Dehn Function of Finitely Presented Metabelian Groups

TL;DR

This paper establishes upper bounds for the Dehn functions of finitely presented and finitely generated metabelian groups, and demonstrates embeddings of wreath products into metabelian groups with exponential Dehn functions.

Contribution

It provides new bounds for Dehn functions in metabelian groups and shows how wreath products embed into groups with exponential Dehn functions.

Findings

Upper bounds for Dehn functions of finitely presented metabelian groups.

The same bounds apply to the relative Dehn function of finitely generated metabelian groups.

Embedding of wreath products into metabelian groups with exponential Dehn functions.

Abstract

In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. In addition, we prove that the same upper bound works for the relative Dehn function of a finitely generated metabelian group. We also show that every wreath product of a free abelian group of finite rank with a finitely generated abelian group can be embedded into a metabelian group with exponential Dehn function.