Inducing maps between Gromov boundaries

TL;DR

This paper introduces visual and radial functions between Gromov hyperbolic spaces that induce continuous boundary maps, exploring their properties and implications for dimension theory, including dimension raising phenomena.

Contribution

It defines and characterizes visual and radial functions inducing boundary maps, establishing their relationship with Hölder maps and large-scale geometric properties.

Findings

Radial functions induce Hölder boundary maps.

Every Hölder boundary map arises from a radial function.

Radial functions can raise asymptotic dimension.

Abstract

It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov boundaries. In this paper we introduce the class of visual functions $f$ that do induce continuous maps $\tilde f$ between Gromov boundaries. Its subclass, the class of radial functions, induces Hoelder maps between Gromov boundaries. Conversely, every Hoelder map between Gromov boundaries of visual hyperbolic spaces induces a radial function. We study the relationship between large scale properties of f and small scale properties of $f$, especially related to the dimension theory. In particular, we prove a form of the dimension raising theorem. We give a natural example of a radial dimension raising map and we also give a general class of radial functions that raise asymptotic dimension.