Subgroup Distortion and the Relative Dehn Functions of Metabelian Groups

TL;DR

This paper explores the relationship between the relative Dehn function and subgroup distortion in metabelian groups, showing polynomial bounds for certain extensions and exponential bounds for finitely presented groups.

Contribution

It establishes a connection between relative Dehn functions and subgroup distortion in metabelian groups, providing bounds based on group extensions.

Findings

Relative Dehn function linked to subgroup distortion in wreath products

Polynomial bounds for Dehn functions in certain metabelian group extensions

Exponential bounds for Dehn functions in finitely presented metabelian groups

Abstract

We show the connection between the relative Dehn function of a finitely generated metabelian group and the distortion function of a corresponding subgroup in the wreath product of two free abelian groups of finite rank. Further, we show that if a finitely generated metabelian group $G$ is an extension of an abelian group by $\mathbb Z$ the relative Dehn function of $G$ is polynomially bounded. Therefore, if $G$ is finitely presented, the Dehn function is bounded above by the exponential function up to equivalence.