Cyclically presented groups with length four positive relators

1 Cyclically presented groups and shift dynamics

One purpose of this article is to provide classifications of finiteness and asphericity (modulo two unresolved cases) for cyclic group presentations where the relators are positive words of length four. More generally, we relate finiteness and asphericity to the dynamics of the shift action that arises from the cyclic symmetry in these presentations, thus extending results from [3].

Let n be a positive integer and let F be the free group of rank n with basis x0,…,xn-1. The shift automorphism is given by θF⁢(xi)=xi+1, where subscripts are taken modulo n. A word w∈F determines the cyclic (group) presentation

which in turn defines the cyclically presented groupG=Gn⁢(w). The shift determines an automorphism θG∈Aut⁢(G). Viewed in Aut⁢(G), the shift θG has order dividing n and may or may not be an inner automorphism of G.

The following two classification problems have received attention in the literature. See [6, 10, 22, 23], for example.

See Section 2 for a discussion of combinatorial asphericity. A group presentation is called aspherical if its two-dimensional cellular model K is aspherical, or equivalently the second homotopy module π2⁢K is trivial. Combinatorial asphericity is a more general concept that allows for the presence of proper power and repeated relators, in which case π2⁢K≠0. See [7] for a comprehensive treatment of asphericity concepts for group presentations. Asphericity and finiteness are related in that any finite subgroup of a group defined by a combinatorially aspherical presentation is finite cyclic [12]. This generalizes the well-known fact that if a two-complex K satisfies π2⁢K=0, then the fundamental group π1⁢K is torsion-free.

In [6, 10], the finiteness and asphericity classification problems were completely solved for cyclic presentations 𝒫n⁢(w) where the defining relator is a positive word of length three. In this article we deal with the length four case. Thus we consider presentations of the form

where the integer parameters j,k,l are to be considered modulo the positive integer n. (Note that the presentation 𝒫n⁢(w) is unchanged if we replace w by any of its shifts.) Modeling an approach developed in [10], our classifications are framed in terms of conditions on the parameters (n,j,k,l) starting with A, B, C as follows; all congruences are considered modulo n:

1. 2⁢k≡0 or 2⁢j≡2⁢l,

2. k≡2⁢j or k≡2⁢l or j+l≡2⁢k or j+l≡0,

3. l≡j+k or l≡j-k,

Table 1 provides an overview of our main results regarding asphericity and finiteness in terms of truth values for conditions A, B, C.

We now discuss the specialized notation in Table 1, where bracketed references [-] are internal to this article. The cyclic group of order n acts on a cyclically presented group G=Gn⁢(w) by automorphisms and so there is the shift extensionEn⁢(w)=Gn⁢(w)⋊θGℤn. This group admits a two-generator two-relator presentation En⁢(w)≅〈a,x|an,W〉, where W is obtained from w via the substitutions xi=ai⁢x⁢a-i, so the shift θG∈Aut⁢(G) arises via conjugation by a in En⁢(w). When w=x0⁢xj⁢xk⁢xl, the relator W takes the form

Conditions A, B, C enable simplifications of relation (1.2) as shown in the right-hand column of Table 1. For example, if k≡2⁢j⁢mod⁡n, as in B, then the substitution u=x⁢aj transforms W to u3⁢al-3⁢j⁢u⁢a-l-j. Simplifications of this sort are developed in Section 5. Section 2 discusses the concept of combinatorial asphericity and its relation to the small cancellation property C⁢(4)-T⁢(4) [15, Chapter V.2]. Section 3 discusses isomorphisms among groups of the form Gn⁢(x0⁢xj⁢xk⁢xl) that arise from an action of the finite group Γn=D4×(ℤn⋊ℤn∗) on the set of length four positive words in the generators x0,…,xn-1. Here, D4 is the dihedral group of order 8 and ℤn∗ is the multiplicative group of units acting naturally on the additive group ℤn. Thus the group Γn has order 8⁢n⁢ϕ⁢(n) where ϕ is the Euler totient function. The secondary divisor

is introduced in Section 4.

The various combinations of conditions represented in the left-hand column of Table 1 are analyzed individually in Sections 6–9. As in [3, 6, 10], we depend on previous work concerning asphericity of relative presentations, in this case [2]. There, the nonaspherical cases fall into well-defined infinite families, along with a handful of isolated cases. In addition, there are two cases treated in [2] where the asphericity status remains unresolved even when we incorporate recent progress due to Aldwaik and Edjvet [1] and Williams [24]. See the proof of Theorem 9.2 for these updates.

The bottom row of Table 1 is the most complex, where we encounter eight isolated conditions and the two unresolved conditions, which are collectively referred to as (I∗) and (U∗), respectively. As we now describe, these conditions are defined in terms of exemplars that are listed in Tables 2 and 3.

2 Combinatorial asphericity

Given a presentation 𝒫=〈𝐱|𝐫〉 for a group G, the free group F=F⁢(𝐱) with basis 𝐱 acts by automorphisms on the free group 𝔽=F⁢(F×𝐫) with basis F×𝐫 by v⋅(u,r)=(v⁢u,r). The homomorphism ∂:𝔽→F given by ∂⁡(u,r)=u⁢r⁢u-1 is F-equivariant where F acts on itself by (left) conjugation. The image of ∂ is the normal closure of 𝐫 in F so the cokernel of ∂ is isomorphic to G. Elements of the kernel 𝕀=ker⁢∂ are called identities (or identity sequences) for 𝒫. Then we have the exact sequence

for the presentation 𝒫. Among the identities are the so-called Peiffer identities having the form (u,r)ϵ⁢(v,s)δ⁢(u,r)-ϵ⁢(u⁢rϵ⁢u-1⁢v,s)-δ, where u,v∈F, r,s∈𝐫, and δ,ϵ=±1. If ℙ denotes the normal closure of the Peiffer identities in 𝔽, then the F-action on 𝔽 descends to a G-action on the abelian quotient group 𝕀/ℙ and there is the exact fundamental sequence

that provides a combinatorial description for the long exact homotopy sequence of the skeleton pair (K,K1), where K is the two-complex modeled on the presentation 𝒫. In particular, π2⁢K≅𝕀/ℙ as ℤ⁢G-modules. See [7, Section 1], [20, Section 2], or [21] for details.

As in [7, Proposition 1.4], a presentation 𝒫=〈𝐱|𝐫〉 for a group G is combinatorially aspherical if the homotopy module π2⁢K≅𝕀/ℙ is generated as a ℤ⁢G-module by the classes of identities that have length two in the free group 𝔽 with basis F⁢(𝐱)×𝐫. The two-dimensional cellular model K of the presentation 𝒫 is aspherical, in the sense that π2⁢K=0, if and only if the presentation 𝒫 is combinatorially aspherical and no relator of 𝒫 is a proper power, nor is conjugate to any another relator or its inverse. See [7], [3, Section 3], or the survey [19], which contains a thorough discussion of identity sequences and their relation to pictures under the blanket assumption that no relator is conjugate to any another relator or its inverse. From the point of view of spherical pictures (or dually, spherical diagrams), an arbitrarily given presentation is combinatorially aspherical if and only if its second homotopy module is generated by the classes of spherical pictures that contain exactly two relator discs. See [19, Section 2.3].

Presentations satisfying any of the C⁢(p)-T⁢(q) small cancellation conditions are combinatorially aspherical if 1/p+1/q≤1/2 (see [7, 14]). In fact, they satisfy even stronger conditions such as diagrammatic asphericity [8]. Here we shall be interested in the C⁢(4)-T⁢(4) condition. Recall that a word is a piece for a presentation if it occurs as a common initial subword of two distinct cyclic permutations of the relators or their inverses.

3 Isomorphisms

Classifications involving cyclic presentations are complicated by the fact that there are many isomorphisms that must be accounted for, not all of which are obvious or easily catalogued. See [6, Section 2] and [10, Lemma 2.1], for example.

For a fixed positive integer n, let Φn denote the set of all positive words of length four in the generators x0,…,xn-1 for the free group F of rank n. Thus every element w∈Φn has the form w=xi⁢xj⁢xk⁢xl, where integers i,j,k,l are defined modulo n. We consider the following bijective transformations on Φn:

Here the element u is taken from the multiplicative group ℤn∗ of units modulo n. All subscripts are interpreted modulo n.

The orbit of the word w=xi⁢xj⁢xk⁢xl∈Φn under the action of Γn consists of all words that arise as cyclic permutations of shifts of words of the form u⁢(w)=xu⁢i⁢xu⁢j⁢xu⁢k⁢xu⁢l or σ⁢(u⁢(w))=xu⁢l⁢xu⁢k⁢xu⁢j⁢xu⁢i, where u∈ℤn∗.

Proof.

Let F=F⁢(𝐱) be the free group with basis 𝐱={xp:0≤p<n}. Given c∈Γn and w=xi⁢xj⁢xk⁢xl∈Φn, we denote the relator sets for the presentations 𝒫n⁢(w) and 𝒫n⁢(c⁢(w)) by

𝐫w={θFq⁢(w):0≤q<n}={xi+q⁢xj+q⁢xk+q⁢xl+q:0≤q<n}

and

c⁢(𝐫w)={θFq⁢(c⁢(w)):0≤q<n},

respectively. We first construct homomorphisms cw:F→F, c^w:𝔽w→𝔽c⁢(w), where 𝔽w=F⁢(F×𝐫w) and similarly 𝔽c⁢(w)=F⁢(F×c⁢(𝐫w)). For c=σ,τ,θF or u∈ℤn∗ and w=xi⁢xj⁢xk⁢xl∈Φn, the assignments

σw⁢(xp)=xp-1,σ^w⁢(v,θFq⁢(w))=(σw⁢(v),θFq⁢σ⁢(w))-1,

τw=1F,τ^w⁢(v,θFq⁢(w))=(v⁢θFq⁢(xl)-1,θFq⁢τ⁢(w)),

(θF)w=θF,(θF)^w⁢(v,θFq⁢(w))=(θF⁢(v),θFq+1⁢(w)),

uw⁢(xp)=xu⁢p,u^w⁢(v,θFq⁢(w))=(uw⁢(v),θFu⁢q⁢(u⁢(w)))

define length-preserving homomorphisms cw:F→F, c^w:𝔽w→𝔽c⁢(w). These homomorphisms are constructed to satisfy

∂∘c^w=cw∘∂

for c=σ,τ,θF,u∈Γn=D4×(ℤn⋊ℤn∗). Viewed in Aut⁢(F), the isomorphisms σw,τw,θF,uw are compatible with the relations of Γn=D4×(ℤn⋊ℤn∗). Specifically, one notes that

σw2=τw4=(σw∘τw)2=θFn=1,tw∘uw=(tu)w (t,u∈ℤn∗)

and that the homomorphisms σw,τw commute pairwise with θF and uw (u∈ℤn∗). Since Γn acts on Φn, this shows that Γn acts on F by automorphisms and in fact this action is independent of w. (We retain the notation cw in order to distinguish between c∈Γn and cw∈Aut⁢(F).) For each c∈Γn we now have an automorphism cw∈Aut⁢(F) that is length-preserving and which is compatible with the boundary maps ∂ from the exact sequences (2.1) for the presentations 𝒫n⁢(w) and 𝒫n⁢(c⁢(w)):

It follows that for any c∈Γn, the automorphism cw∈Aut⁢(F) induces an isomorphism Gn⁢(w)→Gn⁢(c⁢(w)), which we also denote by cw. As for shift equivariance, we have

σw∘θGn⁢(w)=θGn⁢(σ⁢(w))∘σw,

τw∘θGn⁢(w)=θGn⁢(τ⁢(w))∘τw,

uw∘θGn⁢(w)=θGn⁢(u⁢(w))u∘uw.

Moreover, Gn⁢(θF⁢(w))=Gn⁢(w) and (θF)w=θGn⁢(w). This verifies equivariance as in (3.1).

It remains to verify that combinatorial asphericity is preserved under the action of Γn on the set of presentations of the form 𝒫n⁢(w), w∈Φn. Note that the homomorphism τ^w depends on w. It turns out that the homomorphisms σ^w, τ^w, (θF)^w, u^w are compatible with some of the relations of Γn but not all. The fact that σ^w2=1𝔽w is straightforward:

σ^w2⁢(v,θFq⁢(w))=σ^w⁢(σw⁢(v),θFq⁢σ⁢(w))-1

=((σw2(v),θFq(σ2(w))-1)-1

=(v,θFq⁢(w)).

However, τ^w4:𝔽w→𝔽w is not the identity. For example,

τ^w4⁢(1,xi⁢xj⁢xk⁢xl)=τ^w3⁢(xl-1,xl⁢xi⁢xj⁢xk)

=τ^w2⁢(xl-1⁢xk-1,xk⁢xl⁢xi⁢xj)

=τ^w⁢(xl-1⁢xk-1⁢xj-1,xj⁢xk⁢xl⁢xi)

=(xl-1⁢xk-1⁢xj-1⁢xi-1,xi⁢xj⁢xk⁢xl).

Nevertheless, the fact that

xl-1⁢xk-1⁢xj-1⁢xi-1=1

in the group Gn⁢(w)=Gn⁢(xi⁢xj⁢xk⁢xl) implies that

(1,xi⁢xj⁢xk⁢xl)≡(xl-1⁢xk-1⁢xj-1⁢xi-1,xi⁢xj⁢xk⁢xl)

modulo the group ℙw of Peiffer identities for the presentation 𝒫n⁢(w). See, for instance, [21, Lemma 2.4]. In fact, one can verify that the homomorphisms σ^w, τ^w, (θF)^w,u^w are compatible with all of the relations of Γn if we work modulo Peiffer identities. For each c∈Γn and w∈Φn, we obtain a commutative diagram that compares the fundamental sequences (2.2) for the presentations 𝒫n⁢(w) and 𝒫n⁢(c⁢(w)); the vertical arrows all represent isomorphisms:

It remains to note that the isomorphism 𝕀w/ℙw→𝕀c⁢(w)/ℙc⁢(w) is length-preserving and so it follows that 𝕀w/ℙw is generated by classes of length two identity sequences if and only if the same is true for 𝕀c⁢(w)/ℙc⁢(w). ∎