Large scale dimension theory of metric spaces

Abstract

This thesis studies the large scale dimension theory of metric spaces. Background on dimension theory is provided, including topological and asymptotic dimension, and notions of nonpositive curvature in metric spaces are reviewed. The hyperbolic dimension of Buyalo and Schroeder is surveyed. Miscellaneous new results on hyperbolic dimension are proved, including a union theorem, an estimate for central group extensions, and the vanishing of hyperbolic dimension for countable abelian groups. A new quasi-isometry invariant called weak hyperbolic dimension (abbreviated $\wdim$) is introduced and developed. Weak hyperbolic dimension is computed for a variety of metric spaces, including the fundamental computation $\wdim \Hyp^n = n-1$. An estimate is proved for (not necessarily central) group extensions. Weak dimension is used to obtain the quasi-isometric nonembedding result $\Hyp^4 \not \rightarrow \Sol \times \Sol$ and possible directions for further nonembedding applications are explored.