Birman’s Conjecture is True for I_{2}(p)

James East

December 22, 2003

Abstract In 1993, Birman conjectured that the desingularization map from the singular braid monoid to the integral group ring of the braid group determined by σ_{i}^{±1} ↦ σ_{i}^{±1} and τ_{i} ↦ σ_{i} − σ_{i}^{−1} is injective. The conjecture, which has recently been proven true by Paris (2003), may be generalised to all Artin groups. In this article we prove that the conjecture holds for one of the infinite families of Artin groups of spherical type, namely I_{2}(p).

1 Introduction

A Coxeter graph Γ is a complete graph on a finite vertex set S, whose edges are labelled from the set {2, 3, …, ∞}. For s, t ∈ S, let m_{st} denote the label on the edge {s, t}, and for convenience let m_{ss} = ∞ for all s ∈ S. The Coxeter group of type Γ is

W_{Γ} = ⟨S | s^{2} = 1 (∀s ∈ S) , ⟨st⟩^{m_{st}} = ⟨ts⟩^{m_{st}} if m_{st} ≠ ∞ (∀s, t ∈ S)⟩.

Here ⟨gh⟩^{m} denotes the alternating product ghg ⋯ with m terms if m ∈ ℕ, or (⟨gh⟩^{−m})^{−1} if −m ∈ ℕ. Let Σ = {σ_{s} | s ∈ S} be a set in one-one correspondence with S. The Artin group of type Γ is

B_{Γ} = ⟨Σ | ⟨σ_{s}σ_{t}⟩^{m_{st}} = ⟨σ_{t}σ_{s}⟩^{m_{st}} if m_{st} ≠ ∞ (∀s, t ∈ S)⟩.

There is a natural surjective homomorphism π : B_{Γ} → W_{Γ} : σ_{s} ↦ s, the kernel of which, P_{Γ}, is known as the pure Artin group of type Γ. Put Σ^{−1} = {σ_{s}^{−1} | s ∈ S} and T = {τ_{s} | s ∈ S}. The singular Artin monoid of type Γ is the monoid SB_{Γ} generated by Σ ∪ Σ^{−1} ∪ T subject to defining relations, for all s, t ∈ S,

σ_{s}^{±1}σ_{s}^{∓1} = 1
σ_{s}τ_{s} = τ_{s}σ_{s}
τ_{s}τ_{t} = τ_{t}τ_{s}  if m_{st} = 2
⟨σ_{s}σ_{t}⟩^{m_{st}} = ⟨σ_{t}σ_{s}⟩^{m_{st}}  if m_{st} ≠ ∞
τ_{s}⟨σ_{t}σ_{s}⟩^{m_{st}−1} = ⟨σ_{t}σ_{s}⟩^{m_{st}−1}τ_{u}  if m_{st} ≠ ∞,

where u = s if m_{st} is even, or u = t if m_{st} is odd. The τ_{s} are known as the singular generators. See [5] for more details. The map π extends to a monoid homomorphism π : SB_{Γ} → W_{Γ} by further defining π(τ_{s}) = s. Analogously, the pure singular Artin monoid of type Γ is SP_{Γ} = π^{−1}(1) = {β ∈ SB_{Γ} | π(β) = 1}.

An important class of Artin groups are those of spherical type (ie. the associated Coxeter group is finite). For a classification of finite Coxeter groups see for example [9].

When Γ = A_{n−1}, W_{Γ} = 𝔖_{n} is the symmetric group on n letters [11], B_{Γ} = B_{n} is the n-string braid group [2], and SB_{Γ} = SB_{n} is the n-string singular braid monoid [3, 4].

Denote by ℤB_{Γ} the integral group ring of the Artin group B_{Γ}. For s ∈ S, define η(σ_{s}^{±1}) = σ_{s}^{±1} and η(τ_{s}) = σ_{s} − σ_{s}^{−1}. Then η extends to a well defined homomorphism η : SB_{Γ} → ℤB_{Γ}. Birman [4] conjectured that η is injective for Γ = A_{n}. This is simple when n = 1. Járai [10] demonstrated that the conjecture holds when n = 2. Recently, Paris [12] proved the conjecture for all n, and Godelle and Paris [8] have proved the conjecture in the case that Γ is right angled (ie. m_{st} ∈ {2, ∞} for all s, t). With this in mind, it is natural to ask for which other types the conjecture holds. The purpose of this article is to show how to extend the methods of Járai, and demonstrate the injectivity of η when Γ = I_{2}(p) for any 3 ≤ p ∈ ℕ. Along the way we prove some structural results about SB_{Γ} for arbitrary Γ.

I_{2}(p) = s_{1} p s_{2}

2 Preliminary Results

We begin by defining a number of natural homomorphisms which will be used throughout. Define

N : SB_{Γ} → (ℕ, +) : σ_{s}^{±1} ↦ 0 , τ_{s} ↦ 1.

So if β ∈ SB_{Γ}, then N(β) is the number of τ’s in (any word representing) β. Define

ϕ : SB_{Γ} → B_{Γ} : σ_{s}^{±1} ↦ σ_{s}^{±1} , τ_{s} ↦ σ_{s}.

We now introduce some notation, and recall some well known properties of Coxeter groups which may be found in any standard text (eg. [9]). Let S^{∗} denote the set of all words over S. If w, w′ ∈ S^{∗}, we write w ≡ w′ if w and w′ are identical words, and w = w′ if w and w′ represent the same element of W_{Γ}. For w ≡ s_{1} ⋯ s_{k} ∈ S^{∗}, we define ℓ(w) = k, and define

ℓ_{W} (w) = min {ℓ(w′) | w′ ∈ S^{∗} and w = w′}.

So for example, if s ∈ S, then ℓ(ss) = 2 while ℓ_{W} (ss) = 0. If w ∈ S^{∗} and ℓ(w) = ℓ_{W} (w), we say that w is reduced. If w ∈ S^{∗} is reduced and s ∈ S, then either ws (resp. sw) is reduced, or w = w′s (resp. w = sw′) for some reduced w′ ∈ S^{∗} with ℓ(w′) = ℓ(w) − 1. If w ≡ s_{1} ⋯ s_{k} ∈ S^{∗} is reduced, we define w = σ_{s_{1}} ⋯ σ_{s_{k}} ∈ B_{Γ}. If w, w′ ∈ S^{∗} are reduced and w = w′, then w = w′.

For w ∈ S^{∗}, s ∈ S with ws reduced, define

a_{w,s} = wσ_{s}^{2}w^{−1} ∈ P_{Γ}
y_{w,s} = wσ_{s}τ_{s}w^{−1} ∈ SP_{Γ},

and put

A = {a_{w,s} | w ∈ S^{∗} , s ∈ S , ws is reduced}
Y = {y_{w,s} | w ∈ S^{∗} , s ∈ S , ws is reduced}.

Proposition 1 ([6], Corollary 6) The pure Artin monoid P_{Γ} is generated by A. □

The a_{w,s} are calculated explicitly in [6] for some types Γ.

Lemma 2 Let t ∈ S and y ∈ Y. Then there exists y′ ∈ Y and p ∈ P_{Γ} such that

σ_{t}yσ_{t}^{−1} = py′p^{−1}.

Proof Suppose y = y_{w,s} where w ∈ S^{∗}, s ∈ S, and ws is reduced. If tws is reduced, then

σ_{t}yσ_{t}^{−1} = (σ_{t}w)σ_{s}τ_{s}(σ_{t}w)^{−1} = y_{tw,s} ∈ Y

and we are done. So suppose tws is not reduced. Then ws = tw′ for some w′ ∈ S^{∗} with ℓ(w′) = ℓ(ws) − 1. Since both ws and tw′ are reduced, we conclude that wσ_{s} = σ_{t}w′, and so w = σ_{t}w′σ_{s}^{−1}. Now

σ_{t}yσ_{t}^{−1} = σ_{t}wσ_{s}τ_{s}w^{−1}σ_{t}^{−1}
= σ_{t}(σ_{t}w′σ_{s}^{−1})σ_{s}τ_{s}(σ_{t}w′σ_{s}^{−1})^{−1}σ_{t}^{−1}
= σ_{t}^{2}w′σ_{s}τ_{s}(w′)^{−1}σ_{t}^{−2}.

If w′s is reduced, then we are done with y′ = y_{w′,s} ∈ Y and p = σ_{t}^{2} ∈ P_{Γ}. So suppose w′s is not reduced. Then w′ = w′′s for some w′′ ∈ S^{∗} with ℓ(w′′) = ℓ(w′) − 1. Since both w′ and w′′s are reduced, we conclude that w′ = w′′σ_{s}. Thus

σ_{t}yσ_{t}^{−1} = σ_{t}^{2}w′σ_{s}τ_{s}(w′)^{−1}σ_{t}^{−2}  from above
= σ_{t}^{2}(w′′σ_{s})σ_{s}τ_{s}(w′′σ_{s})^{−1}σ_{t}^{−2}
= σ_{t}^{2}w′′σ_{s}τ_{s}(w′′)^{−1}σ_{t}^{−2},

and we are done with y′ = y_{w′′,s} ∈ Y and p = σ_{t}^{2} ∈ P_{Γ}. □

Corollary 3 Let β ∈ B_{Γ} and y ∈ Y. Then there exists y′ ∈ Y and p ∈ P_{Γ} such that

βyβ^{−1} = py′p^{−1}.

Proof Suppose y ∈ Y and β = σ_{s_{1}}^{ε_{1}} ⋯ σ_{s_{k}}^{ε_{k}} where s_{1}, …, s_{k} ∈ S and ε_{1}, …, ε_{k} ∈ {±1}. We prove the result by induction on k. If k = 0 then the result is trivial. The case k = 1 and ε_{1} = 1 is covered by Lemma 2. If k = 1 and ε_{1} = −1, then by the same lemma we have

σ_{s_{1}}^{−1}yσ_{s_{1}} = σ_{s_{1}}^{−2}σ_{s_{1}}yσ_{s_{1}}^{−1}σ_{s_{1}}^{2} = σ_{s_{1}}^{−2}py′p^{−1}σ_{s_{1}}^{2} = (σ_{s_{1}}^{−2}p)y′(σ_{s_{1}}^{−2}p)^{−1}

for some y′ ∈ Y and p ∈ P_{Γ}, and we are done since σ_{s_{1}}^{−2}p ∈ P_{Γ}. Suppose now that k ≥ 2, and put β′ = σ_{s_{1}}^{ε_{1}} ⋯ σ_{s_{k−1}}^{ε_{k−1}}. Then by induction we have

βyβ^{−1} = β′σ_{s_{k}}^{ε_{k}}yσ_{s_{k}}^{−ε_{k}}(β′)^{−1}
= β′py′p^{−1}(β′)^{−1}  for some y′ ∈ Y, p ∈ P_{Γ}
= (β′p(β′)^{−1})β′y′(β′)^{−1}(β′p^{−1}(β′)^{−1})
= (β′p(β′)^{−1})p′y′′(p′)^{−1}(β′p^{−1}(β′)^{−1})  for some y′′ ∈ Y, p′ ∈ P_{Γ}
= [(β′p(β′)^{−1})p′]y′′[(β′p(β′)^{−1})p′]^{−1}

and we are done since (β′p(β′)^{−1})p′ ∈ P_{Γ}. □

The following was mentioned in [10] for type A_{n}. Here we give a proof for arbitrary type Γ.

Proposition 4 The pure singular Artin monoid SP_{Γ} is generated by A ∪ A^{−1} ∪ Y.

Proof Let Z = A ∪ A^{−1} ∪ Y. It is clear that Z ⊆ SP_{Γ}. Now let β ∈ SP_{Γ}. To prove the proposition it suffices to prove by induction on N(β) that β ∈ ⟨Z⟩. If N(β) = 0 then the result is true by Proposition 1. Otherwise, write β = β_{1}τ_{s}β_{2} where N(β_{1}) = 0. By inserting σ_{s}^{−1}σ_{s} after β_{1} (if necessary), we may in fact assume that β = β_{1}σ_{s}τ_{s}β_{2}. Notice that π(β_{1}β_{2}) = π(β) = 1 and that N(β_{1}β_{2}) < N(β) so β_{1}β_{2} ∈ ⟨Z⟩ by an inductive hypothesis. Now

β = β_{1}σ_{s}τ_{s}β_{1}^{−1}β_{1}β_{2} = (β_{1}y_{1,s}β_{1}^{−1})(β_{1}β_{2}).

By Corollary 3, β_{1}y_{1,s}β_{1}^{−1} = py′p^{−1} for some y′ ∈ Y, p ∈ P_{Γ} = ⟨A⟩. Thus β ∈ ⟨Z⟩, completing the proof. □

A Coxeter graph Γ is said to have the FRZ property if for all β ∈ SB_{Γ} and s, t ∈ S, the following conditions are equivalent.

βσ_{s} = σ_{t}β  (FRZ1)
βσ_{s}^{m} = σ_{t}^{m}β for some 0 ≠ m ∈ ℤ (FRZ2)
βτ_{s} = τ_{t}β  (FRZ3)
βτ_{s}^{m} = τ_{t}^{m}β  for some 0 ≠ m ∈ ℕ.  (FRZ4)

The property is named after Fenn, Rolfsen, and Zhu, who proved in [7] that the property holds in SB_{n}. It is now known to be true when W_{Γ} is of FC type [8]. A Coxeter group is of FC type if every subset X ⊆ S which satisfies m_{st} ≠ ∞ for all s, t ∈ X generates a finite subgroup of W_{Γ}. See also [1].

Proposition 5 Suppose Γ is a Coxeter graph which has the FRZ property. Let β_{1}, β_{2} ∈ B_{Γ}, s, t ∈ S, and w_{s}, w_{t} ∈ W_{Γ} such that w_{s}s and w_{t}t are reduced. Then the following statements are equivalent:

β_{1}a_{w_{s},s}β_{1}^{−1} = β_{2}a_{w_{t},t}β_{2}^{−1}  (a)
β_{1}y_{w_{s},s}β_{1}^{−1} = β_{2}y_{w_{t},t}β_{2}^{−1}.  (b)

Proof If (a) holds then we have β_{1}w_{s}σ_{s}^{2}w_{s}^{−1}β_{1}^{−1} = β_{2}w_{t}σ_{t}^{2}w_{t}^{−1}β_{2}^{−1}. Rearranging gives

(w_{t}^{−1}β_{2}^{−1}β_{1}w_{s})σ_{s}^{2} = σ_{t}^{2}(w_{t}^{−1}β_{2}^{−1}β_{1}w_{s}).

By the FRZ property, we have

(w_{t}^{−1}β_{2}^{−1}β_{1}w_{s})σ_{s} = σ_{t}(w_{t}^{−1}β_{2}^{−1}β_{1}w_{s})
(w_{t}^{−1}β_{2}^{−1}β_{1}w_{s})τ_{s} = τ_{t}(w_{t}^{−1}β_{2}^{−1}β_{1}w_{s}),

and so

(w_{t}^{−1}β_{2}^{−1}β_{1}w_{s})σ_{s}τ_{s} = σ_{t}τ_{t}(w_{t}^{−1}β_{2}^{−1}β_{1}w_{s}).

Rearranging gives (b). The other implication is immediate from the ϕ map. □

From now on we will concentrate on the case Γ = I_{2}(p). For simplicity we will write σ_{i} for σ_{s_{i}} and τ_{i} for τ_{s_{i}} (i = 1, 2). We will also denote B_{I_{2}(p)} by B_{I}, and similarly for P_{I}, SP_{I}, etc. So SB_{I} has a presentation with generators σ_{i}^{±1}, τ_{i} (i = 1, 2), and defining relations

σ_{i}^{±1}σ_{i}^{∓1} = 1  for i = 1, 2
σ_{i}τ_{i} = τ_{i}σ_{i}  for i = 1, 2
⟨σ_{1}σ_{2}⟩^{p} = ⟨σ_{2}σ_{1}⟩^{p}
τ_{1}⟨σ_{2}σ_{1}⟩^{p−1} = ⟨σ_{2}σ_{1}⟩^{p−1}τ_{1∧2}
τ_{2}⟨σ_{1}σ_{2}⟩^{p−1} = ⟨σ_{1}σ_{2}⟩^{p−1}τ_{2∧1},

where i ∧ j = i if p is even or i ∧ j = j if p is odd.

For i ∈ {1, …, p − 1} let

a_{i} = ⟨σ_{1}σ_{2}⟩^{i−1}σ_{d}^{2}⟨σ_{1}σ_{2}⟩^{−(i−1)}
y_{i} = ⟨σ_{1}σ_{2}⟩^{i−1}σ_{d}τ_{d}⟨σ_{1}σ_{2}⟩^{−(i−1)}

where d = 1 if i is odd or d = 2 if i is even. Also let

b = σ_{2}^{2}
z = σ_{2}τ_{2}.

It is shown in [6] that A = {a_{1}, …, a_{p−1}, b} and so Y = {y_{1}, …, y_{p−1}, z}, and SP_{I} is generated by A ∪ A^{−1} ∪ Y by Proposition 4. Let U_{I} be the subgroup of P_{I} generated by a_{1}, …, a_{p−1}, and let P_{2} be the (infinite cyclic) subgroup generated by b. The following is from [6] where similar results are proved for other types.

Proposition 6 U_{I} is freely generated by a_{1}, …, a_{p−1}. Further, P_{I} = U_{I} ⋊ P_{2}. The action of b on the a_{i} is given by

ba_{i}b^{−1} = (a_{p−1} ⋯ a_{1})^{−1}a_{i}(a_{p−1} ⋯ a_{1}) for i = 1, …, p − 1,

and these are defining relations for P_{I}. □

For example, when p = 3, the classical three-string pure braid group P_{3} = P_{I_{2}(3)} has a presentation

P_{3} = ⟨a_{1}, a_{2}, b | ba_{1}b^{−1} = (a_{2}a_{1})^{−1}a_{1}(a_{2}a_{1}) , ba_{2}b^{−1} = a_{1}^{−1}a_{2}a_{1}⟩

in terms of generators a_{1} = σ_{1}^{2}, a_{2} = σ_{1}σ_{2}^{2}σ_{1}^{−1}, and b = σ_{2}^{2}.

Let V_{I} be the submonoid of SP_{I} generated by a_{i}^{±1}, y_{i} (i = 1, …, p−1). By Propositions 5 and 6, we have

by_{i}b^{−1} = (a_{p−1} ⋯ a_{1})^{−1}y_{i}(a_{p−1} ⋯ a_{1}) for i = 1, …, p − 1.

Thus, if w is a word in the generators of SP_{I} with no occurrence of the letter z, then we may write w = vp where v ∈ V_{I} and p ∈ P_{2}. To see that this expression is unique (up to equivalence of words), suppose that w = v′p′ is also such an expression. Then ϕ(v)p = ϕ(w) = ϕ(v′)p′. Since ϕ(v), ϕ(v′) ∈ U_{I} and p, p′ ∈ P_{2}, we see that p = p′, and consequently v = v′.

Let β ∈ V_{I}. Since a_{i} and y_{i} commute for each i, we may assume that in β, no a_{i}^{±1} immediately precedes a y_{i}. Thus β may be written as a product of words of the form a_{i}^{ℓ} (with 0 ≠ ℓ ∈ ℤ), and y_{i}^{m}a_{i}^{n} (with 1 ≤ m ∈ ℤ and n ∈ ℤ). We call these words simple words. We may also assume that the subscripts appearing in adjacent simple words are different. As in [10], we call this a freely reduced form of β, which we will shortly see is unique.

3 The Proof of the Conjecture for Type I_{2}(p)

We begin this section by stating some results which are proved in [10]. There they are proved for type A_{n}, but the proofs work unmodified for any Γ.

Define a homomorphism ψ : SB_{Γ} → ℤB_{Γ} by ψ(σ_{i}^{±1}) = σ_{i}^{±1} and ψ(τ_{i}) = σ_{i} + σ_{i}^{−1}.

Lemma 7 (i) If β, γ ∈ SB_{Γ}, then η(β) = η(γ) ⟺ ψ(β) = ψ(γ). Thus η is injective if and only if ψ is injective.

(ii) The image of SB_{Γ} under ψ contains no zero divisors. Thus, if β, γ, δ ∈ SB_{Γ}, then ψ(βδ) = ψ(γδ) ⟺ ψ(β) = ψ(γ) ⟺ ψ(δβ) = ψ(δγ). □

Lemma 8 The map ψ is injective if and only if the restriction of ψ to SP_{Γ} is injective.

Proof Suppose ψ|_{SP_{Γ}} is injective, and take β, γ ∈ SB_{Γ} with ψ(β) = ψ(γ). By looking at the nonzero terms in ψ(β) = ∑_{x∈B_{Γ}} c_{x}x = ψ(γ), we see that π(β) = π(γ). Choose δ ∈ B_{Γ} such that π(δ) = π(β). Then ψ(βδ^{−1}) = ψ(γδ^{−1}). But βδ^{−1}, γδ^{−1} ∈ SP_{Γ}, so that βδ^{−1} = γδ^{−1} and β = γ. □

The following was proved as part of Theorem 3 in [10].

Theorem 9 Let C = {c_{1}, …, c_{n}} and D = {d_{1}, …, d_{n}}. Denote by F (C) the free group (freely) generated by C. Suppose that M is a monoid generated by C ∪ C^{−1} ∪ D and that c_{i}d_{i} = d_{i}c_{i} in M for all i. Suppose also that there is a well defined monoid homomorphism ξ : M → ℤF (C) such that ξ(c_{i}^{±1}) = c_{i}^{±1} and ξ(d_{i}) = c_{i} + 1. Then ξ is injective. □

We again restrict our attention to Γ = I_{2}(p). Notice that ψ(a_{i}^{±1}) = a_{i}^{±1} and that ψ(y_{i}) = a_{i} + 1. So V_{I} is an example of a monoid satisfying the assumptions of the theorem. The proof of Theorem 9 given in [10] makes use of a freely reduced form for elements of M which is defined analogously to that defined above for V_{I}. There it is shown that the freely reduced form of m ∈ M may be constructed simply by knowing ξ(m) = ∑_{x∈F (C)} c_{x}x.

Corollary 10 The restriction of ψ to V_{I} is injective. Further, the freely reduced form of an element of V_{I} is unique. □

So far we know that if ψ(β) = ψ(γ) where β, γ ∈ SP_{I} do not involve b or z, then β = γ. We now allow b to appear, but not z.

Corollary 11 If ψ(β) = ψ(γ) where β, γ ∈ SP_{I} have no appearances of z, then β = γ.

Proof Suppose ψ(β) = ψ(γ) where β = v_{1}p_{1} and γ = v_{2}p_{2} with v_{1}, v_{2} ∈ V_{I} and p_{1}, p_{2} ∈ P_{2}. Since ϕ(β) and ϕ(γ) are the unique monomials of maximal exponent sum in ψ(β) and ψ(γ) respectively, we see that ϕ(β) = ϕ(γ). That is, ϕ(v_{1})p_{1} = ϕ(v_{2})p_{2}. As before, we conclude that p_{1} = p_{2}. But then ψ(v_{1}) = ψ(βp_{1}^{−1}) = ψ(γp_{1}^{−1}) = ψ(v_{2}). By Corollary 10, v_{1} = v_{2} and so β = γ. □

We now proceed to words which may involve z, but not as their first singular generators. Define ε(b) = 1 and ε(a_{i}) = 0 for i = 1, …, p − 1. As a consequence of the defining relations in Proposition 6, ε extends to a well defined homomorphism.

Lemma 12 Suppose that β, γ ∈ SP_{I} with ψ(β) = ψ(γ). Suppose also that N(β) = N(γ) ≥ 1 and that the first singular generator in a word over A ∪ A^{−1} ∪ Y representing β (resp. γ) is y_{i} (resp. y_{j}).Then i = j and there exists α ∈ SP_{I} with N(α) = 1, and β′, γ′ ∈ SP_{I} such that β = αβ′ and γ = αγ′.

Proof Write

β = β_{0}y_{i}β′,  (1)
γ = γ_{0}y_{j}γ′,  (2)

where β_{0}, γ_{0} ∈ P_{I} and β′, γ′ ∈ SP_{I}. By Proposition 6 and the comments following it, we may assume that β_{0}, γ_{0} ∈ U_{I}. We may also assume that any power of a_{i} (resp. a_{j}) at the end of β_{0} (resp. γ_{0}) has been absorbed by β′ (resp. γ′). Replace all generators z appearing in β′ and γ′ by b, and denote the resulting words by β′_{+} and γ′_{+}. Also, let β_{+} = β_{0}y_{i}β′_{+} and γ_{+} = γ_{0}y_{j}γ′_{+}.

Write ψ(β) = ∑_{x∈P_{I}} c_{x}x. We group terms according to their image under ε to obtain

ψ(β) = ∑_{r∈ℤ}( ∑_{{x∈P_{I} \atop ε(x)=r}} c_{x}x).

The inner sum corresponding to the maximal value of r is precisely ψ(β_{+}). Since ψ(β) = ψ(γ), we see that this sum is also ψ(γ_{+}) so that ψ(β_{+}) = ψ(γ_{+}). This implies that β_{+} = γ_{+} by Corollary 11. Now

β_{0}y_{i}β′_{+} = β_{+} = γ_{+} = γ_{0}y_{j}γ′_{+}.

Since both sides do not contain any generator z, we may write

β′_{+} = β′′_{+}p_{1}
γ′_{+} = γ′′_{+}p_{2},

where β′′_{+}, γ′′_{+} ∈ V_{I} are in freely reduced form, and p_{1}, p_{2} ∈ P_{2}. So

β_{0}y_{i}β′′_{+}p_{1} = γ_{0}y_{j}γ′′_{+}p_{2}.

As before, p_{1} = p_{2} and β_{0}y_{i}β′′_{+} = γ_{0}y_{j}γ′′_{+}. By the uniqueness of the freely reduced form, i = j and β_{0} = γ_{0} since y_{i} and y_{j} are the first singular generators appearing on both sides. Now put α = β_{0}y_{i}. Then equations (1) and (2) give the statement in the lemma. □

Lemma 13 If j ∈ {1, …, p − 1}, then σ_{2}y_{j}σ_{2}^{−1} = (a_{p−j−1} ⋯ a_{1})^{−1}y_{p−j}(a_{p−j−1} ⋯ a_{1}).

Proof To see this it is most convenient to consider separate cases depending on the parity of p and j. Here we treat the case when p and j are both even, the others being similar. Now

σ_{2}y_{j}σ_{2}^{−1} = σ_{2}⟨σ_{1}σ_{2}⟩^{j−1}σ_{2}τ_{2}⟨σ_{1}σ_{2}⟩^{−(j−1)}σ_{2}^{−1}
= ⟨σ_{2}σ_{1}⟩^{j}σ_{2}τ_{2}⟨σ_{2}σ_{1}⟩^{−j}
= ⟨σ_{1}σ_{2}⟩^{−(p−j−1)}⟨σ_{1}σ_{2}⟩^{p−j−1}⟨σ_{2}σ_{1}⟩^{j}σ_{2}τ_{2}⟨σ_{2}σ_{1}⟩^{−j}⟨σ_{1}σ_{2}⟩^{−(p−j−1)}⟨σ_{1}σ_{2}⟩^{p−j−1}
= ⟨σ_{1}σ_{2}⟩^{−(p−j−1)}⟨σ_{1}σ_{2}⟩^{p}τ_{2}⟨σ_{1}σ_{2}⟩^{−(p−1)}⟨σ_{1}σ_{2}⟩^{p−j−1}
= ⟨σ_{1}σ_{2}⟩^{−(p−j−1)}⟨σ_{2}σ_{1}⟩^{p}⟨σ_{1}σ_{2}⟩^{−(p−1)}τ_{2}⟨σ_{1}σ_{2}⟩^{p−j−1}
= ⟨σ_{1}σ_{2}⟩^{−(p−j−1)}σ_{2}τ_{2}⟨σ_{1}σ_{2}⟩^{p−j−1}

= ⟨σ_{1}σ_{2}⟩^{−(p−j−1)}⟨σ_{1}σ_{2}⟩^{−(p−j−1)}⟨σ_{1}σ_{2}⟩^{p−j−1}σ_{2}τ_{2}⟨σ_{1}σ_{2}⟩^{−(p−j−1)}⟨σ_{1}σ_{2}⟩^{p−j−1}⟨σ_{1}σ_{2}⟩^{p−j−1}

= (⟨σ_{1}σ_{2}⟩^{p−j−1})^{−2}y_{p−j}(⟨σ_{1}σ_{2}⟩^{p−j−1})^{2}.

We now prove by induction that

(⟨σ_{1}σ_{2}⟩^{k})^{2} = a_{k} ⋯ a_{1}  (∗)

for any odd integer 1 ≤ k ≤ p − 1. This is clear when k = 1. Suppose (∗) holds for some odd integer 1 ≤ k ≤ p − 3. Then

a_{k+2}a_{k+1}a_{k} ⋯ a_{1} = ⟨σ_{1}σ_{2}⟩^{k+1}σ_{1}^{2}⟨σ_{1}σ_{2}⟩^{−(k+1)}⟨σ_{1}σ_{2}⟩^{k}σ_{2}^{2}⟨σ_{1}σ_{2}⟩^{−k}(⟨σ_{1}σ_{2}⟩^{k})^{2}
= ⟨σ_{1}σ_{2}⟩^{k+1}σ_{1}^{2}σ_{2}⟨σ_{1}σ_{2}⟩^{k}
= (⟨σ_{1}σ_{2}⟩^{k+2})^{2}.

The reader is invited to consider the three other cases. □

We are now ready to prove our main result.

Theorem 14 The map ψ : SB_{I} → ℤB_{I} is injective.

Proof Suppose that β, γ ∈ SP_{I} with ψ(β) = ψ(γ) but β ≠ γ. Suppose also that the number N = N(β) is minimal among all such counterexamples. In particular we must have N ≥ 1. Write

β = β_{1}xβ_{2}
γ = γ_{1}x′γ_{2}

where β_{1}, γ_{1} ∈ P_{I}, β_{2}, γ_{2} ∈ SP_{I}, and x and x′ are the first singular generators appearing in β and γ respectively. We break the proof up into a number of cases.

Case 1 Suppose x = y_{i} and x′ = y_{j} for some i, j. We apply Lemma 12. Write β = αβ′ and γ = αγ′ in the notation of the lemma. Then by Lemma 7 Part (ii), ψ(β′) = ψ(γ′). But β′ ≠ γ′, contradicting the minimality of N.

Case 2 Suppose x = x′ = z. Note that σ_{1}zσ_{1}^{−1} = y_{2}. Now ψ(σ_{1}βσ_{1}^{−1}) = ψ(σ_{1}γσ_{1}^{−1}), and

σ_{1}βσ_{1}^{−1} = (σ_{1}β_{1}σ_{1}^{−1})(σ_{1}zσ_{1}^{−1})(σ_{1}β_{2}σ_{1}^{−1}) = β′_{1}y_{2}β′_{2},

where β′_{1} = σ_{1}β_{1}σ_{1}^{−1} ∈ P_{I} and β′_{2} = σ_{1}β_{2}σ_{1}^{−1} ∈ SP_{I}. Similarly, σ_{1}γσ_{1}^{−1} = γ′_{1}y_{2}γ′_{2} with γ′_{1} ∈ P_{I} and γ′_{2} ∈ SP_{I}. By Case 1 we have σ_{1}βσ_{1}^{−1} = σ_{1}γσ_{1}^{−1} and so β = γ, a contradiction.

Case 3 We now consider the case in which x = z and x′ = y_{i} for some i ≠ 2. If i = 1, then note that σ_{1}y_{1}σ_{1}^{−1} = y_{1} and, as in Case 2, we see that

σ_{1}βσ_{1}^{−1} = β′_{1}y_{2}β′_{2}
σ_{1}γσ_{1}^{−1} = γ′_{1}y_{1}γ′_{2},

and we are done by Case 1. Suppose i ≥ 3. Then, again using the notation d = 1 if i is odd, while d = 2 if i is even, we have

σ_{1}y_{i}σ_{1}^{−1} = σ_{1}⟨σ_{1}σ_{2}⟩^{i−1}σ_{d}τ_{d}⟨σ_{1}σ_{2}⟩^{−(i−1)}σ_{1}^{−1}
= σ_{1}^{2}σ_{2}⟨σ_{1}σ_{2}⟩^{i−3}σ_{d}τ_{d}⟨σ_{1}σ_{2}⟩^{−(i−3)}σ_{2}^{−1}σ_{1}^{−2}
= a_{1}σ_{2}y_{i−2}σ_{2}^{−1}a_{1}^{−1}
= a_{1}(a_{p−i+1} ⋯ a_{1})^{−1}y_{p−i+2}(a_{p−i+1} ⋯ a_{1})a_{1}^{−1}  by Lemma 13.

Thus, with notation as in Case 2,

σ_{1}βσ_{1}^{−1} = β′_{1}y_{2}β′_{2}
σ_{1}γσ_{1}^{−1} = γ′_{1}(a_{p−i−1} ⋯ a_{2})y_{p−i+2}(a_{p−i+1} ⋯ a_{2})γ′_{2},

and we are done again by Case 1.

Case 4 Finally we consider the case in which x = z and x′ = y_{2}. Let κ be the automorphism of the graph I_{2}(p) which switches the two vertices. Then κ induces automorphisms of ℤB_{I} and SB_{I} both of which we denote simply by κ, and the following diagram commutes:

SB_{I}  ^{SB_{I}} ℤB_{I}  ℤB_{I} κ κ  ψ ψ

So ψ(κ(β)) = κ(ψ(β)) = κ(ψ(γ)) = ψ(κ(γ)). Notice that κ(z) = y_{1}. But then κ(β) = κ(β_{1})y_{1}κ(β_{2}), and so the first singular generator of κ(β) is y_{1}. Similarly, κ(y_{2}) = σ_{2}y_{1}σ_{2}^{−1} = (a_{p−2} ⋯ a_{1})^{−1}y_{p−1}(a_{p−2} ⋯ a_{1}), so the first singular generator in κ(γ) is y_{p−1}. By Case 1, we have κ(β) = κ(γ) and β = γ. □

Finally we remark that Case 4 may almost be proved by reduction to Case 3. Notice that σ_{1}zσ_{1}^{−1} = y_{2} and σ_{1}y_{2}σ_{1}^{−1} = a_{1}za_{1}^{−1} so conjugation by σ_{1} does not help. However σ_{2}zσ_{2}^{−1} = z, while σ_{2}y_{2}σ_{2}^{−1} = (a_{p−3} ⋯ a_{1})^{−1}y_{p−2}(a_{p−3} ⋯ a_{1}) by Lemma 13, and we may reduce to Case 3 unless p − 2 = 2 (ie. p = 4).

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