Intrinsic tame filling functions are equivalent to intrinsic diameter functions

Intrinsic tame filling functions are quasi-isometry invariants that are refinements of the intrinsic diameter function of a group. The main purpose of this paper is to show that every finite presentation of a group has an intrinsic tame filling function that is equivalent to its intrinsic diameter function. We also introduce some alternative filling functions–based on concepts similar to those used to define intrinsic tame filling functions–that are potential proper refinements of the intrinsic diameter function.