Redundant relators in cyclic presentations of groups

2.1 Presentations and cyclic presentations of groups

Given a positive integer 𝑛, let F n be the free group with basis X = { x 0 , … , x n − 1 } . A non-empty word w ∈ F n is said to be positive or negative if all of the exponents of generators are positive or negative, respectively. (In particular, generators x i are positive, and their inverses x i − 1 are negative.) We shall say that a word 𝑤 of length at least 2 is alternating if it has no subword of the form ( x i ⁢ x j ) ± 1 and that it is cyclically alternating if it has no cyclic subword of that form. Thus a word is cyclically alternating if and only if it is alternating and has even length. A word 𝑤 is reduced if it does not contain a subword of the form x i ⁢ x i − 1 or x i − 1 ⁢ x i ; it is cyclically reduced if all cyclic permutations of it are reduced. If w ∈ F n is a cyclically reduced, non-empty, word, then the unique word v ∈ F n such that w = v p with 𝑝 maximal is called the root of 𝑤. We shall write l ⁢ ( w ) to denote the length of 𝑤 in F n . Throughout this article, equality of words will refer to equality of elements of the free group F n .

When considering group presentations ⟨ X | R ⟩ , 𝑅 will be a set of relators (so does not contain any relator more than once), all of whose elements are cyclically reduced. Let θ : F n → F n be the shift automorphism given by θ ⁢ ( x i ) = x i + 1 , where (as throughout this article) subscripts are taken mod ⁢ n . Given a non-empty cyclically reduced word 𝑤 representing an element in F n , the words θ i ⁢ ( w ) , where 0 ≤ i < n and 1 ≤ i < n , are the shifts and proper shifts of 𝑤, respectively, the presentation

is a cyclic presentation, the group G n ⁢ ( w ) it defines is a cyclically presented group, and 𝑤 is the defining word [22]. Without loss of generality, we may assume that the generator x 0 is a letter of 𝑤, and we make this assumption throughout this article. Then P n ⁢ ( w ) is said to be irreducible if the greatest common divisor of 𝑛 and the subscripts of the generators that appear in 𝑤 is equal to 1 [11]. The shift automorphism 𝜃 satisfies θ n = 1 , and the resulting Z n -action on G n ⁢ ( w ) determines the shift extension E = G n ⁢ ( w ) ⋊ θ Z n , which admits a presentation of the form ⟨ x , t | t n , W ⁢ ( x , t ) ⟩ , where W ⁢ ( x , t ) is obtained by rewriting 𝑤 in terms of the substitutions x i = t i ⁢ x ⁢ t − i , 0 ≤ i < n (see, for example, [23, Theorem 4]).

For 1 ≤ t ≤ n , we define the 𝑡-truncation of P n ⁢ ( w ) to be the presentation

and denote by G n , t ⁢ ( w ) the group that it defines. Let ϕ : F n → F n be the cyclic permutation function that cyclically permutes a word by one generator (or the inverse of a generator), with inverse ϕ − 1 . That is, ϕ ⁢ ( x p 1 ϵ 1 ⁢ x p 2 ϵ 2 ⁢ … ⁢ x p k ϵ k ) = x p 2 ϵ 2 ⁢ … ⁢ x p k ϵ k ⁢ x p 1 ϵ 1 ( 0 ≤ p i < n , ϵ i = ± 1 , 1 ≤ i ≤ k ). Then 𝜙 and 𝜃 commute, and if 𝑤 is a cyclically reduced word of length 𝑘, then ϕ k ⁢ ( w ) = w . It follows that if 𝑤 is a cyclically reduced word of length 𝑘 that is equal to a cyclic permutation of the shift θ h ⁢ ( w ) , then a cyclic permutation of 𝑤 is equal to the shift θ ( n , h ) ⁢ ( w ) , and if a shift of 𝑤 is equal to a cyclic permutation ϕ t ⁢ ( w ) of 𝑤, then a shift of 𝑤 is equal to the cyclic permutation ϕ ( k , t ) ⁢ ( w ) . Consider, for example, P 6 ⁢ ( x 0 ⁢ x 1 ⁢ x 2 ⁢ x 3 ⁢ x 4 ⁢ x 5 ) with h = 4 and t = 4 . (To see the first claim, let α , β ∈ Z satisfy α ⁢ h + β ⁢ n = ( n , h ) , and observe that 𝑤 is equal to a cyclic permutation of

Similar arguments hold for the second claim.) Moreover, ϕ r ⁢ ( v p ) = ( ϕ r ⁢ ( v ) ) p and θ h ⁢ ( v p ) = ( θ h ⁢ ( v ) ) p for any h , r , p ≥ 1 , so if w = v p , where 𝑣 is the root of 𝑤, then θ h ⁢ ( w ) = ϕ r ⁢ ( w ) if and only if θ h ⁢ ( v ) = ϕ r ⁢ ( v ) . We will write ι ⁢ ( w ) , τ ⁢ ( w ) to denote the initial and terminal letters of a word 𝑤, respectively.

Following [2, page 160], given a group presentation P = ⟨ X | R ⟩ , an element r ∈ R is said to be freely redundant if it is freely trivial or if there exists another element s ∈ R such that 𝑟 and 𝑠 are elements of the free group with basis 𝑋 and either 𝑟 is freely conjugate to 𝑠 or 𝑟 is freely conjugate to s − 1 (that is, either 𝑟 is a cyclic permutation of 𝑠 or of s − 1 ). A presentation is said to be redundant if it contains a freely redundant relator and is concise otherwise [9, page 4]. (Concise presentations are referred to as slender presentations in [3, page 5] or irredundant presentations in [28, page 82].) If a presentation P ′ is obtained from a presentation 𝑃 by removing freely redundant relators, then we say that P ′ is a refinement of 𝑃, and if, in addition, P ′ is concise, we say that it is a concise refinement of 𝑃 (compare [9, page 4]). A cyclic presentation P n ⁢ ( w ) is orientable if 𝑤 is not a cyclic permutation of the inverse of any of its shifts [2, page 155] and is non-orientable otherwise. Thus an orientable cyclic presentation is redundant if and only if 𝑤 is equal to a cyclic permutation of one of its proper shifts, and if a cyclic presentation is non-orientable, then it is redundant. As examples, the cyclic presentation P 2 ⁢ ( x 0 ⁢ x 1 ) is orientable and redundant, P 3 ⁢ ( x 0 ⁢ x 1 ) is orientable and concise, and P 2 ⁢ ( x 0 ⁢ x 1 − 1 ) is non-orientable (and therefore redundant).

The deficiency of the presentation P = ⟨ X | R ⟩ is defined as

and the deficiency of a group 𝐺, def ⁡ ( G ) , is defined to be the maximum of the deficiencies of all finite presentations defining 𝐺. A group 𝐺 is large if it has a finite index subgroup that has a non-abelian free homomorphic image, and it is SQ-universal if every countable group embeds in a quotient of 𝐺. Every large group is SQ-universal, and every SQ-universal group contains a non-abelian free subgroup [27]. Moreover, as noted in [32], the converse to neither of these statements holds in the class of cyclically presented groups: the Higman group G 4 ⁢ ( x 0 ⁢ x 1 ⁢ x 0 − 2 ⁢ x 1 − 1 ) is SQ-universal [29] but has no proper finite index subgroup [18] so is not large, and the group G 7 ⁢ ( x 0 ⁢ x 1 ⁢ x 3 ) , identified in [12, Example 3.3] and [20, Example 6.3], contains a non-abelian free subgroup [12] but is just-infinite [13, Theorem 2] so is not SQ-universal. A class of groups is said to satisfy the Tits alternative if every group in that class either contains a non-abelian free subgroup or is virtually solvable.

2.2 Star graphs

Let P = ⟨ X | R ⟩ be a group presentation and let R ~ denote the symmetrised closure of 𝑅, that is, the set of all cyclic permutations of elements in R ∪ R − 1 . The star graph of 𝑃 is the undirected vertex-labelled graph Γ where the vertex set is in one-one correspondence with X ∪ X − 1 , vertices are labelled by the corresponding element of X ∪ X − 1 , and where there is an edge joining vertices labelled 𝑥 and 𝑦 for each distinct word x ⁢ y − 1 ⁢ u in R ~ (see [25, page 61]). Such words occur in pairs, that is, x ⁢ y − 1 ⁢ u ∈ R ~ implies that y ⁢ x − 1 ⁢ u − 1 ∈ R ~ . These pairs are called inverse pairs, and the two edges corresponding to them are identified in Γ. It follows that replacing any relator of a presentation by its root, or removing a redundant relator from a presentation, leaves the star graph unchanged; in particular, the star graphs of a presentation and any concise refinement of it are equal. We refer to vertices in 𝑋 as positive vertices and vertices in X − 1 as negative vertices.

We now set out our graph theoretic terminology. We allow graphs to have loops and to have more than one edge joining a pair of vertices. Given a graph Γ, we write V ⁢ ( Γ ) to denote its vertex set. If Γ is bipartite with vertex partition V ⁢ ( Γ ) = V 1 ∪ V 2 , where each edge connects a vertex in V 1 to a vertex in V 2 , then V 1 , V 2 are called the parts of V ⁢ ( Γ ) . Two adjacent vertices are said to be neighbours, and the set of neighbours of a vertex 𝑣 in a graph Γ is denoted N Γ ⁢ ( v ) . A graph Γ is 𝑟-regular if | N Γ ⁢ ( v ) | = r for all v ∈ V ⁢ ( Γ ) , and it is regular if it is 𝑟-regular for some 𝑟.

A path of length𝑙 in Γ is a sequence of vertices ( u = u 0 , u 1 , … , u l = v ) with edges u i − u i + 1 for each 0 ≤ i < l ; it is a closed path if u = v . The path is reduced if the edge u i + 1 − u i + 2 is not equal to the edge u i + 1 − u i ( 0 ≤ i < l − 1 ). The distance d Γ ⁢ ( u , v ) between vertices u , v of Γ is l ≥ 0 if there is a path of length 𝑙 from 𝑢 to 𝑣, but no shorter path, and d Γ ⁢ ( u , v ) = ∞ if there is no path from 𝑢 to 𝑣. The girth, girth ⁡ ( Γ ) of a graph Γ is the length of a reduced closed path of minimal length if Γ contains a reduced closed path, and girth ⁡ ( Γ ) = ∞ otherwise. The diameter, diam ⁡ ( Γ ) of a graph Γ is the greatest distance between any pair of vertices of the graph (which may be infinite). If Γ is a graph with finite girth, then girth ⁡ ( Γ ) ≤ 2 ⁢ diam ⁡ ( Γ ) + 1 . For an integer n ≥ 2 and a (multi-)set A ⊆ { 0 , 1 , … , n − 1 } , the circulant graph circ n ⁡ ( A ) is the graph with vertices v 0 , … , v n − 1 and edges v i − v i + a for all 0 ≤ i < n , a ∈ A (subscripts mod ⁢ n ). We define the graph circ n ′ ⁡ ( A ) to be circ n ⁡ ( A ) if n / 2 ∉ A and to be circ n ⁡ ( A ) with exactly one edge v i − v i + n / 2 removed for each n / 2 ≤ i < n otherwise. For later use, we note that circ n ⁡ ( A ) is the complete bipartite graph K n / 2 , n / 2 if and only if n / 2 is even and A = { ± 1 , ± 3 , … , ± ( n / 2 − 1 ) } , and circ n ′ ⁡ ( A ) is the complete bipartite graph K n / 2 , n / 2 if and only if n / 2 is odd and A = { ± 1 , ± 3 , … , ± ( n / 2 − 2 ) , n / 2 } .

2.3 Special presentations

An ( m , k , ν ) -special presentation is a group presentation in which the relators have length 𝑘 and whose star graph has 𝜈 isomorphic components, each of which is the incidence graph of a generalised 𝑚-gon. Formally, we have the following definition.

This generalises the concept of ( m , k ) -special cyclic presentations introduced in [13], which corresponds to the case ν = 1 , which in turn generalises the concept of special presentations introduced in [20] (which corresponds to the case m = k = 3 ). The concise cyclic presentations that are ( m , k , ν ) -special were classified in [8]. We refer the reader to [13, 8] for background and further references on properties of ( m , k , ν ) -special presentations and the groups they define.

Note that if a presentation is ( m , k , ν ) -special, then it has at least 3 generators. As in the proof of [13, Theorem 2], a group 𝐺 defined by an ( m , k , ν ) -special presentation with 2 / k + 1 / m < 1 is non-elementary hyperbolic and hence SQ-universal. If w = v p , where 𝑣 is the root of 𝑤, then the star graph Γ of P n ⁢ ( w ) is equal to the star graph of P n ⁢ ( v ) . Moreover, if 𝑣 has length 2, then the vertices of Γ have degree at most 2, so neither P n ⁢ ( v ) nor P n ⁢ ( w ) are ( m , k , ν ) -special. Therefore, P n ⁢ ( w ) is ( m , p ⁢ k , ν ) -special if and only if P n ⁢ ( v ) is ( m , k , ν ) -special. Thus, in classifying ( m , k , ν ) -special cyclic presentations P n ⁢ ( w ) , we can assume that 𝑤 is not a proper power. For our characterisation of ( 3 , k , ν ) -special cyclic presentations, we recall that a set of 𝑘 integers d 1 , … , d k is called a perfect difference set (of order 𝑘) if, among the k ⁢ ( k − 1 ) differences d i − d j ( i ≠ j ), each of the residues 1 , 2 , … , ( k 2 − k ) mod ( k 2 − k + 1 ) occurs exactly once.

3.1 Classification of orientable redundant cyclic presentations

Suppose that P n ⁢ ( w ) is orientable and that 𝑣 is the root of 𝑤, and recall from Section 2.1 that a cyclic permutation of 𝑤 is equal to one of its proper shifts if and only if a cyclic permutation of 𝑣 is equal to one of its proper shifts. It follows that P n , t ⁢ ( w ) is a concise refinement of P n ⁢ ( w ) if and only if P n , t ⁢ ( v ) is a concise refinement of P n ⁢ ( v ) . Therefore, in this context, we may assume that 𝑤 is not a proper power.

Every non-empty word w ∈ F n has an expression of the form

for some u ∈ F n of length l ⁢ ( u ) ≥ 1 and some 0 ≤ h < n , even if only trivially, with u = w and h = 0 . However, as we shall see, if P n ⁢ ( w ) is orientable and redundant, then there is an expression (3.1) with u ≠ w and h > 0 . (For example, P 3 ⁢ ( x 0 ⁢ x 1 ⁢ x 2 ) has such an expression with u = x 0 and h = 1 .) Therefore, 𝑤 satisfies ϕ l ⁢ ( u ) ⁢ ( w ) = θ h ⁢ ( w ) , and hence a cyclic permutation of 𝑤 is equal to the shift θ ( n , h ) ⁢ ( w ) , which is proper if 0 < h < n .

Note that, in (3.1), 𝑤 is positive if and only if 𝑢 is positive, 𝑤 is cyclically alternating if and only if 𝑢 is cyclically alternating, and if w ≠ u , then 𝑤 is alternating if and only if it is cyclically alternating. For later use, we record that, given a word 𝑤 of the form (3.1), every cyclic permutation of 𝑤 has an expression of the same form, where the length of 𝑢 and the value of ℎ are preserved.

Theorem 3.2 identifies the word 𝑢 in an expression (3.1) in terms of the cyclic permutations of 𝑤 that are equal to a shift of 𝑤. Corollary 3.3 shows that if 𝑢 is chosen to be the shortest possible, then (for the corresponding value of ℎ) the ( n , h ) -truncation P n , ( n , h ) ⁢ ( w ) is a concise refinement of P n ⁢ ( w ) .

3.2 Classification of non-orientable cyclic presentations

Given any non-empty word u ∈ F n , where n ≥ 2 is even, and the word

the cyclic presentation P n ⁢ ( w ) is non-orientable. Lemma 3.6 of [2] characterises the non-orientable cyclic presentations P n ⁢ ( w ) as those for which 𝑛 is even and 𝑤 is equal to u ⁢ θ n / 2 ⁢ ( u ) − 1 for some reduced word 𝑢. Unfortunately, that statement is not quite correct: a presentation that demonstrates this is P 4 ⁢ ( w ) , where w = x 0 ⁢ x 2 − 1 ⁢ x 3 ⁢ x 1 − 1 . (However, the remaining results from [2] appear to be unaffected.) In Theorem 3.4, we instead characterise such presentations as those for which 𝑛 is even and some cyclic permutation of𝑤 is equal to u ⁢ θ n / 2 ⁢ ( u ) − 1 for some reduced word 𝑢, and we show that the n / 2 -truncation P n , n / 2 ⁢ ( w ) is a concise refinement of P n ⁢ ( w ) .

In considering non-orientable cyclic presentations P n ⁢ ( w ) in contexts where relators can be replaced by their cyclic permutations, it thus suffices to consider cyclic presentations P n ⁢ ( u ⁢ θ n / 2 ⁢ ( u ) − 1 ) for some reduced word 𝑢 (even though 𝑤 itself may not be of the form u ⁢ θ n / 2 ⁢ ( u ) − 1 ). Note that, for such a 𝑤 and 𝑢, the word 𝑤 is non-positive, non-negative, and 𝑤 is cyclically alternating if and only if 𝑤 is alternating if and only if 𝑢 is alternating.