The Dehn function of Baumslag's Metabelian Group

TL;DR

This paper proves that Baumslag's metabelian group has an exponential Dehn function, highlighting a significant difference from its torsion analogues which have quadratic Dehn functions, thus advancing understanding of its geometric properties.

Contribution

The paper establishes that Baumslag's group exhibits an exponential Dehn function, contrasting with the quadratic Dehn functions of its torsion analogues, and provides a detailed proof of this property.

Findings

Baumslag's group has an exponential Dehn function.

Torsion analogues of Baumslag's group have quadratic Dehn functions.

The result differentiates the geometric complexity of Baumslag's group from its torsion variants.

Abstract

Baumslag's group is a finitely presented metabelian group with a Z \wr Z subgroup. There is an analogue with an additional torsion relation in which this subgroup becomes C_m \wr Z. We prove that Baumslag's group has an exponential Dehn function. This contrasts with the torsion analogues which have quadratic Dehn functions. v.2: version for publication, incorporating referee's suggestions.