Presentations and isoperimetric functions of finitely generated metabelian groups

This work is motivated by Philip Hall's result in 1954 that finitely generated metabelian groups satisfy maximal condition for normal subgroups. It turns out that they have finite metabelian presentations. We investigate metabelian presentations for a varied collection of finitely generated metabelian groups and the relative isoperimetric functions associated with these presentations. The most general result in this thesis is that the isoperimetric function of the wreath product of a finitely generated abelian group by another is bounded above by a polynomial which depends only on the rank of the second group. We also introduce the notion of the isoperimetric function for a finitely generated module over a polynomial ring in finitely many variables. We do not have enough knowledge finding lower bounds for groups having been dealt with in this research but the relative centralized isoperimetric functions are discussed in general.