Characterising trees and hyperbolic spaces by their boundaries

TL;DR

This paper investigates how various hyperbolic spaces and trees can be uniquely characterized and reconstructed from their boundary structures using cross ratios, unifying previous theoretical frameworks.

Contribution

It provides a uniform proof that these spaces can be reconstructed from their boundary cross ratios, integrating work by Tits and Bourdon.

Findings

Spaces can be reconstructed from boundary cross ratios

Unified approach for trees and hyperbolic spaces

Bridges work of Tits and Bourdon

Abstract

We use the language of proper CAT(-1) spaces to study thick, locally compact trees, the real, complex and quaternionic hyperbolic spaces and the hyperbolic plane over the octonions. These are rank 1 Euclidean buildings, respectively rank 1 symmetric spaces of non-compact type. We give a uniform proof that these spaces may be reconstructed using the cross ratio on their visual boundary, bringing together the work of Tits and Bourdon.