Mediterranean Journal of Mathematics

, Volume 13, Issue 5, pp 2733–2752 | Cite as

Inducing Maps Between Gromov Boundaries

Article

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 Hölder maps between Gromov boundaries. Conversely, every Hölder 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 \({\tilde{f}}\), 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.

Keywords

Gromov boundary hyperbolic space dimension raising map visual metric coarse geometry 

Mathematics Subject Classification

Primary 53C23 Secondary 20F67 20F65 20F69 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.University of TennesseeKnoxvilleUSA
  2. 2.University of LjubljanaLjubljanaSlovenia

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