 1 cannot both be of degree 4 and sô ∆ must contain at least two vertices of degree ≥ 6 therefore c * (∆ ) ≤ c(4, 4, 4, 6, 6) + π 4 + 4. π 12 < 0. Finally, let d(∆ ) = 4. If∆ contains at most one vertex of degree 4, then c * (∆ ) ≤ c(4, 6, 6, 6) + 4. π 12 < 0. The case when ∆ contains three vertices of degree 4 is dealt with by If ab = ba, we obtain the exceptional case (E2), so assume otherwise. Checking Γ (and using the assumption ab ̸ = ba together with the hypotheses on |a|, |b| and |ab −1 |) shows that if d(v) ≤ 8, then In particular, any given region has at most two vertices of degree 4, and Figure 4.11 (iv) shows that two such vertices cannot be adjacent.