Stephen J. Pride

On theT(q)-conditions of small cancellation theory

In this paper we give a graphical method which can be used to determine whether or not a group presentation satisfies the small cancellation conditionT(q). We use this method to determine all 2- and 3-generator presentations satisfyingT(4).

Rewriting Systems, Finiteness Conditions, and Associated Functions

Associated with any rewriting system P there is a certain two-dimensional complex D(P)independently introduced by a number of authors, and now known as the Squier complex. We adopt a geometric approach to this complex here in terms of “pictures” as in [24], [25].

On the Left and Right Cohomological Dimension of Monoids

In this paper we construct, for any 1 m; n 1, a nitely presented monoid with left cohomological dimension m and right cohomological dimension n.

On the Asphericity of Length Four Relative Group Presentations

Excluding five unresolved cases, asphericity is classified in relative group presentations of the form (H,x: xaxbxcxd).

The (CO)homology of groups given by presentations in which each defining relator involves at most two types of generators

Our set-up will consist of the following: (i) a graph with vertex set V and edge set E ; (ii) for each vertex ∈ V a non-trivial group G v given by a presentation (x ν ; r ν ); (iii) for each edge e = { u, ν } ∈ E a group G e given by a presentation (x u , x v ; r e ) where r e consists of the elements of r u ∪ r v , together with some further words on x u ∪ x v . We let G = (x; r) where x = ∪ v∈v x v , r = ∪ e∈E r e . Ouraim is to try to describe the structure of G in terms of the groups G v , ( v ∈ V ), G e ( e ∈ E ). Under suitable conditions the natural homomorphisms G v , → G ( ν ∈ V ), G e → G e ( e e E ) are injective; and there is a short exact sequence (where, for any group H , IH is the augmentation ideal). Some (co)homological consequences of these resultsare derived.

On the Efficiency of Coxeter Groups

If G is a finitely presented group and [Kscr ] is any ( G ,2)-complex (that is, a finite 2-complex with fundamental group G ), then it is well known that χ([Kscr ]) [ges ] ν( G ), where ν( G ) = 1−rk H 1 G+dH 2 G . We define χ( G ) to be min{χ([Kscr ]): [Kscr ] a ( G , 2)-complex}, and we say that G is efficient if χ( G ) = ν( G ). In this paper we give sufficient conditions for a Coxeter group to be efficient (Theorem 4.2). We also give examples of inefficient Coxeter groups (Theorem 5.1). In fact, we give an infinite family G n ( n = 2, 3, 4, . . . ) of Coxeter groups such that χ( G n )−ν( G n ) [xrarr ] ∞ as n [xrarr ] ∞.

Homological finiteness conditions for groups, monoids, and algebras

Recently, Alonso and Hermiller (2003) introduced a homological finiteness condition bi − FP n (here called “weak bi-FP n ”) for monoid rings, and Kobayashi and Otto (2003) introduced a different property, also called bi − FP n (we adhere to their terminology). From these and other articles we know that: bi − FP n ⇒ left and right FP n ⇒ weak bi − FP n ; the first implication is not reversible in general; the second implication is reversible for group rings. We show that the second implication is reversible in general, even for arbitrary associative algebras (Theorem 1′), and we show that the first implication is reversible for group rings (Theorem 2). We also show that the all four properties are equivalent for connected graded algebras (Theorem 4).

Low-dimensional (co)homology of free Burnside monoids

In this paper we compute the second and the third integral (co)homology groups of the free Burnside monoid satisfying Tm = Tm + d, m ≥ 3, d ≥ 1.

FINITENESS CONDITIONS FOR REWRITING SYSTEMS

Monoids that can be presented by a finite complete rewriting system have both finite derivation type and finite homological type. This paper introduces a higher dimensional analogue of each of these invariants, and relates them to homological finiteness conditions.

For Rewriting Systems the Topological Finiteness Conditions FDT and FHT are Not Equivalent

A anite rewriting system is presented that does not satisfy the homotopical aniteness condition FDT, although it satisaes the homological aniteness condition FHT. This system is obtained from a group G and a anitely generated subgroup H of G through a monoid extension that is almost an HNN-extension. The FHT property of the extension is closely related to the FP2 property for the subgroup H, while the FDT property of the extension is related to the anite presentability of H. The example system separating the FDT property from the FHT property is then obtained by applying this construction to an example group considered by Bestvina and Brady (1997).