Dehn Functions of Metabelian Groups

Abstract

In this thesis, we focus on topics related to the Dehn function of a finitely presented metabelian group and relative Dehn function of a finitely generated metabelian groups. We establish a commutative algebra approach to estimate upper bounds for the Dehn function of a given finitely presented metabelian group. This approach yields much wilder results than estimating the upper bound. It first gives a uniform upper bound for Dehn functions of all finitely presented metabelian groups. Secondly, we give a similar result for Dehn functions relative to the variety of metabelian groups. For a finitely presented metabelian group, we also analyze contributions of different parts, including the metabelian part and the commutative algebra part, to its Dehn function. Finally, we use this technique to compute and estimate the Dehn functions as well as the relative Dehn functions for various examples.