The BNS-invariant for some Artin groups of arbitrary circuit rank

Kisnney Almeida

doi:10.1515/jgth-2016-0057

Article Publicly Available

The BNS-invariant for some Artin groups of arbitrary circuit rank

Kisnney Almeida

Published/Copyright: January 12, 2017

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From the journal Journal of Group Theory Volume 20 Issue 4

Abstract

We classify the Bieri–Neumann–Strebel invariant Σ1⁢(G) for a class of Artin groups with minimal graphs of arbitrary circuit rank.

1 Introduction

The class of groups G of homotopical type Fm was first studied by C. T. C. Wall and later a homological version, i.e. the class of groups of type FPm, was introduced by Bieri and Eckmann [4]. By definition a group G is of type FPm if the trivial ℤ⁢G-module ℤ has a projective resolution with all projectives finitely generated in dimensions ≤m and G is of type Fm if there is a K⁢(G,1) CW-complex with finite m skeleton. The homological and homotopical Σ-invariants of a group G are monoid versions of the types Fm and FPm for some special submonoids of G associated to non-zero homomorphisms from G to ℝ.

The first Σ-invariant was introduced originally by Bieri and Strebel for the class of metabelian groups in [9] and was used to classify all finitely presented metabelian groups. This definition was later extended by Bieri, Neumann and Strebel to all finitely generated groups [7], forming the invariant now known as Σ1. New homological invariants {Σm⁢(G,ℤ)}m≥1 were defined by Bieri and Renz in [8] and a series of homotopical Σ-invariants {Σm⁢(G)}m≥1 was studied by Renz [13] and Meinert [12]. It is interesting to note that the invariant Σm⁢(G,ℤ) (resp. Σm⁢(G)) classifies which subgroups of G above the commutator have homological type FPm (resp. Fm) provided the group G is of type FPm (resp. Fm), see [8]. As shown by Bieri and Groves in the case of metabelian groups, Σ1⁢(G) has deep links with valuation theory from commutative algebra [6].

The Σ-invariants are calculated for very few groups in all dimensions m, for example the case of the R. Thompson group F was treated by Bieri, Geoghegan and Kochloukova in [5] but the case of the generalised Thompson groups Fn,∞ is known only in dimension m=2 for n≥3 (see [10]). The case of metabelian groups of finite Prüfer rank was solved completely by Meinert [12].

Let 𝒢 be a finite simplicial graph with edges labeled by integer numbers greater than one. The Artin group G associated to 𝒢 has a finite presentation, with generators the vertices V⁢(𝒢) of 𝒢 and relations

u⁢v⁢u⁢⋯⏟n⁢ factors=v⁢u⁢v⁢⋯⏟n⁢ factors⁢ for each edge of ⁢𝒢⁢ with vertices ⁢u,v⁢ and label ⁢n≥2.

A subgraph ℋ of 𝒢 is called dominant if, for each v∈V⁢(𝒢)∖V⁢(ℋ), there is an edge e∈E⁢(𝒢) such that σ⁢(e)=v and τ⁢(e)∈V⁢(ℋ), where σ⁢(e) is the beginning and τ⁢(e) is the end of e.

Let χ:G→ℝ be a non-zero character of an Artin group G with underlying graph 𝒢. An edge e of 𝒢 is called dead if e has an even label greater than 2 and

χ⁢(σ⁢(e))=-χ⁢(τ⁢(e)).

Set ℒℱ=ℒℱ⁢(χ) as the complete subgraph of 𝒢 generated by the vertices v∈V⁢(𝒢) such that χ⁢(v)≠0. Define the living subgraph

ℒ=ℒ⁢(χ)⊂ℒℱ

as the subgraph obtained from ℒℱ after removing the dead edges.

If 𝒢 is connected, the fundamental group π1⁢(𝒢) is free of rank n for some n≥0. In this case we say that G is an Artin group of circuit rankn. It was shown in [2] that for Artin groups G of circuit rank 1

Σ1⁢(G)={[χ]∈S⁢(G)∣ℒ⁢(χ)⁢ is a connected dominant subgraph of 𝒢}.

The case of Artin groups G of circuit rank 2 will be similarly treated in [1]. To prove these results, we use some strategies to reduce the general graph to as few minimal graphs as possible with the same relevant properties. However, even these minimal cases are very hard to work with as the circuit rank of the graph increases. For instance we have treated the case of circuit rank 3 with the full graph on four vertices as underlying graph in [3], which was only possible by applying major restrictions on the labels.

In this paper we prove the description of Σ1 for a class of Artin groups whose underlying graphs are minimal graphs of arbitrary rank. This could be used, together with the techniques we show in [2], to validate the same description to a much larger class of groups. In fact, we use a special case (n=3) of this result to prove the circuit rank 2 case in [1]. This result is also a generalization of [2, Theorem 3.2].

Proposition A.

Let G=G⁢(k1,k2,…,kn,l2,l3,…,ln) be the Artin group with underlying graph

such that ki,li are positive integers, with ki>1 for all i and [χ]∈Hom⁡(G,R) such that -1=χ⁢(u1)=-χ⁢(u). Besides, suppose

(1.1)∑ki⁢ is a power of 212⁢li+1<1.

Then [χ]∈Σ1⁢(G)c.

The proof of Proposition A is quite technical. It is given in section 3, where it is split in many small subsections. By Lemma 1.3, Proposition A is equivalent to “N=ker⁡(χ) is not finitely generated” and the idea of the proof of Proposition A is to find a quotient of N that is not finitely generated.

Write 𝒢 for the underlying graph of the group G above. Using Proposition A, we classify Σ1⁢(G) for the class of groups considered in Proposition A.

Theorem B.

Let G=G⁢(k1,k2,…,kn,l2,l3,…,ln) be an Artin group such that ki,li are positive integers and ki>1 for all i. Then

Σ1⁢(G)={[μ]∈S⁢(G)∣ℒ⁢(μ)⁢ is a connected dominant subgraph of 𝒢}.

1.1 On the Σ-invariants

In this subsection we discuss several results from Σ-theory. Let G be a finitely generated group. By definition S⁢(G) is the set of equivalence classes [χ] of non-zero real characters χ:G→ℝ with respect to the equivalence relation ∼, where χ1∼χ2 precisely when there exists a positive real number r such that χ1=r⁢χ2, i.e. [χ]=ℝ>0⁢χ. Thus S⁢(G) can be identified with the unit sphere Sn-1, where n=dimℚ⁡G/G′⊗ℤℚ. By definition for a finitely generated ℤ⁢G-module A

Σm⁢(G,A)={[χ]∈S⁢(G)∣A⁢ has type ⁢FPm⁢ as left ⁢ℤ⁢[Gχ]⁢-module},

where Gχ={g∈G∣χ⁢(g)≥0}. The homotopical invariant Σm⁢(G) has a more complicated definition. Here we will need only the case m=1, where the situation is much simpler, since for any finitely generated group G we have

Σ1⁢(G)=Σ1⁢(G,ℤ),

where ℤ is the trivial ℤ⁢G-module. For a subset S⊆S⁢(G) we write Sc for the complement S⁢(G)∖S. The following result is a monoid version of the fact that a quotient of a finitely generated group is finitely generated.

Lemma 1.1.

Let π:G→G¯ be a group epimorphism and let χ¯ be a non-trivial character of G¯ such that [χ¯∘π]∈Σ1⁢(G). Then [χ¯]∈Σ1⁢(G¯).

The next result is one of the most important results on Σ-theory and one of the main motivations of the study of Σ-invariants.

Theorem 1.2 ([8]).

Let G be a group of homological type FPm and let H be a subgroup of G containing the commutator [G,G]. Then H is of type FPm if and only if S⁢(G,H):={[χ]∈S⁢(G)∣χ⁢(H)=0}⊂Σm⁢(G,Z).

Finally, the next result is an easy consequence of the application of Bieri–Renz’s Theorem on Artin groups.

Lemma 1.3 ([2, Corollary 2.11]).

Let G be an Artin group and χ a discrete character of G. Then [χ]∈Σ1⁢(G) if and only if ker⁡(χ) is finitely generated.

2 Proposition A implies Theorem B

Note that by [11] if G is an Artin group with underlying graph 𝒢 and μ:G→ℝ a non-zero real character of G such that ℒ⁢(μ) is a connected dominant subgraph of 𝒢, then [μ]∈Σ1⁢(G). Observe that under the assumptions of Theorem B if μ:G→ℝ is a non-zero homomorphism, then μ⁢(u1)=μ⁢(u2)=⋯=μ⁢(un). If the restriction of μ on all the vertices is non-zero and there is not a dead edge, then ℒ⁢(μ)=𝒢, so trivially ℒ⁢(μ) is a connected dominant subgraph of 𝒢 and hence [μ]∈Σ1⁢(G). If the restriction of μ on all the vertices is non-zero and there is a dead edge, then μ⁢(u)=1=-μ⁢(u1)=-μ⁢(u2)=⋯=-μ⁢(un) or μ⁢(u)=-1=-μ⁢(u1)=-μ⁢(u2)=⋯=-μ⁢(un). In this case by Proposition A we have that [μ]∉Σ1⁢(G).

It remains to consider the following cases: μ⁢(u1)=μ⁢(u2)=⋯=μ⁢(un)=0 or μ⁢(u)=0. In both cases ℒ⁢(μ) is a connected dominant subgraph of 𝒢 hence [μ]∈Σ1⁢(G).

Recall that as stated above by [11] if ℒ⁢(μ) is a connected dominant subgraph of 𝒢, then [μ]∈Σ1⁢(G). By the last two paragraphs if L⁢(μ) is not a connected dominant subgraph of 𝒢, then either μ=χ or μ=-χ, where χ is the character considered in Proposition A. Then Lemma 1.3 completes the proof.

3 The proof of Proposition A

The proof of Proposition A is quite technical and is split in several subsections.

3.1 An infinite presentation for N=ker⁡(χ)

By definition G has a presentation

G=〈u,u1,u2,…,un∣⁢(u⁢u1)k1=(u1⁢u)k1,

(u⁢u2)k2=(u2⁢u)k2,…,

(u⁢un)kn=(un⁢u)kn,

(u1⁢u2)l2⁢u1=(u2⁢u1)l2⁢u2,

(u1⁢u3)l3⁢u1=(u3⁢u1)l3⁢u3,…,

(u1un)lnu1=(unu1)lnun〉.

Observe that if k~i≥2 is a divisor of ki for all 1≤i≤n, there is an epimorphism of groups

π:G⁢(k1,k2,…,kn,l2,l3,…,ln)→G⁢(k~1,k~2,…,k~n,l2,l3,…,ln)

induced by the identity on the vertices. Then, by Lemma 1.1, to prove Proposition A we can work with the groups G⁢(k~1,k~2,…,k~n,l2,l3,…,ln) instead of G⁢(k1,k2,…,kn,l2,l3,…,ln), i.e. substitute ki with k~i. Thus we can assume that ki is prime for all i. For a technical reason, if the original ki is not a power of 2, we choose the new ki as a prime factor other than 2.

By Lemma 1.3, it is enough to prove that N:=ker⁡(χ) is not finitely generated. Suppose that N is finitely generated. Our strategy will be to find a finite sequence of quotients of N in such a way that the last one is not finitely generated, leading to a contradiction. Define

x0,i:=u⁢ui,i=1,…,n,y0:=x0,1

which means ui=u-1⁢x0,i for all i. Then N=〈{x0,i}1≤i≤n〉G. Define also

xj,i:=x0,iuj i=1,…,n,j∈ℤ,

which are generators of N as a subgroup of G. We will also use the notation

yj:=xj,1.

Conjugating by powers of u, we can rewrite the relations of G in the following way: For i=1,2,…,n,

(3.1)(ui⁢u)ki=(u⁢ui)ki⇔xj,iki=x0,iki,∀j∈ℤ.

For i=2,3,…,n,

(u1⁢ui)li⁢u1=(ui⁢u1)li⁢ui

⇔xj,i=yj-1-1⁢xj-2,i-1⁢yj-3-1⁢xj-4,i-1⁢⋯⁢yj-2⁢li+3-1⁢xj-2⁢li+2,i-1⁢yj-2⁢li+1-1⁢xj-2⁢li,i-1

⋅yj-2⁢li⁢xj-2⁢li+1,i⁢yj-2⁢li+2⁢xj-2⁢li+3,i

(3.2)⁢⋯⁢xj-3,i⁢yj-2⁢xj-1,i⁢yj for all ⁢j∈ℤ.

Therefore N has the following infinite presentation:

N=〈{xj,i}0≤i≤n,j∈ℤ∣(3.1), (3.2), 1≤i≤n,j∈ℤ〉.

3.2 A special quotient N¯ of N

Define the following quotient of N:

N¯:=〈N∣x0,iki=1, 1≤i≤n〉.

Note that, in N¯,

xj,iki=1,1≤i≤n,j∈ℤ,

by (3.1). Define the groups

Ki:=〈x0,i,x1,i,…,x2⁢li-1,i∣xj,iki=1,j=0,1,…,2⁢li-1〉

=∗0≤j≤2⁢li-1〈xj,i∣xj,iki=1〉≃∗0≤j≤2⁢li-1ℤki,2≤i≤n,

K:=K1=〈{yj}j∈ℤ∣yjk1=1,j∈ℤ〉=∗j∈ℤ〈yj∣yjk1=1〉≃∗j∈ℤℤk1,

M:=∗1≤i≤nKi,Mi:=K∗Ki,2≤i≤n,

D:=×2≤i≤nℤki,A:=K∗D.

Note that by (3.2) {xj,i}j∈ℤ may be seen as a subset of Mi for each i=2,…,n. So we can think of N¯ as a quotient of M. As a quotient of N, we have that N¯ is also finitely generated.

3.3 A technical lemma

By (3.2) we can define elements {xj,i}j∈ℤ⊂Mi. The following is a technical Lemma about these elements.

Lemma 3.1.

For each 2≤i≤n and for all j∈Z, there are unique x~j,i∈Ki and vj,i∈KKi such that

xj,i=x~j,i⁢vj,i.

Note that if j=0,1,…,2⁢li-1, we have x~j,i=xj,i so vj,i=1. Besides, for each 2≤i≤n,

x~j,i=x~j-2,i-1⁢x~j-4,i-1⁢⋯⁢x~j-2⁢li+2,i-1⁢x~j-2⁢li,i-1⁢x~j-2⁢li+1,i

(3.3)⋅x~j-2⁢li+3,i⁢⋯⁢x~j-3,i⁢x~j-1,i,j∈ℤ,

and

vj,i⁢yj-1∈〈{ykK1}0≤k<j〉,j≥2⁢li,

vj,i-1⁢yjx~j,i∈〈{ykKi}j+1≤k≤2⁢li-1〉,j<0.

Proof.

Choose 2≤i≤n. The first part is an easy consequence of the normal form theorem for free products. We obtain (3.3) by applying conjugations on (3.2) and then using the uniqueness of the first part.

Suppose j≥2⁢li. To ease notation, let ζk,i be the product in Ki of the last k factors of the right side of (3.3), so x~j,i=ζ2⁢li,i. Substituting xk,i=x~k,i⁢vk,i in (3.2) and conjugating the factors, from right to left, in such a way that we move the x~k,i to the left, we obtain

xj,i=ζ2⁢li,i⁢(yj-1ζ2⁢li,i)-1⁢(vj-2,iζ2⁢li,i)-1⁢(yj-3ζ2⁢li-1,i)-1⁢(vj-4,iζ2⁢li-1,i)-1

⁢⋯⁢(yj-2⁢li+3ζli+2,i)-1⁢(vj-2⁢li+2,iζli+2,i)-1⁢(yj-2⁢li+1ζli+1,i)-1⁢(vj-2⁢liζli+1,i)-1

⋅(yj-2⁢liζli,i)⁢(vj-2⁢li+1,iζli-1,i)⁢(yj-2⁢li+2ζli-1,i)⁢(vj-2⁢li+3,iζli-2,i)

⁢⋯⁢(vj-3,iζ1,i)⁢(yj-2ζ1,i)⁢(vj-1,i)⁢(yj).

Then the uniqueness of the decomposition gives us the desired result for j≥2⁢li.

For j<0 we use a similar argument. Translating the indexes in (3.3) from j to j+2⁢li and reorganizing the equation we obtain

x~j,i=x~j+1,i⁢x~j+3,i⁢⋯⁢x~j+2⁢li-3,i⁢x~j+2⁢li-1,i⁢x~j+2⁢li,i-1

(3.4)⋅x~j+2⁢li-2-1⁢⋯⁢x~j+4,i-1⁢x~j+2,i-1,j∈ℤ.

By doing the same to (3.2), we obtain

xj,i=yj⁢xj+1,i⁢yj+2⁢xj+3,i⁢⋯⁢xj+2⁢li-3,i⁢yj+2⁢li-2⁢xj+2⁢li-1,i⁢yj+2⁢li

⋅xj+2⁢li,i-1⁢yj+2⁢li-1-1⁢xj+2⁢li-2,i-1⁢yj+2⁢li-3-1

(3.5)⁢⋯⁢xj+4,i-1⁢yj+3-1⁢xj+2,i-1⁢yj+1,i-1,j∈ℤ.

To ease notation, consider ζk,i′ as being the product in Ki of the last k factors on the right side of (3.4), so

x~j,i=ζ2⁢l1,i′.

Then substituting xk,i=x~k,i⁢vk,i in (3.5) and conjugating the factors, from the right to the left, in such a way that we move the x~k,i to the left, we obtain

xj,i=(ζ2⁢li,i′)⁢(yjζ2⁢li,i′)⁢(vj+1,iζ2⁢li-1,i′)⁢(yj+2ζ2⁢li-1,i′)⁢(vj+3,iζ2⁢li-2,i′)

⁢⋯⁢(vj+2⁢li-3,iζli+1,i′)⁢(yj+2⁢li-2ζli+1,i′)⁢(vj+2⁢li-1,iζli,i′)⁢(yj+2⁢liζli,i′)

⋅(vj+2⁢li,iζli,i′)-1⁢(yj+2⁢li-1ζli-1,i′)-1⁢(vj+2⁢li-2,iζli-1,i′)-1⁢(yj+2⁢li-3ζli-2,i′)-1

⁢⋯⁢(vj+4,iζ2,i′)-1⁢(yj+3ζ1,i′)-1⁢(vj+2,iζ1,i′)-1⁢(yj+1)-1.

The result then follows from the uniqueness of the decomposition. ∎

3.4 The epimorphism θ

We will now define an epimorphism of groups

θ:M=K∗(∗2≤i≤nKi)↠A=K*D

such that θ|K=IdK and such that θ projects each Ki onto ℤki. To do that, for each 2≤i≤n define θ as follows:

θ⁢(x0,i)=θ⁢(x1,i)=⋯=θ⁢(x2⁢li-2,i)=1¯∈ℤki,θ⁢(x2⁢li-1,i)=2¯∈ℤki.

By the first part of Lemma 3.1, we have

θ⁢(x~j+2⁢li,i)=-θ⁢(x~j,i)+θ⁢(x~j+1,i)-θ⁢(x~j+2,i)+⋯+θ⁢(x~j+2⁢li-1,i),j∈ℤ,

from which it follows that

θ⁢(x~2⁢li,i)=1¯,θ⁢(x~2⁢li+1,i)=θ⁢(x~2⁢li+2,i)=⋯=θ⁢(x~4⁢li-1,i)=-1¯,

θ⁢(x~4⁢li,i)=-2¯,θ⁢(x~4⁢li+1,i)=-1¯,

(3.6)θ⁢(x~4⁢li+2,i)=θ⁢(x~4⁢li+3,i)=⋯=θ⁢(x~6⁢li,i)=1¯,θ⁢(x~6⁢li+1,i)=2¯,…

and so

θ⁢(xj+4⁢li+2,i)=θ⁢(xj,i),j∈ℤ.

By using the decomposition xk,i=x~k,i⁢vk,i, we obtain

θ⁢(xj,i)ki=(θ⁢(x~j,i)⁢θ⁢(vj,i))⁢(θ⁢(x~j,i)⁢θ⁢(vj,i))⁢⋯⁢(θ⁢(x~j,i)⁢θ⁢(vj,i))⏟ki⁢ terms,θ⁢(xj,i)ki=θ⁢(x~j,i)ki⁢(θ⁢(vj,i)θ⁢(x~j,i)ki-1)⋯⁢(θ⁢(vj,i)θ⁢(x~j,i)2)⁢(θ⁢(vj,i)θ⁢(x~j,i))⁢(θ⁢(vj,i)1),θ⁢(xj,i)ki=(θ⁢(vj,i)θ⁢(x~j,i)ki-1)⁢⋯⁢(θ⁢(vj,i)θ⁢(x~j,i)2)⁢(θ⁢(vj,i)θ⁢(x~j,i))⁢(θ⁢(vj,i)1).

Now define

xi:=1¯=θ⁢(x0,i)=θ⁢(x~0,i)∈ℤki.

Suppose ki≠2. Then, by (3.6), θ⁢(x~j,i) is a generator of ℤki for all j∈ℤ. So, possibly after a permutation of exponents {xiki-1,xiki-2,…,xi2,xi,1} we have

(3.7)θ⁢(xj,i)ki=(θ⁢(vj,i)xiki-1)⁢⋯⁢(θ⁢(vj,i)xi2)⁢(θ⁢(vj,i)xi)⁢(θ⁢(vj,i)1).

If ki=2 we have

(3.8)θ⁢(x~j,i)={eA,if ⁢j≡-2⁢mod⁡ 2⁢li+1,xi,else,

where eA is the neutral element of A. So if j≡-2⁢mod⁡ 2⁢li+1, then

(3.9)θ⁢(xj,i)2=θ⁢(vj,i)2.

Note that this is the only case for which θ⁢(x~j,i)=eA.

3.5 One commutative diagram and the group B

Consider the canonical projection

δ:A↠A〈{θ⁢(xj,i)ki}j∈ℤ,2≤i≤n〉A=:A¯

and the following commutative diagram of group homomorphisms:

where θ¯ is induced by θ. Note that since N¯ is finitely generated, A¯ is finitely generated. Note also that KA is a finite index subgroup of A, since

AKA≃×2≤i≤nℤki=D.

Then

B:=δ⁢(KA)

is a finite index subgroup of A¯, so B is a finitely generated group. Note that A¯/B≃A/KA≃D, furthermore A=KA⋊D, hence A¯=B⋊D. Since x is a generator of ℤk1 and z is a generator of ℤk3 as groups, it follows that

ℤ⁢[D]≃ℤ⁢[x±1,z±1].

As xi is a generator of ℤki for each 2≤i≤n, we have

ℤk1⁢[D]≃ℤk1⁢[x2±1,x3±1,…,xn±1]

as a ℤk1- algebra.

3.6 The abelianization Bab of B

Consider the abelianization Bab of B. The ℤ-module Bab is finitely generated, because B is a finitely generated group. We will think of Bab as a ℤ⁢[D]-module, generated by the classes of

ej:=δ⁢θ⁢(yj),

where the action (denoted by ∘) of D≃A¯/B is induced by conjugation (denoted by ∘). Define

pk⁢(a):=1+a+⋯+ak-2+ak-1

and x^j,i,e^j,v^j,i as being the images in Bab of δ⁢θ⁢(x~j,i),ej,δ⁢θ⁢(vj,i), respectively. By (3.7) we have that, as ℤk1⁢[D]-modules,

Bab≃⊕j∈ℤej⁢(ℤk1⁢[x2±1,x3±1,…,xn±1])I+I′,

in such a way that ej⁢ℤk1⁢[D]≃ℤk1⁢[D], I is the ℤk1⁢[D]-submodule generated by

{v^j,i∘(pki⁢(xi))∣θ⁢(x~j,i)≠eA}

and I′ is the ℤk1⁢[D]-submodule generated by

{ki⁢v^j,i∣θ⁢(x~j,i)=eA}={2⁢v^j,i∣ki=2,θ⁢(x~j,i)}.

3.7 A contradiction: Bab is not finitely generated

Without loss of generality, suppose the elements of {ki}2≤i≤n which are powers of 2 are

k2=k3=⋯=km+1=2,

so, by (1.1),

(3.10)∑2≤i≤m+112⁢li+1<1.

Define

Li={j∈ℤ∣j≡-2⁢mod⁡ 2⁢li+1}.

By (3.7), (3.8) and (3.9), I is the ℤk1⁢[D]-submodule generated by

{v^j,i∘pki⁢(xi)∣(j,i)∈(ℤ×{m+2,m+3,…,n})∪((ℤ∖Li)×{2,3,…,m+1})}

and I′ is the ℤk1⁢[D]-submodule generated by

{2⁢v^j,i∣(j,i)∈Li×{2,3,…,m+1}}.

Now define the ℤk1-algebra

R:=ℤk1⁢[x2,x3,…,xn]({xi+1}2≤i≤m+1∪{pki⁢(xi)}m+2≤i≤n)

(3.11)≃ℤk1⁢[xm+2,xm+3,…,xn]({pki⁢(xi)}m+2≤i≤n),

so that

x2=x3=⋯=xm+1=-1

in R. Note that I′ is contained in the ℤk1-vector subspace of R generated by

⊔(j,i)∈Li×{2,3,…,m+1}v^j,i⁢R,

with the equality holding if 2≠k1. So the ℤk1-vector space

W:=⊕i∈ℤe^i⁢R〈⊔(j,i)∈Li×{2,3,…,m+1}v^j,i⁢R〉,where ⁢e^i⁢R≃R,

is a quotient of Bab and hence finite dimensional. Let

l:=lcm⁡({2⁢li+1}2≤i≤m+1)≥3.

Define also, for each s≥1, the set

Λs:={0,1,…,s⁢l-1}

and the following subspace of W:

Es:=⊕i∈Λse^i⁢R〈⊔(j,i)∈Li×{2,3,…,m+1}v^j,i⁢R〉∩[⊕i∈Λse^i⁢R].

By Lemma 3.1,

v^j,i-e^j∈⊕0≤k<je^k⁢R,if ⁢j≥2⁢li,

v^j,i=0,if ⁢0≤j≤2⁢li-1,

v^j,i-e^j⁢x^j,i∈⊕j+1<k≤2⁢li-1e^k⁢R,if ⁢j<0.

Es=⊕i∈Λse^i⁢R〈⊔(j,i)∈[Li∩Λs]×{2,3,…,m+1}v^j,i⁢R〉.

Note that by (3.11) we have the following equality of ℤk1-vector spaces:

〈v^j,i⁢R〉=〈{v^j,i⁢x2j2⁢x3j3⁢⋯⁢xnjn}0≤ji≤ki-2,2≤i≤n〉.

Then dimℤk1⁡〈v^j,i⁢R〉=dim⁡R=δ, hence

dimℤk1⁡〈⊔(j,i)∈[Li∩Λs]×{2,3,…,m+1}v^j,i⁢R〉≤δ⁢∑2≤i≤m+1♯⁢[Li∩Λs]

≤δ⁢∑2≤i≤m+1s⁢l2⁢li+1

≤δ⁢s⁢l⁢(∑2≤i≤m+112⁢li+1).

As dimℤk1⁡⊕i∈Λsei⁢R=δ⁢s⁢l, we have

dimℤk1⁡Es≥δ⁢s⁢l⁢(1-∑2≤i≤m+112⁢li+1)>0,

where the last inequality comes from (3.10). The above implies

lims→∞⁡dimℤk1⁡Es=∞.

As Es≤W for each s>0, it follows that W is infinite dimensional, which is a contradiction.

Communicated by James Howie

Funding statement: The author was supported by a PhD scholarship first by Capes and later by CNPq, Brazil.

Acknowledgements

The author thanks Dessislava Kochloukova for the valuable ideas and advisoring.

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Received: 2016-4-24

Revised: 2016-11-25

Published Online: 2017-1-12

Published in Print: 2017-7-1

Articles in the same Issue

https://doi.org/10.1515/jgth-2016-0057