Quasi-isometries need not induce homeomorphisms of contracting boundaries with the Gromov product topology

TL;DR

The paper demonstrates that in non-hyperbolic spaces, quasi-isometries may not induce homeomorphisms on the contracting boundary with the Gromov product topology, challenging assumptions from hyperbolic geometry.

Contribution

It provides explicit examples showing that quasi-isometries do not always preserve the topology of the contracting boundary in non-hyperbolic spaces.

Findings

Quasi-isometries can fail to induce homeomorphisms on the contracting boundary.

Continuity of boundary maps can fail even in CAT(0) spaces.

Explicit counterexamples are constructed to illustrate these phenomena.

Abstract

We consider a `contracting boundary' of a proper geodesic metric space consisting of equivalence classes of geodesic rays that behave like geodesics in a hyperbolic space. We topologize this set via the Gromov product, in analogy to the topology of the boundary of a hyperbolic space. We show that when the space is not hyperbolic, quasi-isometries do not necessarily give homeomorphisms of this boundary. Continuity can fail even when the spaces are required to be CAT(0). We show this by constructing an explicit example.