We find a polynomial (n6) isoperimetric function for Artin groups, the defining graph of which contains no edges labelled by 3. This in particular shows that even Artin groups have solvable word problem. We use small cancellation theory of relative extended presentations.
Let m,n∈ℕ (ℕ – the natural numbers), m≥0, n≥1, and let Γ be a simple graph without loops, with vertex set V={v1,…,vn} and edge set E={e1,…,em}. Label the edges by natural numbers via a labelling function λ:E→ℕ∖{1}. Denote λ(e)=nij, where the edge e connects vi to vj, i≠j (nij=nji), and let nij=0 if vi and vj are not connected by an edge. For every such labelled graph corresponds a group presentation
𝒫(Γ)=⟨X∣ℛ⟩
such that
X={xv,v∈V}, ℛ={Re,e∈E}, denote xvi by xi,
Re=(xixj)12nij(xi−1xj−1)12nij, if nij is even and
Re=(xixj)nij−12xi(xi−1xj−1)nij−12xj−1, if nij is odd.
The group presented by 𝒫(Γ) is denoted by A(Γ) and is called the Artin group defined by Γ.
Examples 1.
nij=0 for every i,j, 1≤i<j≤n.
Then A(Γ) is the free group on X.
nij=2 for every i,j, 1≤i<j≤n.
Then A(Γ) is the free abelian group on X.
nij∈{0,2} for every i,j, 1≤i<j≤n.
Then A(Γ) is called right-angled Artin group.
nij∈2ℕ for every i,j, 1≤i<j≤n.
Then A(Γ) is called an even Artin group.
nij≠2 for every i, j, 1≤i<j≤n.
Then A(Γ) is said to be of large type.
To each Artin group A(Γ), there corresponds a Coxeter group WA, obtained by adding the relators xi2, i=1,…,n. An Artin group is said to be of spherical type if WA is a finite group. The graphs of the spherical type Artin groups are classically known. An Artin group is said to be of FC type if it belongs to the smallest class of Artin groups which are closed under amalgamation along standard parabolic subgroups and contain all the spherical type Artin groups. Standard parabolic subgroups are the subgroups which are generated by subsets of X.
We recall isoperimetric functions. Let G be a group presented by 𝒫=⟨X∣ℛ⟩. Let W∈F(X), F(X) be the free group freely generated by X, W cyclically reduced (i.e., WW contains neither xx−1 nor x−1x, x∈X). Then W represents 1 if and only if
W=C1⋯Ckw,kw≥1,Ci=fiRiεifi−1,fi∈F(X),Ri∈ℛ and ε∈{1,−1}.
A function g:ℕ→ℕ is an isoperimetric function for the presentation 𝒫=⟨X∣ℛ⟩ if for every word W which represents 1 of the group G, kW in (1.2) satisfies kW≤g(|W|). Here |W| denotes the length of W in F(X).
The isoperimetric functions are known for spherical type, FC type, large type, right-angled Artin groups and some other classes of Artin groups. See [1, 4, 5].
Main Theorem.Let A be an Artin group, A=A(Γ). If nij≠3 for every nij, 1≤i<j≤n, then f(n)=n6 is an isoperimetric function for 𝒫(Γ).
We believe that “6” can be replaced by “2”.
Corollary.
Let A(Γ) be an Artin group. If nij≠3 for every i,j, 1≤i<j≤n, then A has solvable word problem. In particular, even Artin groups have solvable word problem.
The corollary has been established recently also in [9], using a completely different method.
Our basic method is small cancellation theory of extended (relative) presentations via van Kampen and Howie diagrams (see [6, Lemma 1.2.2, Remark 1.2.3]). (Recall that a van Kampen ℛ-diagram over F(X) is a connected, simply connected labelled planar 2-complex, labelled by elements of F(X) such that the labels of the boundary of the 2-cells are elements of ℛ. We say that the corresponding 2-cell (region) realises the element of ℛ.)
The original version of this article appeared in [6] and has been approved by the referee. The present article is a shortened version of it.
Outline of the proof of the Main Theorem. In order to find an isoperimetric function for the presentation 𝒫, we have to find an upper bound on the number kw in (1.2) in terms of the length |W| of W for every W representing 1 in the group defined by 𝒫. By the basic theorem of van Kampen diagrams, for every cyclically reduced W representing 1, there is a van Kampen diagram M with boundary label W such that the number of regions in M is at most kW. See [10, Chapter V]. So our problem can be considered as a counting problem in van Kampen diagrams, and this is our approach. We use the following simple basic principles, together with the well-known result in Proposition 1. Thus let S be a finite set. In order to count the elements of S, we subdivide S into subsets S1,…,Sm such that we know m and know the maximal possible number n of elements in Si, i=1,…,m. Then S has at most mn elements. In our case, Proposition 1 provides the first approximation to the numbers m and n, in terms of the length of W.
Proposition 1.
Let M be a reduced van Kampen ℛ-diagram over F(X). Let W be a boundary label of M, cyclically reduced. Assume M satisfies conditions C(4) and T(4). Suppose that each piece has length 1. Then each of the following holds:
For each regionΔofM, we have|∂Δ|≤|W|.
The number|M|of regionsΔinMis at most|W|2.
Hence if our diagram M would be a C(4) and T(4) diagram, then by part (b) of the proposition, the number of regions in M would be bounded by |W|2. Being this not the case, we first modify our approach. The core idea is not to try to estimate the number of all the regions via a result like Proposition 1 but to use that result for the estimation of a part of the regions only, namely, |Reg4+(M)|, the number of regions of M having boundary cycle of length at least 8, and then relate |Reg2(M)|, the number of remaining regions, to |Reg4+(M)|.
In this direction, we have the following.
Theorem 1.
Let M be a van Kampen ℛ-diagram, ℛ given by (1.1). Assume
ℛ={Re∣e∈E},λ(e)≠3for every e∈E.
Then there is a van Kampen diagram M′ with the same boundary label as M such that |Reg4+(M′)|≤|∂M|4.
For the proof, see [6, pp. 5–11,105].
Now, we relate the number of regions in Reg2(M) to the number of regions in Reg4+(M). First, let us state a definition.
Definition 1
(a-Bands). Let D1,D2∈Reg2(M), and let Supp(D1)={a,x},Supp(D2)={a,y} (x=y is not excluded). Say that D1 and D2 are a-neighbours if θ:=∂D1∩∂D2 contains a subpath θ0 with label a±1. (We consider Di as a neighbour of itself, i=1,2.) The transitive closure of “a-neighbourhood” is an equivalence relation. For D∈Reg2(M) with Supp(D)={a,x}, denote by L^a(D) the a-equivalence class of D. Let La(D) be the subdiagram of M generated by L^a(D). Call La(D) the a-band generated by D. Denote by Str(M) the collection of all the bands in M. Clearly,
Reg2(M)=⋃L∈Str(M)Reg(L).
We make the observation that every band either starts on the boundary of a region in Reg4+(M) or on the boundary of M. The same is true for the ends of bands. Hence we can estimate the number of bands in terms of the number of regions from Reg4+(M). Next, we show that the number of regions in a band is bounded by the number of bands. Relying on Theorem 1, this easily implies that |Reg2(M)| is bounded by |∂M|6. See [6, pp. 10–12].
The proof of the Main Theorem is reduced to the proofs of Theorems A–C and some results on the way to their proofs. In the rest of the work, we concentrate on their proofs. We assume that ℛ is given by (1.3).
See Definition 2.1.4 for ℳ3(W).
Theorem A.
Let M be a van Kampen ℛ-diagram over F with cyclically reduced boundary label W. If M∈ℳ3(W), then 𝕄~t satisfies the small cancellation conditions C(4) and T(4) for every t∈T(M).
Theorem B.
Let M be a van Kampen ℛ-diagram over F with cyclically reduced boundary label W. If M∈ℳ3(W), then for every region D in Reg(M) realising a relation R(a,t), ∥W∥t≥n(a,t) and ∥W∥a≥n(a,t), a and t in T.
The main theorem obtained from the proof of Theorem 1 and the main result is Theorem C, which is a variant of Greendlinger’s lemma.
Theorem C
(Greendlinger’s lemma). Let A(Γ) be an Artin group given by (1.3). Let M0∈ℳ3(W) (see Definition2.1.4) be a connected, simply connected minimal ℛ-diagram over F with cyclically reduced boundary label W,W≠1 in A(Γ2).
Let W=W1W2−1 be reduced as written. Then there is a van Kampen diagram M,M∈ℳ3(W), with boundary cycle ω, such that ω decomposes into ω=ω1ω2−1, ωi labelled with Wi, i=1,2 and one of the following holds:
There is a boundary regionK∈Reg4+(𝕄)with∂K∩ωiconnected,∥∂K∩ωi∥≥n(K), fori=1ori=2, whereReg4+(𝕄)={K∈Reg(𝕄)∣n(K)≥4}. Moreover,|ξ|≤|∂K∩ωi|, whereξis the complement of∂K∩ωion∂K.
There are boundary regionsD1,D2∈Reg2(M)with∂Di∩ωjconnected,|∂Di∩ωj|≥2, forj=1or forj=2.
There is a regionD∈Reg2(M)with∂D∩ωjconnected,|∂D∩ωj|≥2, and there is a regionK∈Reg4+(𝕄)with∂K∩ωjconnected,∥∂D∩ωj∥≥2, forj=1or forj=2.
Overview of the present work. The goals of the work are
to estimate the number of regions in Reg4+(M),
to estimate the number of regions in Reg2(M).
The idea behind (1) is to count the number of regions in 𝕄~t,t∈T(M) (for notations, see [6, the three constructions on pp. 6, 7, 11]) where due to Theorem A, conditions C(4) and T(4) are satisfied, hence we may do this, and then to show that the number of regions in 𝕄~t and Reg4+,t𝕄 is the same, via a natural mapping ψt of diagrams which sends a region Δ in 𝕄t to a uniquely defined region Δ~ in 𝕄t~. For Ψt, see [6, pp. 79–81]. One of the problems is that 𝕄~t is obtained from 𝕄t by shrinking edges and identifying vertices; hence in principle, Δ~ may shrink to an edge or a vertex. Thus our task is to show that ψt (and ψt−1) does not cause deformation and collapse of regions. One of the classical methods to avoid deformations in diagrams is by Greendlinger’s lemma. It roughly says that a diagram with at least two regions has at least two Greendlinger regions, the boundary of which contains a big portion of the relators. See [6, Definition 1.3.1]. In general terms, it shows that if there is a kind of deformation in certain subdiagrams, then certain events are simultaneously unavoidable and forbidden, which of course is absurd. Hence no deformation occurs. For example, consider a reduced diagram in which every region has boundary label Un, U cyclically reduced, not proper power, n≥7. Suppose that the boundary of a region E is not simple closed. (This is the deformation which we would like to avoid.) Then there is a loop in the boundary of the region which surrounds a disc Q, the boundary of which is labelled by a subword of the cyclic word Un. Suppose that in Q, we have Greendlinger’s lemma. Then Q has a boundary region D with ∂D∩∂Q connected such that |∂D∩∂Q|≥3|U|. Hence D and E necessarily have a common (unavoidable) piece of length at least 3|U|. But such a word is forbidden, because due to U not being a proper power, this would mean that D cancels E, violating that the diagram is reduced. (Though we do not use this example, we use its underlying idea. See the proofs of Propositions 2.1.7 and 2.1.8 in [6].) Coming back to the context of our work, notice that by Theorem C, every bigonal ℛ-diagram M, that is, ∂M=ω1ω2−1 has a generalised Greendlinger region on one of the sides ωi of the bigon. Since 𝕄t~ does satisfy C(4) and T(4), hence it has Greendlinger regions. The natural candidates for Greendlinger’s regions in M are the images (by Ψt−1) of the Greendlinger regions of 𝕄t~. The problem is that while Greendlinger regions are boundary regions, our candidates in M need not be. So we need a technique to “move” inner regions to the boundary. More precisely, to show that there is a diagram with the same boundary label as M in which the candidates are boundary regions. We do this by observing that since the generators of the second homotopy group of the defining complex are prisms, we can replace one half of the prism by its complement by a rotation of the prism. In M, this has the effect of moving a region which occurs both in M and in the prism. See [6, Section 3]. The involvement of the second homotopy group requires the introduction of extended presentations.
Finally, we have to count |Reg2(M)|. To this end, it is enough to show that two bands cannot intersect more than once (see Section 4). Essentially, we prove this simultaneously with the other main proposition (see Section 4).
The ideas developed here lead to further results for Artin groups A dealt with in the present work. Recall that a parabolic subgroup of an Artin group is a conjugate of a standard parabolic subgroup. In [7], we show that the intersection of parabolic subgroups is parabolic. Also, we describe fusion in A. In particular, we show that in even Artin groups, every standard parabolic subgroup P controls fusion in P (i.e., if two elements of P are conjugate in A, then they are already conjugate in P).
Together with further ideas, we show in [8] that locally reducible Artin groups (i.e., no (2.3.3), (2.3.4) and (2.3.5) type standard parabolics occur) have polynomial (n6) isoperimetric functions.
The work is organised as follows:
In Section 2, we introduce the diagrams 𝕄 and 𝕄t and show that they are adequate. Also, M is adequate.
In Section 3, we consider extended presentations, Howie diagrams, ℐ-moves and banded diagrams.
In Section 4, we consider the connection between 𝕄t and 𝕄t~ and prove the main theorems.
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Let M be a simply connected ℛ-diagram over F, ℛ given by (1.3), and let t∈T(M). Denote by RegHt(M) the set of all the regions of M the boundaries of which are labelled only by letters from Ht (Ht=⟨X−{t}⟩). Denote by Regt(M) the set of all the regions of M the boundaries of which have labels from ⟨t⟩ and also from Ht. Thus Reg(M)=Regt(M)∪RegHt(M). We subdivide Regt(M) further by
Regt(M)=Reg4+,t(M)∪Reg2,t(M),
where Reg4+,t(M) is the set of regions in Regt(M) with boundary length at least 8 and Reg2,t(M) is the set of regions in Regt(M) with boundary length 4. We have for t∈T
Reg(M)=Reg4+,t(M)∪RegHt(M)∪Reg2,t(M).
For the definition of the diagrams 𝕄 and 𝕄t, we introduce equivalence relations “∼t”, “∼Ht” and “∼Bt” on Reg4+,t(M), RegHt(M) and Reg2,t(M), respectively.
Definition 2.1.1
(of ∼t). Let D1 and D2 be regions in Reg4+,t(M). Say that they are t-friends if the following hold:
∂D1∩∂D2 contains an edge.
Supp(∂D1)=Supp(∂D2).
Let ∼t be the transitive closure of t-friendness. It is easy to see that ∼t is an equivalence relation on Reg4+,t(M). Denote by [D]t the ∼t equivalence class of D.
Definition 2.1.2
(of ∼Ht). Let D1 and D2 be regions in RegHt(M). Say that they are Ht-friends if ∂D1∩∂D2 contains an edge. Let ∼Htbe the transitive closure of Ht-friendness. Then ∼Ht is an equivalence relation on RegHt(M). Denote by [D]Ht the equivalence class of D∈RegHt(M).
Definition 2.1.3
(of ∼Bt). Let D1 and D2 be regions in Reg2,t(M). Say that D1 and D2 are Bt-friends if ∂D1∩∂D2 consists of an edge with label t±1. Let ∼Btbe the transitive closure of Bt-friendness. Then ∼Btis an equivalence relation. For D∈Reg2,t(M), denote by [D]Bt the equivalence class of D in Reg2,t(M).
Definition 2.1.4
(The class ℳ3(W) of diagrams). The following statements hold:
Let W be a cyclically reduced word which represents 1 in A(Γ). Assume that W≠1 in A(Γ2). Here Γ2 is the graph obtained from γ by removing all edges not labelled by 2. Denote by ℳ(W) the set of all the van Kampen diagrams with boundary label W. By [10, Chapter V], ℳ(W)≠∅. Let ℳ1(W) denote the set of all the diagrams in ℳ(W) for which |Reg4+(M)| is minimal possible. Clearly, ℳ1(W)≠∅. Let ℳ2(W) be the set of the diagrams in ℳ1(W) which contain minimal number of equivalence classes [D]M of regions in Reg4+(M). Clearly, ℳ2(W)≠∅. Finally, let ℳ3(W) be the set of the diagrams in ℳ2(W) with adequate bands. Say that M is minimal if M∈ℳ2(W) and |Reg2(M)| is minimal possible. It follows that if M is minimal, then M∈ℳ3(W). Hence ℳ3(W)≠∅.
Definition 2.1.1 makes sense without the requirement that D1 and D2 are in Reg4+,t(M), namely, D1∼D2 if Supp(∂D1)=Supp(∂D2) and ∂D1∩∂D2 contains an edge. We shall call the corresponding equivalence relation “equivalence”. For D∈RegZ(M), define κ0(D)=Int(∪{E¯∣E∈[D]Z}). Then κ0(D) is connected. If Z=t4+, then define n(κ0(D))=n(D). Notice that n(κ0(D)) is well defined. Also, denote κ0(D) by Δ(D) for this case. Denote the diagram obtained by 𝕄.
Definition 2.1.5
(Adequate). Let M be a connected, simply connected ℛ-diagram over F, M∈ℳ3(W). Say that M is adequate, if for every D∈Reg(M),D is homeomorphic to the open unit disc, ∂D is simple such that the boundary label is not 1 and is cyclically reduced.
Definition 2.1.6
(𝕄t). Let M be a van Kampen ℛ-diagram over F, and let t∈T(M). If κ0(D) is homeomorphic to the open unit disc, for every D∈Reg(M), then we may consider {κ0(D)∣D∈Reg(M)} as a set of regions such that their totality generates a diagram which we denote by 𝕄t. Thus Reg(𝕄t)={κ0(D)∣D∈Reg(M)}. We have ∂𝕄t=∂M. Also,
Reg(𝕄t)=Reg4+,t(𝕄t)∪RegBt(𝕄t)∪RegHt(𝕄t)for every t∈T(M).
The main results of this section are Propositions 2.1.7 and 2.1.8.
Proposition 2.1.7.
Let M∈ℳ3(W). Then 𝕄t is adequate for every t∈T(M). Moreover, if K1 and K2 are regions of 𝕄t, not both bands, then ∂K1∩∂K2≠∅ implies that ∂K1∩∂K2 is connected.
Proposition 2.1.8.
Let M∈ℳ3(W). Then M is adequate. Moreover, if ∂D1∩∂D2≠∅, then ∂D1∩∂D2 is connected.
Remark 2.1.9.
Let K∈Reg4+,t(𝕄t). When passing from 𝕄t to 𝕄t~ by shrinking edges labelled by elements of Ht to a point and shrinking t-bands to edges, it may happen that the interior of the obtained region Kt~ in 𝕄t~ becomes disconnected. Since M is adequate, we can replace Kt~ by a disc (and we shall do so), the boundary of which is labelled by Φ(∂(Kt~)).
We prove Propositions 2.1.7 and 2.1.8 by simultaneous induction on the size of M. See [6, Sections 2.2–2.4].
Extended presentations, <inline-formula><alternatives><mml:math display="inline"><mml:mi>ℐ</mml:mi></mml:math><tex-math><![CDATA[\(\mathcal{I}\)]]></tex-math></alternatives></inline-formula>‑moves and banded diagramsIdentities among relations
We recall identities among relations from [2, p. 157]. Let C be a 2-complex. Let c0 be a basepoint for C (some designated vertex of C). A sequence σ over C at v0 is σ=(q1,…,qn), where qi is a path in C consisting of a concatenation γi∂Δiγi−1, Δi being a region (face) of C and γi a path from v0 to v(Δi), 1≤i≤n, where v(Δi) is the basepoint of Δi. We call σ an identity sequence if the concatenation q1⋯qn is freely equivalent to the trivial path at v0.
Given a presentation 𝒫=⟨X∣ℛ⟩, following [10], we construct a bouquet of tailed circles C with basepoint v0. Let σ=(q1,…,qn) be a sequence over C at v0, qi=γiΔiγi−1, γi tails. Label ∂Δi by Ri∈ℛ and label γi by a word in F(X), via a labelling function Φ0:C→F. After carrying out all free cancellations in Φ0(σ)=Φ0(q1)⋯Φ0(qn), we get an ℛ-diagram M over F the boundary label of which is the cyclically reduced word for Φ0(q1⋯qn) in F.
If Φ0(q1⋯qn)≠1F, then M is a connected and simply connected planar diagram (van Kampen diagram), and if Φ0(q1⋯qn)=1 in F, then M is the union of a finite number of spines and tessellated spheres (≅S2⊆ℝ3) by the Δi. In this case, we call M a spherical diagram. We call M a singular sphere or just a sphere if it consists of a single sphere. When the qi are labelled with Ri and ϕ(q1⋯qn)=R1⋯Rn=1 in F, then we call the sequence S=(R1,…,Rn) an identity sequence over ℛ and call it a simple identity sequence, if the corresponding spherical diagram is a simple sphere.
Definition 3.1.1
(Extended presentations). Let 𝒫=⟨X∣ℛ⟩ be a finite presentation of a group G, and let ℐ be a set of simple identity sequences over ℛ. The corresponding extended presentation is the triple ⟨X∣ℛ∣ℐ⟩. This notion was introduced in [3] by Roger Fenn. Now let A=A(Γ) be an Artin group given by ⟨X∣ℛ⟩ in (1.3). The identities among relations ℐ we choose are obtained from the 3-generated standard parabolic subgroups A(i,j,k) with defining subgraphs Γi,j,k with the vertex set V(Γi,j,k)={xi,xj,xk} and the edge set E(Γi,j,k)={eij,ei,k,ej,k}, 1≤i<j<k≤n, with labels λ(eij)=λ(ej,k)=2 and λ(ei,k)=m, m≥4. The corresponding tessellated simple spheres Σi,j,k are prisms with two regions of degree 2m (upper and lower) labelled with Rik, which we call the big regions, and a band of length 2n in which the regions are labelled by Rij and Rjk in an alternating manner. See Figure 1 for n=4.
A big region realising Rik in a half sphere for n=4.
We introduce one of our main tools, ℐ-moves.
Let Σ=Σi,j,k, and let ω be a simple closed curve on Σ. Then ω subdivides Σ into two simply connected subdiagrams, ωR and ωL, with connected interiors, where ωR is the submap of all regions to the right of ω and ωL is the submap of all the regions to the left of ω. Clearly, ωR¯∩ωL¯=ω and ωR∪ω∪ωL=Σ.
Let S(Σ) be the set of all the connected, simply connected submaps of Σ with connected interior, and let βω:S(Σ)→S(Σ) be the function which for every closed curve ω on Σ sends ωR to ωL and ωL to ωR and leaves ω unaltered.
Let M be a van Kampen ℛ-diagram over F, and let M0 be a simply connected subdiagram of M with connected interior and simple boundary cycle μ. Suppose that M0¯ can be embedded into Σ=Σi,j,k for some (uniquely defined) i, j and k as an ℛ-diagram with image N in Σ with ∂N simple closed such that μ is mapped onto ∂N. Then N∈S(Σ).
Definition 3.1.2
(ℐ-moves). Let ℐ be a simple identity among relations realised by a uniquely defined sphere Σ=Σi,j,k. Let M, M0, N, S(Σ) and βω be as defined above for some ω. An ℐ-move at M0 is a surgery in M by which we cut out M0 and fill in instead by βω(N). Since ∂N=∂(βωN), this is well defined.
Below we consider some examples which illustrate the way ℐ-moves are used.
Example 3.1.3.
Let X={a,b,c}, ℛ={R1,R2,R3}, where R1=aba−1b−1,
R2=aca−1c−1, R3=bcb−1c−1. Here G≅ℤ⊕ℤ⊕ℤ. Let W=abca−1b−1c−1. Then W represents 1 in G. Figure 2 shows the corresponding van Kampen diagrams.
Application of an ℐ-move on ⟨D1,D2,D3⟩.
Example 3.1.4.
Now let ℰ=⟨a,b,c∣ℛ∣I⟩, ℛ={R1,R2,R3}, I=R1, R2b, R3, (R1−1)c, R2−1, (R3−1)a be an extended presentation, where Rx=xRx−1 and R1, R2, R3 are as in Example 3.1.3. The effect of I on the diagrams of Figure 2 is the replacement of the diagram in Figure 2 (a) by the diagram in Figure 2 (b).
This changes the positions of the regions realising R1, R2 and R3 and may reduce the number of regions. See Figure 3.
An ℐ-move followed by a free reduction which reduces the number of regions.
Banded subdiagrams and transfer of regions
In this subsection, we assume M∈ℳ3(W) and assumption ℋ.
Lemma 3.2.1.
Let M∈ℳ2(W), and let I be an I-move or extended I-move. Then MI∈ℳ2(W).
Proof.
We have to show that I increases neither |Reg4+(M)| nor |Reg4+(𝕄)|. Suppose that I replaces the simply connected subdiagram N with connected interior by a diagram N′.
We use notation of Definition 3.1.1. If n=2, then clearly I cannot alter |Reg4+(M)| and |Reg4+(𝕄)|. If n≥4, then N contains exactly one big region Δ and N′ also contains exactly one big region Δ′, which is a copy of Δ. Since M∈ℳ2(W), the result follows.∎
Definition 3.2.2
(Banded subdiagrams). Let B=⟨E1,…,Ek⟩ be a band in M with sides σ1 and σ2, and let θ be a boundary path of a simply connected subdiagram M0 with connected interior.
Say that M0 is banded at θ by B if ∥θ∥≥1 and θ⊆σ1∪σ2. See Figure 4 (a).
Let notation be as in part (a), and let θ=v0θ1v1θ2v2⋯θrvr be a decomposition of θ, vi vertices, r≥1. Say that M0 is banded at (θ1,…,θr) by bands Bi, respectively, if M0 is banded at θi by Bi for every i, i=1,…,r.
Definition 3.2.3
(Transfer of regions from M0). Let M be a connected, simply connected ℛ-diagram over F with connected interior, and let M0 be a connected, simply connected ℛ-subdiagram over F with connected interior. Let D be a boundary region of M0 with μ:=∂D∩∂M0 connected, and let B=B1∪B2∪B3 be such that ∂D∩∂B=∂D∩∂B2. Observe that D∪B2 embeds into the prism Pm, m=2n(D), Pm=D∪B2∪D′∪B2′, where B2∪B2′ is an annulus with m regions from Reg2(M) (a closed band) and D is a region with n(D′)=n(D) and Supp(D)=Supp(D′). (We identify D and B2 with their images in Pm.) Notice that B2 and B2′ have the same poles α and β, hence B′=B1∪B2′∪B3 is a band. Replace B∪D with B′∪D′ (which have the same boundaries) and denote the resulting diagram by M0J, where J is the ℐ-move which replaced B∪D with B′∪D′. Thus M0∪B is replaced by ((M0∪B)∖(D∪B))∪(D′∪B′). Hence
M0J=M0∖D,BJ=B′andM0J is banded by B′.
In this situation, we say that J transferred D from M0. See Figure 4 (b) and 4 (c). If every region can be transferred from M0, we say that M0 is transferable. The removal of D∈Reg(M0) naturally extends to transfer of regions Δ from 𝕄. (See Figure 4 (b′) and 4 (c′).)
(a)–(c): Transfer of the region D via an adjacent band B by the application of the ℐ-move. (b′) and (c′): Transfer of the modified region Δ via an adjacent band B by the application of the ℐ-move.
Notice that if instead of B we have a band-bundle, then D can be transferred by a sequence of ℐ-moves beyond the band-bundle, without destroying the bundle structure. See Figure 5 (a) and 5 (b) for D and Figure 5 (a′) and 5 (b′) for Δ.
(a) and (b): Transfer of the region D via a band-bundle (β1,β2) by the application of the ℐ-move J. (a′) and (b′): Transfer of the modified region Δ via a band-bundle (β1,β2) by the application of the ℐ-move J.
We are interested in a special type of banded subdiagrams (2-banded subdiagrams) which we consider in the next subsection.
Definition 3.2.4
(2-Banded subdiagram). Let W≠1 be a cyclically reduced word in F which represents 1 in A. Let M∈ℳ3(W), and let Q be a connected, simply connected subdiagram of M with connected interior. Say that Q is 2-banded if
there is an a-band Bl and a b-band Br in Q, a,b∈X (a=b is not excluded),
there are connected and simply connected subdiagrams Qr, Ql and P in Q and paths θr and θl such that
Q=Ql∪Bl∪θlP∪θrBr∪Qr and ∂Q=ω1αl−1γαr−1ω3βrβl, where θl=∂Bl∩∂P,θr=∂Br∩∂P,θl and θr are sides of Bl and Br, respectively, ω1=∂Ql∩∂Q,γ=∂P∩∂Q, ω3=∂Qr∩∂Q are connected, αl and βl are poles of Bl and αr and βr are poles of Br,
o(θl)=o(θr),
Int(P)≠∅, then Φ(∂P) is cyclically reduced.
Call Qclosed 2-banded if t(θl)=t(θr). Let U be a cyclically reduced word in F which represents 1 in A. Denote by 𝒬M(U) the collection of all the 2-banded subdiagrams of M with boundary label U.
We shall show that under certain conditions, there is a boundary region of M0 which we can transfer from M0. We wish to repeat the process of transferring until either no regions remain in M0 or M0 has a special structure. The next proposition shows that ℐ-moves preserve these “certain conditions”, hence we can repeat transferring regions as long as there are regions in M0J, or M0J does not have the special structure. Here M0J is the result of applying J on M0, J a sequence of ℐ-moves.
Definition 3.2.5
(Standard ℐ-moves). Say that an ℐ-move J is standard if θ is a maximal piece in the sense that θ in Figure 5 (b) cannot be extended as a common boundary of D and B.
Proposition 3.2.6.
Let W be a cyclically reduced word in F, W≠1, which represents 1 in A, and let M∈ℳ3(W). Let Q be a subdiagram of M with cyclically reduced boundary label U and suppose that Q∈𝒬M(U). (Thus Q is 2-banded.) Let notation be as in [6, Figure 33 (c)].
Assume that P contains a boundary region Δ∈Reg4+,t(P) or D∈Reg2(P) such that ∂Δ∩∂P or ∂D∩∂P, respectively, are connected and ∂Δ∩∂P=∂Δ∩θ, or ∂D∩∂P=∂D∩θ, θ=θr or θ=θl such that ∥∂Δ∩θ∥≥2, or ∥∂θ∩∂D∥≥2. Let I be the standard ℐ-move on B∪Δ, or B∪D, respectively, where B=Br or B=Bl according as θ=θr or θ=θl, respectively, which transfers D or Δ, respectively, from Q. Then QI∈𝒬MI(U). See the proof of Proposition 3.2.6 in [6].
Lifting <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:mi>𝒟</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mover accent="true"><mml:mi>𝕄</mml:mi><mml:mo>~</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math><![CDATA[\(\mathcal{D}(\tilde{\mathbb{M}}^{t})\)]]></tex-math></alternatives></inline-formula> to <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:mrow><mml:msub><mml:mi>𝕄</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math><tex-math><![CDATA[\(\mathbb{M}_{t},t\in T(M)\)]]></tex-math></alternatives></inline-formula>, and the proofs of the main results
The idea behind showing that 𝕄~t satisfies conditions C(4) and T(4) is to use it in M. Obviously, we cannot lift conditions C(4) and T(4) from 𝕄~t to M; however, we can lift a version of Greendlinger’s lemma. This is the content of Theorem C. The first step towards such a lifting is to establish the basic connection between 𝕄~t and 𝕄t. We consider 𝕄t as a map and consider (𝕄t)1, its 1-skeleton. Following the definition of 𝕄~t, we consider (𝕄~t)1 as a quotient of (𝕄t)1. Then we extend this from the 1-skeleton to the whole map and show that |Reg4+,t(𝕄t)|=|Reg(𝕄~t)|. Since 𝕄~t satisfies conditions C(4) and T(4), this allows us to estimate |Reg4+,t(𝕄)|.
Let M be a connected, simply connected ℛ-diagram, M∈ℳ3(W) and t∈T(M). Consider the diagram 𝕄t (defined in Section 2.1). Assume first that it has connected interior. Denote by (𝕄t)1 the 1-skeleton of 𝕄t. Define an equivalence relation “≈1” on the vertices of (𝕄t)1 to be the transitive closure of the relation “∼1” defined as follows: vertices v1 and v2 satisfy v1∼1v2 if either v1=v2 or there is an edge e labelled by a letter from Ht (Ht-edge) with ι(e)=v1 and τ(e)=v2. Here ι(e) denotes the initial vertex of e and τ(e) denotes the terminal vertex of e. Edges e1 and e2 satisfy e1∼1e2 if e1=e2. Then by the definition of 𝕄t, (𝕄t)1/≈1 is isomorphic to (𝕄t)1.
Let Et be the collection of all the edges of 𝕄t which are labelled by t±1 and define EHt accordingly.
Denote the projection mapping of (𝕄t)1 which realises ≈1 by ϕt:(𝕄t)1→(𝕄t)1. Then ϕt sends a vertex v in (𝕄t)1 to the vertex ϕt(v) which we denote by vt. If e∈Et((𝕄t)1), then ϕt(e) is an edge which we denote by et. If e∈EHt((𝕄t)1), then et:=ϕt(e) is a loop with ι(e)=τ(e), which we remove. We can do this because there is no interrelation between a loop and an edge (or loop). So we define ϕt(e)=∅ and ϕt(e¯)=v1t=v2t, where ι(e)=v1 and τ(e)=v2. We define ϕt−1:(𝕄t)1→(𝕄t)1 by its set theoretical definition as follows.
Let vt be a vertex in 𝕄t,vt=ϕt(v),v a vertex of 𝕄t.
Define ϕt−1(vt)={w∣w is Ht-connected to v}.
If et is an edge in (𝕄t)1, then ϕt−1(et) is the unique edge e∈Et((𝕄t)1) such that ϕt(e)=et. (Notice that e is necessarily in Et((𝕄t)1).)
If Int(𝕄t) is not connected, then consider each connected component individually.
Consider now 𝕄~t. Assume first that it has connected interior. Define an equivalence relation “≈2” on (𝕄t)1 to be the transitive closure of the relation “∼2” defined as follows:
Vertices v1t and v2t of (𝕄t)1 satisfy v1∼2v2 if and only if v1=v2.
Edges e1t and e2t satisfy e1t∼2e2t if and only if τ(e1t)=τ(e2t) and ι(e1t)=ι(e2t).
It follows from the definition of 𝕄~t that (𝕄t)1/“≈2” is isomorphic to (𝕄~t)1.
Denote by ξt the projection from (𝕄t)1 onto (𝕄~t)1 which realises ≈2. Then we have
if vt∈V((𝕄t)1), then ξt(vt)=v~,v~ a uniquely defined vertex in 𝕄t~1,
if et∈E((𝕄t)1), then ξt(et)=e~,e~ a uniquely defined edge in 𝕄t~1.
We define ξt−1:𝕄~t→𝕄t as follows:
If v~∈V((𝕄t~)1), then ξt−1(v~)=vt, where vt is the unique vertex in (𝕄t)1 with ξt(vt)=v~.
If e~∈E((𝕄t~)1), then ξt−1(e~)={et∈𝕄t∣ι(et)=ι(e0t) and τ(et)=τ(e0t)}, where ξt(e0t)=e~.
Finally, define Ψt:(𝕄t)1→(𝕄~t)1 by Ψt=ξt∘ϕt. Then Ψt−1=ϕt−1∘ξt−1. It follows by checking Euler characteristics that 𝕄~t is planar. We have the following.
Lemma 4.1.2.
The following statements hold:
Lete∈Et((𝕄t)1), and letBe1be the1-skeleton of thet-band which containse. ThenΨt−1∘Ψt(e¯)=Be1. In particular,e¯⊆Ψt−1∘Ψt(e¯).
Lete∈Et((𝕄t)1). ThenΨt−1∘Ψt(e)=e.
Letv∈V((𝕄t)1). ThenΨt−1∘Ψt(v)=[v]Ht. In particular,v∈Ψt−1∘Ψt(v).
Letv~∈V((𝕄~t)1). ThenΨt∘Ψt−1(v~)=v~.
Ψt(ι(e))=ι(Ψt(e))andΨt(τ(e))=τ(Ψt(e))for every edgee∈Vt((𝕄t)1).
Proof.
Immediate by the definitions of Ψt and Ψt−1.∎
Proposition 4.1.3.
Let notation and assumptions be as above. Then |Reg4+,t(𝕄t)|=|Reg(𝕄~t)|.
Let M∈ℳ3(W) and assume Int(M) is connected. In general, this does not imply that 𝕄~t has connected interior. In the rest of the work, we aim to prove Theorems A–C. It is enough to prove these theorems for every connected component of Int(𝕄~t). Hence without loss of generality, we may work under the assumption that 𝕄~t has connected interior, and we shall do so. Thus for the rest of the work, we shall assume that 𝕄~t has connected interior, unless said otherwise.
Definition 4.2.1
(Opt(Δ),Optr(Δ),Optl(Δ),Opt^(Δ)). Suppose Δ~∈𝒟(𝕄~t), and let Δ=Ψt−1(Δ~). Assume ∂Δ~∩∂𝕄~t=v~1e~1v~2e~2v~3e~3⋯v~k,k≥2, v~i vertices, e~i edges. Define Opt(Δ)=Ψt−1(e~1v~2⋯e~k−1), define Optr(Δ)=Ψt−1(e~1v~2⋯e~k−1v~k) and define Optl(Δ)=Ψt−1(v~1e~1v~2⋯e~k−1). Thus Optr(Δ)=Opt(Δ)∪Ψt−1(v~k) and Optl(Δ)=Opt(Δ)∪Ψt−1(v~1). Define Opt^(Δ)=Ψt−1(v~1e~1⋯v~k). It follows from the definition of ψt−1 that Opt(Δ)=Ψt−1(e~1)∪Ψt−1(v~2)∪Ψt−1(e~2)∪⋯∪Ψt−1(e~k−1)=B1∪Z2∪B2∪Z3∪⋯∪Bk−1, where Bi are t-bands emanating from ∂Δ∩∂Opt(Δ) and Zi are connected unions of regions from RegHt(𝕄t).
Thus Δ∪Opt(Δ), Δ∪Optr(Δ),Δ∪Optl(Δ) and Opt^(Δ) are connected subdiagrams of 𝕄t. Due to Proposition 2.1.7, ∂M∩∂Bi,∂Zi∩∂Δ and ∂Bi∩∂Δ are connected, and it is easy to see that ∂Δ∩∂Opt(Δ),∂Δ∩∂Optr(Δ),∂Δ∩∂Optl(Δ) and ∂Δ∩∂Opt^(Δ) are connected.
We are interested in Optr(Δ)(Optl(Δ)) for the special case when v~k(v~1) has valency 3 in 𝕄~t.
Lemma 4.2.2.
Let M∈ℳ3(W). Assume that ∂M decomposes by ∂M=uω1wω2−1, u,w vertices and t∈T(M). Let Δ~1t and Δ~2t be boundary regions of 𝕄~t, Δ~1t and Δ~2t in N(ω~1),Δi=Ψt−1(Δ~it),Supp(Δi)={a,t},ω~1=Ψt(ω1). Assume that ∂Δ~1∩∂Δ~2∩∂ω~1 is a vertex v~t with valency 3 in 𝕄~t. Let Q=Ψt−1(v~), and let P=Ψt−1(N(v~)). Thus P=Ψt−1(Δt~1∪Δ2t~∪(∂Δt~1∩∂Δt~2)∪e~k∪f~1∪v~), where e~k and f~1 are the boundary paths of Δ~1t and Δ~2t with common endpoint v~. Then each of the following holds:
Qis connected and simply connected with cyclically reduced boundary label such that each connected componentQi,i=1,…,kof Q has cyclically reduced boundary labelsUi. Moreover,Qi∈ℳ3(Ui),i=1,…,k.
P is connected and simply connected with connected interior and cyclically reduced boundary label V, P∈ℳ3(V).
Proof of Lemma <xref rid="j_GGD950_stat_004_029" ref-type="statement">4.2.2</xref>.
See the proof of Lemma 5.2.2 in [6].∎
Definitions and assumptions
Generalised Greendlinger regions. In our earlier work [6, Section 1.3], we introduced (classical) Greendlinger regions. In the rest of the work, we shall need more general notions of Greendlinger regions. Recall from [6, Section 1.3] 𝒟i(M),i=1,2, and 𝒟(M) that M is a connected, simply connected map with connected interior which contains at least two regions.
If ∂M=uμvν−1,μ and ν boundary paths, we shall call Mbigonal with sides μ and ν. We define 𝒟i(μ),𝒟i(ν),i=1,2 and 𝒟(μ) and 𝒟(ν) as follows.
Definitions 4.3.1.
Let M be a bigonal map with sides μ and ν. Define
𝒟j(μ)={D∈Reg(M),∂D∩μ is connected and i(D)=j}j=1,2. 𝒟j(ν) is defined accordingly. Furthermore, define 𝒟(μ)=𝒟1(μ)∪𝒟2(μ)∪𝒟0(μ), where 𝒟0(μ) is defined only if M={D}. In this case, 𝒟0(μ)={D} if |μ|≥|ν| and 𝒟0(ν)={D} if |ν|≥|μ|. In case |μ|=|ν|, we choose 𝒟0(μ)={D}.
Let M be an ℛ-diagram, ℛ given by (1.1). Assume M∈ℳ3(W) and ∂M=μν−1.
Define
𝔻(μ)={Δ∈Reg4+(𝕄)∣∂Δ∩μ is connected,∥∂Δ∩μ∥≥n(Δ)and |Φ(ξ)|≤|Φ(∂Δ∩μ)|}.
Here, ξ is the complement of ∂Δ∩μ on ∂Δ. Define 𝔻(ν) accordingly.
Define
𝒟v(μ)={Δ∈Reg4+(𝕄)∣Δ contains a subdiagram S with S∈O.L.(θ1,θ2),S⊆N(μ),∂S∩μ=θ1,∥θ1∥≥n(Δ)≥4and |Φ(θ2)|≤|Φ(θ1)|}.
Define 𝒟v(ν) accordingly.
Define
𝒟21(μ)={D∈𝒟1(μ),n(D)=2}, 𝒟22(μ)={D∈𝒟2(μ),n(D)=2}.
Define σ:boundary paths→{0,1}
as follows:
σ(μ)=1 if one of the following holds:
𝒟21(μ)≠∅.
|𝒟22(μ)|≥2.
|𝒟22(μ)|=1, and there is a region Δ in 𝕄4+ and a sequence J of ℐ-moves such that ∂ΔJ∩μ is connected and |∂ΔJ∩μ|≥2.
𝒟22(μ)=∅, and 𝔻(μ)∪𝒟v(μ)≠∅.
In all other cases, σ(μ)=0.
Let μ~=Ψt(μ). Define
𝒟′(μ~)={Δ~∈Reg(𝕄~t)∣Δ∈𝒟v(μ),Δ=Φt−1(Δ~)}.
Assumptions 4.3.2.
The proofs of Theorems A–C rely on other results as well (below) which are proved by simultaneous induction. They are
Theorems A–C,
Theorems S, L and T.
Theorem 4.3.3
(Theorem S). Let M∈ℳ3(W) and assume that ∂M decomposes by ω1v1ω2−1v2, where ω1 and ω2 are boundary paths and v1 and v2 are boundary vertices. Let t∈T(M), and let Δ~∈𝒟(ω~1), ω~1=Ψt(ω1), Δ=Ψt−1(Δ~),Supp(Δ)={a,t}. Assume σ(ω1)=0 in parts (a), (b) and (c). Then there is a sequence J of ℐ-moves such that ΔJ=Δ,MJ∈ℳ3(W), and each of the following holds:
If β is an edge of (∂Δ∩∂Opt^(Δ))J, then there is a band which has a pole on β and the other pole on ω1. In particular, (∂Δ∩Bω1)J is connected.
If B1 and B2 are bands emanating from (∂Δ∩Bω1)J with adjacent poles on ∂Δ, then ∂B1∩∂B2 is connected. Moreover, let L1,…,Lk be the bands that emanate from (∂Δ∩Bω1)J. Then ∂Li∩∂Li+1 is connected, i=1,…,k−1, where Li and Li+1 have adjacent poles on ∂Δ.
Let Z:=Opt^(Δ)J∪ω1∖{L1,…,Lk}=Q1∪˙⋯∪˙Ql, where Qi are the connected components.
∂Qiare simple closed curves and ∂Qi decompose by ∂Qi=ρiξiγi−1, where ρi=∂Qi∩∂Lj, ξi=∂Qi∩∂Lj+1 and γi=∂Qi∩ω1, j∈{1,…,k},i=1,…,l.
Up to diamond moves, Ui:=Φ(∂Qi) is cyclically reduced and Qi∈ℳ3(Ui),i=1,…,l, and either Qi={Di},n(Di)=2 or Qi is transferable via Lj∪Lj+1. See Figure 6.
For every D∈Reg(Qi),i=1,…,l,Supp(Φ(∂D))⊆X∖{a,t}.
Let η=(∂Δ∩Bω1)J. Then the following hold:
If Δ∈𝔻(η), then σ(ω1)=1.
If 𝕄~t∈O.L.,𝕄~t=⟨Δ~1,…,Δ~m⟩, m≥2 and Δj∈𝒟v(η), for j=1 and j=m, then σ(ω1)=1.
The possible configurations of the adjacent bands in Opt(Δ) emanating from Δ.
Remarks
(Pushing up Δ).
It follows from part (c) that if 𝒟22(ω1)=∅, then Opt(Δ) is a homogeneous band-bundle, hence we can push up Δ by a sequence J1 of ℐ-moves to ω1, such that Δj∈𝔻(μ), obtaining the classical result on Greendlinger regions.
The same results hold if we use ν in place of μ.
Theorem 4.3.4
(Theorem L). Let M∈ℳ3(W), ∂M=uω1vω2−1,u and v vertices, ω1 and ω2 boundary paths of M. Assume T(M)≠∅, and let t∈T(𝕄). Let ω~1=Ψt(ω1), and let Δ~∈Reg(𝕄~t). Let Δ=Ψt−1(Δ~), and let η=∂Δ∩Bω1. Let Supp(Δ)={a,t}.
IfΔ~∈𝒟(ω~1), thenσ(ω1)=1,∥η∥a≥2and∥η∥t≥2.
IfΔ~∈𝒟′(ω~1), thenσ(ω1)=1,∥η∥a≥2and∥η∥t≥2.
Theorem 4.3.5
(Theorem T). Let M∈ℳ3(W), and let P be a simply connected, connected subdiagram of M with at least one region. Assume that P is closed 2-banded in M by bands Br and Bl. If ∂P has cyclically reduced label U and P∈ℳ3(U), then P is transferable. In particular, there is a sequence J of ℐ-moves such that BrJ∩BlJ is connected and BrJ∪BlJ is a band-bundle. Moreover, |BrJ|=|Br| and |BlJ|=|Bl|.
In each of these theorems, M∈ℳ3(W) and ∂M=uμvν−1, u and v vertices. The induction hypothesis claims that if N is a proper subdiagram of M with cyclically reduced boundary label UN such that N∈ℳ3(UN), then the corresponding result holds for N. We shall denote these hypotheses by ℋ(A),ℋ(B),ℋ(C), ℋ(S),ℋ(L) and ℋ(T), respectively. So we assume in the sequel that
ℋ(A),ℋ(B),ℋ(C),ℋ(S),ℋ(L) and ℋ(T) hold.
The proof of Theorem <xref rid="j_GGD950_stat_006" ref-type="statement">A</xref>Lemma 4.4.1.
Let W be a cyclically reduced word in F , W=1 in A and W≠1 in A(Γ2). Let M be a van Kampen ℛ-diagram over F, ℛ given by (1.1). Assume that M∈ℳ1(W) and all bands are adequate. Let Δ1 and Δ2 be regions in Reg4+(𝕄), with Supp(Δ1)=Supp(Δ2)={a,t},Δ1≠Δ2. Assume that there are bands B1 and B2 connecting adjacent edges e1v1e2 of ∂Δ1 with adjacent edges f1w1f2 of ∂Δ2, v1 and w1 are vertices such that Φ(e1)=a and Φ(e2)=t. Then M∉ℳ2(W). In particular, if M∈ℳ3(W), then the poles of B1 and B2 cannot be adjacent both on ∂Δ1 and on ∂Δ2.
Proof.
See the proof of Lemma 5.4.1 in [6].∎
Lemma 4.4.2.
Let M∈ℳ3(W) and assume T(M)≠∅. Let t∈T(M). Then
𝕄~thas no inner vertices with valency2,
𝕄~tsatisfies conditionC(4).
Proof.
See the proof of Lemma 5.4.2 in [6].∎
Proof of Theorem <xref rid="j_GGD950_stat_006" ref-type="statement">A</xref>
If 𝕄~t does not satisfy condition T(4), then due to Lemma 4.4.2, 𝕄~t contains an inner vertex v~ with valency 3. Let Δ~1,Δ~2 and Δ~3 be the regions of 𝕄~t which contain v~ on their boundary. Let Δi=Ψt−1(Δ~i),i=1,2,3, and let Q=Ψt−1(v~). It is not difficult to show by arguments similar to those used in the proof of Lemma 4.2.2 that ψt−1(⋃i=1i=3Δ~i) is connected and simply connected, and ∂Q=α1θ1α2θ2α3θ3,αi=∂Δi∩∂Q and θi are sides of bands Bi,θi=∂Bi∩∂Q,i=1,2,3, and αi is either a point or a path with label aiki,ki≠0,Supp(Δi)={t,ai},i=1,2,3. Consider U:=Φ(∂Q). Since Supp(Φ(θi))∈CΓ(t) while if αi is not a point, then αi is labelled with aiki∉CΓ(t), the only occurrences of letters in U which do not commute with t are the non-trivial labels of αi. In particular,
there are at most three disjoint occurrences of letters in Uwhich do not commute with t}.
We claim that if αi and αj are not points, then ai=aj. Suppose not. By [6, Lemma 1.1.c], there are at least two disjoint occurrences of ai and two disjoint occurrences of aj in U, hence U has at least four such occurrences violating (*). Hence ai=aj. Denote a=ai=aj. It is easy to see that U is cyclically reduced and Q∈ℳ3(U). In particular, Theorem B applies to Q (|Q|<|M|). Hence if D is a region of Q with a∈Supp(D), then n(D)≤3, due to (*). Since n(D)≠3, hence n(D)=2. Hence Q is a-abelian. Consequently, if αi is not a point, then from each edge of αi, there emanates a band L in Q which has one of its poles on αi and the other on αj. Observe that L and Bi have adjacent poles on Δi and on Δj. But this violates Lemma 4.4.1. Hence all αi are points. But then since three is odd, the three changes of orientations of the poles of βi,i=1,2,3, force t=t−1. This violates [6, Lemma 1.1 (a)]. Hence d(v~)≠3. It follows from Lemma 4.4.2 (b) that 𝕄~t satisfies conditions C(4) and T(4).∎
Preparatory results for the proofs of Theorems <xref rid="j_GGD950_stat_004_003_003" ref-type="statement">S</xref>, <xref rid="j_GGD950_stat_004_003_004" ref-type="statement">L</xref>, <xref rid="j_GGD950_stat_004_003_005" ref-type="statement">T</xref> and <xref rid="j_GGD950_stat_007" ref-type="statement">B</xref>Lemma 4.5.1.
Let M∈ℳ3(W) be a connected, simply connected ℛ-diagram with connected interior. Suppose ∂M=u1ω1u2ω2−1 and T(M)≠∅, where u1 and u2 are vertices.
Let t∈T(M) and consider 𝕄~t. Assume Reg(M~t)={Δ~}. Let Δ=Ψt−1(Δ~), Supp(Δ)={t,a} and define Optωi(Δ):=Ψt−1(ω~i),i=1,2. Assume TheoremSholds for M. Then σ(ωi)=1 for i=1 or i=2, and ∥ωi∥t≥2.
Proof.
See the proof of Lemma 5.5.1 in [6].∎
Lemma 4.5.2.
Let M∈ℳ3(W) with boundary cycle μuν−1v, where u and v are vertices. Assume that T(M)≠∅, and let t∈T(M). Let Δ~∈Reg(𝕄~t), and let Δ=Ψt−1(Δ~). Let η=(∂Δ∩∂Opt^(Δ))∩Bμ, and let η0=(∂Δ∩∂Opt(Δ))∩Bμ. Assume η is connected. Then in each of the following cases, Δ∈𝔻(η) and ∥Φ(μ)∥t≥2:
Δ~∈𝒟1(μ~).
Δ~∈𝒟2(μ~), and there is anHt-bandLemanating fromη∖η0and ending onμ.
Moreover, in case (a), ∥Φ(μ)∥t≥3,
Proof.
See the proof of Lemma 5.5.2 in [6].∎
Remark.
Lemma 4.5.2 remains true if we replace μ by ν.
Lemma 4.5.3.
Let M∈ℳ3(W) be a connected, simply connected ℛ-diagram over F with connected interior. Let ∂M=v1ω1v2ω2−1, v1 and v2 vertices and ω1 and ω2 boundary paths. Let t∈T(M) and assume that |𝕄~t|≥2 and 𝕄~t∈O.L.(ω1~,ω2~),ωi~=Ψt(ωi),𝕄~t=⟨Δ~1,…,Δ~k⟩,k≥2. If σ(ω1)=0 and TheoremSholds for M, then each of the following holds:
σ(∂Opt(Δi)∩Bω2)=1,i=1,k.
∥ω2∥t≥3.
Proof.
See [6, pp. 98–99].∎
Lemma 4.5.4.
Let notation and assumptions be as in Lemma4.2.2. Let ηi=∂Δi∩Bωi, i=1,i=2. Let ω1=γ and ω2=θ1α1θ3α2θ2. (Thus |Q|<|M|, and ω1uω2−1 is a boundary cycle of Q.) If σ(ω1)=0, then Q is a-abelian.
Proof.
See the proof of Lemma 5.5.4 in [6].∎
Corollary.
Let notation and assumptions be as in the lemma. If Δ~∈𝒟2(μ~), then Δ∈𝔻(μ).
Proof.
Due to the lemma, condition (b) of Lemma 4.5.2 is satisfied. Hence the result follows from part (b) of Lemma 4.5.2.∎
Proof of Theorem <xref rid="j_GGD950_stat_004_003_003" ref-type="statement">S</xref>
Let τ1w1αw2τ2 be a subpath of ∂Δ∩Opt(Δ), w1 and w2 vertices, Φ(τ1)=t±1, Φ(τ2)=t±1, Φ(α)=af,f≠0. It follows from the definition of Opt(Δ) that there are t-bands L1 and L2 emanating from τ1 and τ2, respectively, and ending on ∂M. Let θ1 be the side of L1 which has w1 as an endpoint, and let θ2 be the side of L2 which has w2 as an endpoint. Then Supp(Φ(θi))⊆CΓ(t), i=1 and i=2. Let z1 and z2 be the endpoints of θ1 and θ2, respectively, on ∂M, and let γ be the subpath of ∂M∩∂Opt(Δ) which starts at z1 and ends at z2. Then σ(γ)=0 as γ⊆ω1 and σ(ω1)=0. Also, U:=Φ(θ1αθ2γ−1) is cyclically reduced, since a∉CΓ(t). Let Q be the bounded component of 𝔼2∖(θ1αθ2γ−1). Assume first that ∂Q(=θ1αθ2γ−1) is a simple closed path. Hence Int(Q) is connected. It follows from [6, Lemma 2.2.1] that Q∈ℳ3(U). Hence Theorem A applies to Q, and it follows from the proof of Lemma 4.5.4 that if Q is not a-abelian, then one of the four cases mentioned in the proof of Lemma 5.5.4 in [6] occurs. From this point on, we follow the proof of Lemma 4.5.4, with ∥ω2∥a=1 (here) instead of ∥ω2∥a=2, showing that each case leads to a contradiction, hence Q is a-abelian, more precisely, in the first case, by Lemma 4.5.1, in the second case, by Lemma 4.5.3, in the third case, by Lemma 4.5.2 and in the fourth case, by Theorem L. Therefore, if β=β1⋯βk, Φ(βi)=a±1, then from every βi, there emanates an a-band which ends on γ, since a∉CΓ(t). (Notice that it cannot end on β.) If Int(Q) is not connected, then a similar argument applies to each component.
Let B1 and B2 be bands emanating from ∂Δ∩∂Opt(Δ) and ending on ω1. Assume that their poles on ∂Δ are adjacent. Let ξ1 and ξ2 be the sides of B1 and B2, respectively, with w:=ξ1∩ξ2∩∂Δ≠∅ a point. We claim that ξ1∩ξ2 is connected. Suppose not and consider 𝔼2∖(ξ1∪ξ2). Let P be a bounded connected component of it. Then ∂P=(∂P∩ξ1)∪(∂P∩ξ2). Let V:=Φ(∂P). Then applying diamond moves at the endpoints of ∂P∩ξ1 if needed, we may assume that V is cyclically reduced. It follows from [6, Lemma 2.2.1] that P∈ℳ3(V). Hence Theorem T applies to P (|P|<|M|). Hence by a sequence J0 of ℐ-moves, P is transferable beyond B1 and B2 such that PJ contains no regions. Doing so independently for every connected component Pi, by Ji,i=1,…,r, we get that (B1∩B2)J is connected, where J=J1∘J2∘⋯∘Jr.
Consider now 𝔼2∖{L1,…,Lk}. Let Q1,…,Qm be its bounded connected components.
Assume k=3 and ∂L1∩∂L2 is connected. Assume that for some i,1≤i≤m, Qi occurs between L2 and L3 such that ∂Qi=(∂Qi∩∂L2)∪(∂Qi∩∂L3). Suppose that we would like to transfer a region D from Qi via L2. In principle, it is possible that a previous step of transferring regions D occurred between L1 and L2 and was transferred to Q1. Hence now we transfer it back, and by this we create a loop in the process. The point is that because ∂L1∩∂L2 is connected, we may regard L1∪L2 as a single band, and this enables us to transfer D further beyond L1, out of Opt^(Δ), and, by this, to “empty” (by ℐ-moves J) each Qi, so to get L1J∩L2J and L2J∩L3J simultaneously connected. From this, it is clear how to prove for general k.
(i) This is clear from the construction of Z and the definition of Qi.
(ii) and (iii) It follows from [6, Lemma 2.2.1] that Qi∈ℳ3(Ui). It remains to show that either Qi={D} with n(D)=2 or Qi is transferable. Thus Theorem C is applicable to Qi, implying that σ(ρiξi)=1. (Recall that σ(γ)=0.) Consequently, if Qi is not abelian, then N(ρiξi) contains a transferable region. Hence by applying a finite number of ℐ-moves, we can transfer all the regions D with n(D)≥4 beyond Li and/or Li+1 and remaining either with an abelian diagram Qi′ or with a diagram with 𝒟22(ω1′)=∅, where ω1′ is obtained from ω1 by the appropriate I-move. Applying [6, Lemma 4.1.2], the result follows. Finally, since either every region D in Q is transferable via Lj∪Lj+1 or Q={D}, Supp(∂D)⊆CΓ(a)∪CΓ(t). But {a,t}∩(CΓ(a)∪CΓ(t))=∅. Hence {a,t}∩Supp(D)=∅.
We follow notation of part (c).
(i) By assumption, Δ∈𝔻(η). Hence ∥∂Δ∩η∥≥∥n(Δ)∥≥4. Consider Opt^(Δ). Let {L1,…,Lk} be the bands which emanate from η and end on ω1. Consider L1∪Q1∪L2. If Q1≠{D} with n(D)=2, then we can transfer Q1 beyond L1 and L2 via a suitable sequence J of ℐ-moves such that in MJ, configurations (1), (2), (3) and (4) as in [6, Figure 41] occur. In configuration (1), 𝒟22(ω1)≥2, in configurations (2) and (3), 𝒟22(ω1)≥1, and in configuration (4), 𝒟22(ω1)≥0. Similar arguments apply to Lk−1∪Qk−1∪Lk. Notice that if L1∪Q1∪L2 and Lk−1∪Qk−1∪Lk are in configurations (1), (2) or (3), then |𝒟22(ω1)|≥2. This is part (2) in the definition of σ(ω1)=1. Hence we have to show that if 𝒟22(ω1)≤1, then σ(ω1)=1, which contradicts σ(ω1)=0.
Let L1∪L2 be in configuration i and Lk−1∪Lk in configuration j, 1≤i≤4 and 1≤j≤4. If |𝒟22(ω1)|≤1 then either i=4 or j=4 or both. Assume L1∪L2 is in configuration (4) and Lk−1∪Lk in configuration j≤3. Then L1∪L2 is a homogeneous band-bundle. See [6] for the definition. Let 𝔹0 be the maximal band-bundle that contains L1∪L2. Then we claim that 𝔹0=L1∪⋯∪Lk−1. Thus suppose 𝔹0=L1∪⋯∪Ll, l<k−1 and consider Ll+1. If Reg(Ql)≠∅, then we transfer Ql via Ll∪Ll+2 by a sequence J of ℐ-moves such that Reg(QlJ)=∅. Hence Ll∪LL+1 is in configurations (1), (2) or (3), since |𝒟22(ω1)|=1 implies that l=k−1. Observe that we can push up Δ by a sequence J1 of ℐ-moves along 𝔹0 and ∥∂ΔJ∩ω1∥≥n(Δ)−2≥2. This is part (3) in the definition of σ(ω1)=1. So it remains to deal with the case when both L1∪L2 and Lk−1∪Lk are in configuration (4). Let 𝔹1 be the maximal band-bundle which contains Lk−1∪Lk. Then 𝔹0∪𝔹1=Opt(Δ). If 𝔹0=𝔹1, then Opt(Δ) is a band-bundle, and by pushing up Δ via Opt(Δ), we get ΔJ2∈𝒟(ω1). But then by part (4) of the definition of σ(ω1), σ(ω1)=1. Finally, if 𝔹0=𝔹1, then either ∥η∩∂𝔹0∥≥2 or ∥η∩∂𝔹1∥≥2, and we fulfil part (3) of the definition of σ(ω1), showing that σ(ω1)=1. If one of the Qi consists of a single region D with n(D)=2, then |𝒟22(ω1)|≥1, and the above arguments apply to this case.
(ii) The case when 𝕄~t∈O.L.(ω1,ω2) is dealt with similarly.∎
Proof of Theorem <xref rid="j_GGD950_stat_004_003_004" ref-type="statement">L</xref>
If |𝕄~t|=1, then by Lemma 4.5.1, Δ∈𝔻(η). Assume |𝕄~t|≥2. By Theorem S part (d), it is enough to show that Δ∈𝔻(η). By [6, Lemma 1.3.2], i(Δ~)≤2. If i(Δ~)=1, then by Lemma 4.5.2 (a), Δ∈𝔻(η). If i(Δ~)=2, then by [6, Lemma 1.3.4], ∂Δ~∩ω~1 has an endpoint v~ with valency 3. Let Q=Ψt−1(v~). If σ(ω1)=0, then Q is an a-abelian, by Lemma 4.5.4, hence there is an a-band-bundle emanating from α1 and ending at ω1. (We follow the notation of Lemma 4.5.4.) Consequently, Δ∈𝔻(η), by Lemma 4.5.2 (b), as required. Since n(Δ)≥4, the rest follow.
Let 𝕄~t=⟨Δ~1,…,Δ~k⟩,k≥2 and consider Δ1. By [6, Proposition 1.5.2], ∂S∩∂Δ1 is connected, ∥∂S∩∂Δ1∥≥n(Δ1) and |ξ|≤|∂S∩∂Δ1|, where ξ is the complement of ∂S∩∂Δ1 on ∂Δ1. Consequently, S∈𝔻v(η). See Definition 4.3.1 (iii). Therefore, the result follows from Theorem S part (d). The rest follows from the fact that ∥∂S∩∂Δ1∥≥n(Δ1)≥4, hence ∥∂S∩∂Δ1∥a≥2 and ∥∂S∩∂Δ1∥t≥2, due to the alternating nature (in a and t) of the boundary label of Δ1.∎
Proof of Theorem <xref rid="j_GGD950_stat_004_003_005" ref-type="statement">T</xref>
By induction on |P|. Let ωr=∂P∩∂Br, and let ωl=∂P∩∂Bl. Then uωrvωl−1=∂P, u and v boundary vertices. We start with all the transferable regions from Reg2(P) by applying successively a sequence J1 of ℐ-moves. Let P1=PJ1 be the resulting diagram. Assume Reg(P1)≠∅. If P1∉ℳ3(U1), then we can replace P1 by P1′∈ℳ3(U1) and make the same argument. Hence we may assume that P1∈ℳ3(U1), where U1 is a boundary label of P1, which we may assume to be cyclically reduced. Since we removed only regions of Reg2(M) and this cannot increase |Reg4+(M)| and |Reg4+(𝕄)|, Theorem C applies to P1 and implies that σ(ω1∗)=1 or σ(ω2∗)=1, where ωi∗=ωiJ1,i=1,2. Consequently, there is a region Δ with n(Δ)≥4 or a submap S∈𝔻v(ωi∗) which can be transferred from P1. See Remarks following Theorem S. Let the resulting diagram obtained from P1 be denoted by P2. Notice that |P2|<|P|. Then we may continue transferring from P all the regions Δ with n(Δ)≥4, ending up either with a diagram without regions proving the theorem or with an abelian diagram. In this case, [6, Lemma 4.1.2] completes the proof.∎
Proofs of Theorems <xref rid="j_GGD950_stat_007" ref-type="statement">B</xref> and <xref rid="j_GGD950_stat_008" ref-type="statement">C</xref>Proof of Theorem <xref rid="j_GGD950_stat_007" ref-type="statement">B</xref>.
We start with the proof of Theorem B.
Case <inline-formula><alternatives><mml:math display="inline"><mml:mn>1</mml:mn></mml:math><tex-math><![CDATA[\(1\)]]></tex-math></alternatives></inline-formula>.
T(M)=∅.
Then Reg4+(M)=∅, hence Reg(M)=Reg2(M). Hence if Supp(D)={a,b}, D∈Reg(M), then ∥∂D∥a=∥∂D∥b=2. Now, by [6, Lemma 1.1.1 (c)], ∥∂M∥a≥2. Hence ∥∂M∥a≥∥∂D∥a and ∥∂M∥b≥∥∂D∥b, as required.
Case <inline-formula><alternatives><mml:math display="inline"><mml:mn>2</mml:mn></mml:math><tex-math><![CDATA[\(2\)]]></tex-math></alternatives></inline-formula>.
T(M)≠∅.
Let t∈T and consider 𝕄~t. Recall that for every region Δ in 𝕄t, ∥∂Δ∥t=∥∂Δ~∥ and n(Δ)=n(Δ~), where Δ~=Ψt(Δ). Without loss of generality, we may assume that 𝕄~t has connected interior. Hence due to Theorem A and [6, Lemma 1.3.2], either 𝕄~t={Δ~} or |𝒟(𝕄~t)|≥2. In the first case, the result is clear. Assume, therefore, that 𝒟(𝕄~t)={Δ~1,…,Δ~k}, k≥2. We show that
∥∂𝕄~t∥≥n(Δj),j=1,…,k.
From this, the result follows. We show this for j=1. The proof is the same for j=2,…,k. Recall the notation of Lemma 4.5.2. By (3) in the proof of Lemma 4.5.2, ∥η0∥t≥n(Δj)−i(Δ~j), j=1,…,k. Hence by Theorem A and due to [6, Lemma 1.3.2], the individual contribution of Δj to ∥∂M∥t is at least n(Δj)−2. However, due to possible consolidation of labels of adjacent Δj, the actual contribution of Δj to ∥∂M∥t is at least n(Δj)−3. Observe that
∥∂𝕄∥t≥∥η0∥t.
Consequently, due to (4.2),
∥∂𝕄∥t≥∑j=1k[n(Δj)−3].
Now, ∑j=1k[n(Δj)−3]=n(Δ1)+(∑j=2k[n(Δj)−3])−3≥n(Δ1)+(∑j=2k[4−3])−3=n(Δ1)+(k−1)−3=n(Δ1)+k−4, that is,
∥∂𝕄∥t≥n(Δ1)+k−4(recall that n(Δj)≥4).
Hence if k≥4, then ∥∂𝕄∥t≥n(Δ1), as required. If k=3, then due to [6, Lemma 1.3.2], i(Δ~j)=1,j=1,2,3, hence ∥∂𝕄∥t≥(n(Δ1)−2)+(n(Δ2)−2)+(n(Δ3)−2)≥n(Δ1)+(4−2)+(4−2)−2>n(Δ1) n(\Delta_{1})\)]]>. Finally, if k=2, then by [6, Lemma 1.3.2], i(Δj)=1,j=1,2, hence (n(Δ1)−2)+(n(Δ2)−2)=n(Δ1)+[(n(Δ2)−2)−2]≥n(Δ1), that is, due to (4.3), ∥∂𝕄∥t≥n(Δ1), as required.∎
Proof of Theorem <xref rid="j_GGD950_stat_008" ref-type="statement">C</xref>.
We have to show that either σ(ω1)=1 or σ(ω2)=1. Let t∈T(M) and consider 𝕄~t. By [6, Lemma 1.3.2], either 𝕄~t={Δ~} or |𝒟(𝕄~t)|≥2. If |𝒟(𝕄~t)|≥ 3, then by the Pigeon Hole Principle (P.H.P), either |𝒟(ω~1)|≠∅ or |𝒟(ω~2)|≠∅. Assume |𝒟(ω~2)|=∅. Then |𝒟(ω~1)|≠∅, hence σ(ω1)=1, by Theorem L, as required. If |𝒟(𝕄~t)|=2, then by [6, Lemma 1.3.2], 𝕄~t∈O.L. Hence, again, by Theorem L, σ(ω1)=1. Finally, if 𝕄~t={Δ~}, then by Lemma 4.5.3, σ(ω1)=1.∎
Acknowledgements
I am very grateful to the referee for his important remarks on a previous version of the work, in particular for pointing out several omissions.
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