Stackable groups, tame filling invariants, and algorithmic properties of groups

Abstract

We introduce a combinatorial property for finitely generated groups called stackable that implies the existence of an inductive procedure for constructing van Kampen diagrams with respect to a canonical finite presentation. We also define algorithmically stackable groups, for which this procedure is an effective algorithm. This property gives a common model for algorithms arising from both rewriting systems and almost convexity for groups. We also introduce a new pair of asymptotic invariants that are filling inequalities refining the notions of intrinsic and extrinsic diameter inequalities for finitely presented groups. These tame filling inequalities are quasi-isometry invariants, up to Lipschitz equivalence of functions (and, in the case of the intrinsic tame filling inequality, up to choice of a sufficiently large set of defining relators). We show that radial tameness functions are equivalent to the extrinsic tame filling inequality condition, and so intrinsic tame filling inequalities can be viewed as the intrinsic analog of radial tameness functions. We discuss both intrinsic and extrinsic tame filling inequalities for many examples of stackable groups, including groups with a finite complete rewriting system, Thompson's group F, Baumslag-Solitar groups and their iterates, and almost convex groups. We show that the fundamental group of any closed 3-manifold with a uniform geometry is algorithmically stackable using a regular language of normal forms.