  A new pair of asymptotic invariants for finitely presented groups, called
intrinsic and extrinsic tame filling functions, are introduced. These filling
functions are quasi-isometry invariants that strengthen the notions of
intrinsic and extrinsic diameter functions for finitely presented groups. We
show that the existence of a (finite-valued) tame filling function implies that
the group is tame combable. Bounds on both intrinsic and extrinsic tame filling
functions are discussed for stackable groups, including groups with a finite
complete rewriting system, Thompson's group F, and almost convex groups.
