Homological and homotopical Dehn functions are different

TL;DR

This paper constructs the first known examples of finitely-presented groups where the homological and homotopical Dehn functions differ, highlighting fundamental differences in their geometric and algebraic properties.

Contribution

It provides the first explicit examples of finitely-presented groups with different homological and homotopical Dehn functions, using amalgamation techniques.

Findings

Homological and homotopical Dehn functions can differ in finitely-presented groups.

Constructed examples involve amalgamating free-by-cyclic groups with Bestvina-Brady groups.

Demonstrates fundamental differences in filling functions related to group geometry.

Abstract

The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, while the homotopical Dehn function measures fillings of curves by disks. Since the two definitions involve different sorts of boundaries and fillings, there is no a priori relationship between the two functions, but prior to this work there were no known examples of finitely-presented groups for which the two functions differ. This paper gives the first such examples, constructed by amalgamating a free-by-cyclic group with several Bestvina-Brady groups.