Geometry and topology of horofunction compactifications

This thesis concerns the global geometry and topology of the horofunction compactification of various metric spaces. In Chapters 4-6 we study the horofunction compactification of various homogeneous Finsler metric spaces, and establish a homeomorphism between the horofunction compactification and the dual unit ball in the tangent space at the base point. This homeomorphism establishes a one-to-one correspondence between the geometric parts of the horoboundary and the relative interiors of faces of the dual ball. In Chapter 7 we build on the work of Gutiérrez, and explore the topology and geometry of the horofunction compactification of infinite dimensional ℓᵖ spaces for 1 ≤ p < ∞. We show a clear disconnect between the global geometry and topology of the horofunction compactification in the infinite dimensional case versus the finite dimensional case. We also establish a marked difference in the behaviour of the horoboundary of ℓ¹ versus ℓᵖ for 1 < p < ∞. Chapter 8 deals with the horofunction compactification of infinite-dimensional spin factors considered as JB-algebras. We show that the exponential map extends to a geometry preserving homeomorphism on the boundary, mapping the horofunction compactification of the spin factor homeomorphically onto the horofunction compactification of the positive cone equipped with the Thompson metric. We conclude by showing that, considering an infinite dimensional Hilbert space as the tangent space at the identity of infinite dimensional real hyperbolic space, the exponential extends to a homeomorphism between the horofunction compactification of infinite dimensional Hilbert space and infinite dimensional real hyperbolic space.