Tame combings, almost convexity and rewriting systems for groups

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. 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 tame 1-combable.