Rewriting systems for Coxeter groups

A finite complete rewriting system for a group is a finite presentation which gives a solution to the word problem and a regular language of normal forms for the group. In this paper it is shown that the fundamental group of an orientable closed surface of genus g has a finite complete rewriting system, using the usual generators a1,.., ag, b1,.., bg along with generators representing their inverses. Constructions of finite complete rewriting systems are also given for any Coxeter group G satisfying one of the following hypotheses. 1) G has three or fewer generators. 2) G does not contain a special subgroup of the for

Tame combings, almost convexity, and rewriting systems for groups

Abstract: A finite complete rewriting system for a group is a finite presentation which gives an algorithmic solution to the word problem. Finite complete rewriting systems have proven to be useful objects in geometric group theory, yet little is known about the geometry of groups admitting such rewriting systems. We show that a group G with a finite complete rewriting system admits a tame 1-combing; it follows (by work of Mihalik and Tschantz) that if G is an infinite fundamental group of a closed irreducible 3-manifold M, then the universal cover of M is R 3. We also establish that a group admitting a geodesic rewriting system is almost convex in the sense of Cannon, and that almost convex groups are tam

Homological finite derivation type

In 1987, Squier defined the notion of finite derivation type for a finitely presented monoid. To do this, he associated a 2-complex to the presentation. The monoid then has finite derivation type if, modulo the action of the free monoid ring, the 1-dimensional homotopy of this complex is finitely generated. Cremanns and Otto showed that finite derivation type implies the homological finiteness condition left FP3, and when the monoid is a group, these two properties are equivalent. In this paper we define a new version of finite derivation type, based on homological information, together with an extension of this finite derivation type to higher dimensions, and show connections to homological type FPn for both monoids and groups

Abstract Artin groups, rewriting systems and three-manifolds

We construct finite complete rewriting systems for two large classes of Artin groups: those of finite type, and those whose defining graphs are based on trees. The constructions in the two cases are quite different; while the construction for Artin groups of finite type uses normal forms introduced through work on complex hyperplane arrangements, the rewriting systems for Artin groups based on trees are constructed via three-manifold topology. This construction naturally leads to the question: Which Artin groups are three-manifold groups? Although we do not have a complete solution, the answer, it seems, is “not many.” 1