  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.
