Isoperimetric functions of groups and their subgroups

Abstract

The objects of interest in this thesis are various isoperimetric functions of non{ compact topological spaces. Such functions are of classical interest in Riemannian Geometry, and are used as group{theoretic invariants in Geometric Group Theory. Very little is known about the general relationship between isoperimetric functions of a group and its subgroups, and it is this problem which this thesis aims to address. By using algebraic techniques, we will give sufficient conditions for an isoperimetric function of a subgroup H [less than or equal to] G to be bounded above by that of the ambient group. This result contrasts with known examples illustrating that this relationship does not always hold when our conditions are relaxed. We conclude our investigation by discussing applications to hyperbolic groups and how our main result can be used as a \negativity test" to show that a group does not contain particular subgroups.