THEORY OF Zn-STRUCTURES

Abstract

The goal of this dissertation is to extend the notion of $\mathcal{Z}$-structures to the family of Type $F_n$ groups; those being groups whose $K(G,1)$ complex has a finite $n$-skeleton. At the heart of this extension lies the idea of passing from groups acting on contractible space to groups acting on $n$-connected spaces. This means the foundational theorems of boundary swapping and shape equivalence need to be revised and make up a substantial portion of this work. These revised theorems are stated in terms of a generalized structure coined $\\mathcal{Z}_n$-structures. After laying the foundation work for a theory of $\\mathcal{Z}_n$-structures, we explore how these new structures relate to other well-known compactifications such as the one-point and end-point compactifications along with $\mathcal{Z}$-compactifications themselves.