Groupoids and the algebra of rewriting in group presentations

Presentations of groups by rewriting systems (that is, by monoid presentations), have been fruitfully studied by encoding the rewriting system in a $2$--complex -- the Squier complex -- whose fundamental groupoid then describes the derivation of consequences of the rewrite rules. We describe a reduced form of the Squier complex, investigate the structure of its fundamental groupoid, and show that key properties of the presentation are still encoded in the reduced form.


INTRODUCTION
The study of the relationships between presentations of semigroups, monoids, and groups, and systems of rewriting rules has drawn together concepts from group and semigroup theory, low-dimensional topology, and theoretical computer science. In [17], Squier addressed the question of whether a finitely presented monoid with solvable word problem is necessarily presented by a finite, complete, string rewriting system. He proved that a monoid presented by a finite, complete, string rewriting system must satisfy the homological finiteness condition F P 3 : indeed, an earlier result of Anick [1] implies that such a monoid satisfies the stronger condition F P ∞ . These ideas are concisely surveyed by Cohen [6], and more extensively by Otto and Kobayashi in [14]. Since examples are known of finitely presented monoids with solvable word problem that do not satisfy F P 3 , Squier's work shows that such monoids need not be presented by finite, complete, string rewriting systems.
Squier, Otto and Kobayashi [18] studied finite, complete, string rewriting systems for monoids and proved that the existence of such a system presenting a monoid M implies a homotopical propertyfinite derivation type -defined for a graph that encodes the rewriting system. Moreover, they show that having finite derivation type does not depend on the particular rewriting system used to present M , and so is a property of M itself and a necessary condition that M should be presented by a finite, complete string rewriting system.
Finite derivation type is naturally thought of as a property of a 2-complex, the Squier complex associated to a monoid presentation P, and obtained by adjoining certain 2-cells to the graph of [18]. This point of view was introduced independently by Pride [15] and Kilibarda [13], and then extensively developed by Guba and Sapir in terms of both stringrewriting systems [10] and more geometrically, in terms of directed 2-complexes [11]. The theory developed by Kilibarda and then by Guba and Sapir focusses on the properties of diagram groups, which are fundamental groups of the Squier complex.
Kilibarda [13] studied the fundamental groupoid of the Squier complex associated to a monoid presentation [X : R]. Gilbert [9] showed that the fundamental groupoid is a monoid in the category of groupoids, and used this enriched structure to explain Pride's corresponding theory of diagram groups for monoid presentations of groups [16].
Pride's approach in [16] is based upon the addition of extra 2-cells to a Squier complex so as to realise a homotopy relation introduced by Cremanns and Otto [7]. This augmented Squier complex was called the Pride complex in [9] and denoted by K + . Beginning with a group presentation P = X : R of a group G, we obtain a monoid presentation of G by adding relations xx −1 = 1 = x −1 x for each x ∈ X, and the additional 2-cells correspond to possible overlaps in the use of such relations in the free reduction of words on A = X ∪ X −1 . The outcome is that if u and v are freely equivalent then any two edgepaths in K + from u to v that record this free equivalence are fixed-end-point homotopic, as required for a homotopy relation as defined in [7]. Gilbert investigated the structure of the fundamental groupoid π(K + , A * ) and showed that there is a retraction map π(K + , A * ) → π(K + , F (X)) to the fundamental groupoid with vertex set the free group F (X).
In this paper -which reconfigures the approach to monoid presentation of groups in [7] and is a somewhat belated sequel to [9,16] -we adopt a similar approach, but use a different modification of the Squier complex, defining the reduced Squier complex Sq ρ (P) of a group presentation P = X : R as a 2-complex having vertex set F (X). We can then work directly with the fundamental groupoid π(Sq ρ (P), F (X)) and so avoid some of the technicalities from [9]. In particular, we show that the set star 1 (Sq ρ (P)) of homotopy classes of paths in π(Sq ρ (P), F (X)) that begin at 1 ∈ F (X) has a natural group structure, and the end-of-path map r : star 1 (Sq ρ (P)) → F (X) is a crossed module, as defined by J.H.C. Whitehead (see [19]). We give a presentation for star 1 (Sq ρ (P)), and use it to show that the crossed module is isomorphic to that usually associated to a group presentation, as in [3]. It then follows that the fundamental group π 1 (Sq ρ (P ), 1) can be interpreted as the kernel of a free presentation of the relation module of P, and as in [7] we may link the module structure of π 1 (Sq ρ (P ), 1) to the homological finiteness condition FP 3 , and as in [8] to Cockcroft properties of P.
A version of these results is presented in the second author's PhD thesis at Heriot-Watt University, Edinburgh. The generous financial support of a PhD Scholarship from the Carnegie Trust for the Universities of Scotland is duly and gratefully acknowledged.

BACKGROUND NOTIONS AND NOTATION
1.1. Groupoids. A groupoid G is a small category in which every morphism is invertible. We consider a groupoid as an algebraic structure as in [12]: the elements are the morphisms, and composition is an associative partial binary operation. The set of vertices of G is denoted V (G), and for each vertex x ∈ V (G) there exists an identity morphism 1 x . An element g ∈ G has domain gd and range gr in V (G), with gg −1 = 1 gd and g −1 g = 1 gr . For e ∈ V (G) the star of e in G is the set star e (G) = {g ∈ G : gd = e}, and the local group at e is the set G(e) = {g ∈ G : gd = e = gr}.
1.2. Crossed modules. Crossed modules will be the algebraic models of group presentations that we shall use in our formulation of the relation module and the module of identities for a group presentation. For a more detailed account of these topics , we refer to [3].
A crossed module is a group homomorphism ∂ : T → Γ together with an action of Γ on T (written (t, g) → t g ) such that ∂ is Γ-equivariant, so that for all t ∈ T and g ∈ Γ we have and such that for all t, u ∈ T , we have: We shall say that (T, ∂) is a crossed Γ-module.
Example 1.1. Examples of crossed modules include the following: • any Γ-module M with the trivial map M 0 → Γ, • the inclusion of any normal subgroup N ֒→ Γ, • the map T → Aut T that associates to t ∈ T the inner automorphism of T defined by a → t −1 at, • any surjection T → Γ with central kernel, where Γ acts on T by lifting and conjugation, • the boundary map π 2 (X, Y ) → π 1 (Y ) from the second relative homotopy group of a pair of spaces (X, Y ) with Y ⊆ X.
The last example motivated the introduction of crossed modules by J.H.C. Whitehead [19].
Let ∂ : T → Γ be a crossed module, and let N be the image of ∂. The following properties are easy consequences of (1.1) and (1.2).
• N is normal in Γ, and so if we set G = Γ/N we get the short exact sequence of groups: , the center of T , so ker ∂ is abelian.
• ker ∂ is invariant under the Γ-action on T , and so is a Γ-module. • N acts trivially on Z(T ) and thus on ker ∂, hence ker ∂ inherits an action of G to become a G-module. • the abelianisation T ab of T inherits the structure of a G-module.

Free crossed modules.
Definition 1.2. Let (T, ∂) be a crossed Γ-module , let R be a set, and let ρ : R → T be a function. We say (T, ∂) is a free crossed Γ-module with basis ρ if for any crossed Γ-module (T ′ , ∂ ′ ) and function σ : R → T ′ such that σ∂ ′ = ρ∂, that is, such that the square We may also choose to emphasise ω = ρ∂ : R → Γ by saying that a free crossed module (T, ∂) with basis ρ is a free crossed module on ω.
The construction of free crossed modules is due to Whitehead [19], and is also discussed in [3].
Let Γ be a group, R a set, and ω : R → Γ a function. Then a free crossed Γ-module on ω exists and is unique up to isomorphism.
Proof. We sketch the construction, following [3, Proposition 5]. Let F be the free group on the basis R × Γ. Then Γ acts on F by right multiplication of basis elements: for r ∈ R and u, v ∈ Γ we have (r, u) v = (r, uv). We map (r, u) → u −1 (rω)u and this induces a group homomorphism δ : F → Γ. The subgroup P of F generated by all elements of the form with r, s ∈ R and u, v ∈ Γ is normal in F , invariant under the Γ-action, and contained in ker δ. It follows that δ induces ∂ : F/P → Γ, and this is a free crossed Γ-module on ω. Uniqueness up to isomorphism follows from the usual universal argument.
Whitehead also observed the following:

Crossed modules from group presentations.
A group presentation P = X : R of a group G, consists of a set of generators X, and a set of be the canonical map, and we define ρ : R → F (X) by (ℓ, r) ρ = (ℓ −1 r)ρ. We let R be the image of ρ in F (X), and define N = R to be the normal closure of R in F , so that a typical element of N has the form u −1 1 (r 1 ρ) ε1 u 1 · · · u −1 k (r k ρ) ε k u k , where, for 1 j k, we have u j ∈ F , r j ∈ R, and ε j = ±1. Then G is the quotient group F (X)/N , and we have a canonical presentation map θ : F (X) → G.
We now let (C(P), ∂) be the free crossed F (X)-module on the function ρ : R → F (X). An element of C = C(P) is represented by a product For (r, w) ∈ C we have ∂ : (r, w) → w −1 (r ρ)w, and the image of ∂ is N . We denote ker ∂ by π = π(P). We therefore have short exact sequences of groups with π central in C and a G-module.
Corollary to Proposition 7] The free crossed module C is isomorphic as a group to π × N . Its abelianisation C ab is a free G-module, and the induced map π → C ab is injective, so that we have a short exact sequence of G-modules.
In the sequence (1.5), the G-module N ab is the relation module of the presentation P, and the G-module π is the module of identities. The sequence (1.5) then gives a free presentation of the relation module.

REGULAR GROUPOIDS
We now introduce some additional structure on a groupoid. This idea originates in work of Brown and Gilbert [4], and was further developed by Gilbert in [9] and by Brown in [5]. Brown uses the terminology whiskered groupoid for what Gilbert had called a semiregular groupoid. We shall use the semiregular terminology, and will discuss in detail the special case of regular groupoids.
Definition 2.1. Let G be a groupoid, with vertex set V (G) and domain and range maps d, r : • there are left and right actions of V (G) on G, denoted x ⊲ α, α ⊳ x, which for all x, y ∈ V (G) and α, β ∈ G satisfy: Our first result collates some simple facts from [9, section 1].
(a) Let G be a semiregular groupoid. Then there are two everywhere defined binary operations on G given by: Proof. We remark only on the proof of (c), since it is mis-stated in [9]. The inverse of α with respect to * is where • is the inverse of α with respect to the groupoid operation, and −1 is the inverse in the group V (G).

THE SQUIER COMPLEX OF A GROUP PRESENTATION
Let P = X : R be a group presentation. Recall from section 1.2.2 that relations (l, r) ∈ R may involve words in (X ∪ X −1 ) * that are not freely reduced. However, to reduce notational clutter, we shall suppress mention of the free reduction map ρ : (X ∪ X −1 ) * → F (X) in what follows. Hence if p, q ∈ F (X) and (l, r) ∈ R, we shall write prq for p(rρ)q, and so on.
Definition 3.1. The reduced Squier complex Sq ρ (P) is the 2-complex defined as follows: • the vertex set of Sq ρ (P) is the free group F (X) on X, • the edge set of Sq ρ (P) consists of all 3-tuples (p, l, r, q) with p, q ∈ F (X) and (l, r) ∈ R. Such an edge will start at plq and end at prq, so each edge corresponds to the application of a relation in F (X). • the 2-cells correspond to applications of non-overlapping relations, and so a 2-cell is attached along every edge path of the form: will therefore be homotopic in Sq ρ (P).
Lemma 3.1. The fundamental groupoid π(Sq ρ (P), F (X)) of the Squier complex Sq ρ (P) of a group presentation P is a regular groupoid.
Proof. The vertex set of π = π(Sq ρ (P), F (X)) is the group F (X). We need to define left and right actions of F (X) on homotopy classes of paths in Sq ρ (P). We first define such actions for single edges. Let u, v ∈ F (X) and suppose that (p, l, r, q) is an edge in Sq ρ (P). We define u ⊲ (p, l, r, q) = (up, l, r, q) (3.1) (p, l, r, q) ⊳ v = (p, l, r, qv) .

(3.2)
It is then clear that these actions can be extended to edge-paths in Sq ρ (P), and induce actions of F (X) on homotopy classes of paths.
In what follows it will be convenient to work directly with edge paths in Sq ρ (P), even though these are to be interpreted as representatives of homotopy classes in the fundamental groupoid π(Sq ρ (P), F (X)). In particular, we shall apply the operations * and ⊛ directly to edge paths.

From Proposition 2.2 we have:
Corollary 3.3. The subset star 1 (π(Sq ρ (P), F (X)) of the fundamental groupoid of the Squier complex Sq ρ (P) is a group under the binary operation * , and the restriction of the range map is a crossed module r : star 1 (π(Sq ρ (P), F (X))) → F (X) .

The crossed module of a Squier complex.
Our aim is now to show that the crossed module in Corollary 3.3 is isomorphic to the free crossed module C ∂ − → F (X) derived from the presentation P, as in Section 1.2.2. Furthering our blurring of the distinction between an edge path and its homotopy class in the fundamental groupoid, we shall abbreviate the group star 1 (π(Sq ρ (P), F (X))) as star 1 (Sq ρ (P)). We denote by S 1 the set of all edges e ∈ Sq ρ (P) with ed = 1, that is We shall denote the edge (q −1 l −1 , l, r, q) by λ l,r,q .
We now want to understand the effect of homotopy of edge paths in Sq ρ (P) on the *products defined in Proposition 3.4. We first consider a 1-homotopy, that is, the insertion of deletion of a pair of inverse edges. Let ξ = ρ • σ in Sq(P), with ρ ∈ star 1 (Sq ρ (P)).
If this 2-cell is involved in a 2-homotopy between edge paths ξ and ξ ′ , we may assume using 1-homotopies where necessary, that we have Then, using ≃ to denote homotopy of edge paths in Sq ρ (P), we have The above considerations show that, for a given homotopy class in star 1 (Sq ρ (P)), we may select a representative edge path ξ in the form of its * -product ξλ and that this product will be unique up to changes induced by the 2-cells in Sq ρ (P), which may modify the product as in equations (3.5) above. We can be more precise.
Then the following are a set of defining relations for the group (star 1 (Sq ρ (P)), * ) on the generating set S 1 : where (l, r), (s, d) ∈ R and u, v ∈ F (X).
Theorem 3.7. The crossed F (X)-module star 1 (Sq ρ (P)) r − → F (X) derived from the Squier complex Sq ρ (P) of a group presentation P = X : R , is isomorphic to the free Proof. Recall from section 1.2.2 that the free crossed module C ∂ − → F has basis function v : R → C, v : (l, r) → (l, r, 1). We define v : R → star 1 (Sq ρ (P)) by v : (l, r) → (l −1 , l, r, 1). Then v∂ = vr, and thus by freeness of (C, ∂), we have a crossed module morphism φ : C → star 1 (Sq ρ (P)), defined on generators by (l, r, u) → (u −1 l −1 , l, r, u) = λ l,r,u . We note that this is a bijection from the group generating set of C to S 1 .
To obtain an inverse to φ, we therefore wish to map λ l,r,u → (l, r, u). This will be welldefined and a homomorphism if and only if the defining relations given in (3.6) in Proposition 3.6 are mapped to an equation that holds in the group C. Now the left-hand side of (3.6) maps to (l, r, vsu)(s, d, u) and the right-hand side to (s, d, u)(l, r, vdu) . and in the crossed F (X)-module C we do indeed have (s, d, u) −1 (l, r, vsu)(s, d, u) = (l, r, vsu(u −1 s −1 du)) = (l, r, vdu) .
The kernel of the map r : star 1 (Sq ρ (P)) → F (X) is the local group at 1 ∈ F (X) of the groupoid π(Sq ρ (P), F (X)), that is the fundamental group π 1 (Sq ρ (P), 1). Then from Proposition 1.4 we obtain: Proposition 3.8. Let P = X : R be a presentation of a group G with presentation map θ : F (X) → G and let N = ker θ, so that N ab is the relation module of P. Then we have a short exact sequence of G-modules: Example 3.9. Let P = x : xx −1 = 1 presenting the infinite cyclic group x . Then the relation modle is trivial, and (3.7) reduces to an isomorphism π 1 (Sq ρ (P), 1) ∼ = Z x . We can also see this from the construction of Sq ρ (P). The Squier complex Sq ρ (P) has vertex set x and each edge is a loop. The generating set S 1 in Proposition 3.4 is and we write λ q = (x −q , xx −1 , 1, x q ). By Proposition 3.6 we have a presentation for π 1 (Sq ρ (P), 1) = star 1 (Sq ρ (P)) given by π 1 (Sq ρ (P), 1) = λ q (q ∈ Z) : λ p+q * λ q = λ q * λ p+q (p, q ∈ Z) and so π 1 (Sq ρ (P), 1) is free abelian of countably infinite rank, and the x -action is defined by λ x q = λ q+1 .

3.2.
Properties of π 1 (Sq ρ (P), 1). We show in two Corollaries of Proposition 3.8 how properties of the presentation P and the group G are reflected in properties of the fundamental group of the reduced Squier complex.
The illustrative examples that we give are drawn from [7] and [8].
The first result was proved for the Squier complex of [18]  (a) π 1 (Sq ρ (P), 1) is a finitely generated G-module, Proof. There is an exact sequence of G-modules (see [2, Proposition II.5.4]), and if π 1 (Sq ρ (P), 1) is a finitely generated as a G module by a set S this extends, using (3.7), to a partial free resolution of finite type which shows that G has type FP 3 . Conversely, if G has type FP 3 then π 1 (Sq ρ (P), 1) is the kernel (at dimension 2) in a partial free resolution of Z of finite type and so is finitely generated as a consequence of the generalized Schanuel Lemma, see [2,Proposition 4.3].
The second result characterizes the Cockcroft properties of P. Following Dyer [8, Theorem 4.2] we make the following definition. Let L be a subgroup of G, and apply the tensor product − ⊗ L Z to the sequence (3.7) to obtain the sequence