Tame filling invariants for groups

TL;DR

This paper introduces intrinsic and extrinsic tame filling functions as new quasi-isometry invariants for finitely presented groups, linking them to tame combability and analyzing bounds for specific classes of groups.

Contribution

It defines and explores the properties of tame filling functions, establishing their invariance under quasi-isometry and their relation to tame combability.

Findings

Tame filling functions are quasi-isometry invariants.

Existence of finite-valued tame filling functions implies tame combability.

Bounds are provided for stackable groups, including groups with rewriting systems, Thompson's group F, and almost convex groups.

Abstract

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.