To state the main result of the paper we start with two definitions: A metric space \(X\) has “bounded growth at some scale” if there are constants \(R>r>0\) and a positive integer \(N\) such that every open ball of radius \(R\) in \(X\) can be covered by \(N\) open balls of radius \(r\). A metric space \(X\) is “roughly similar” to a metric space \(Y\), if there is a map \(f:X\to Y\) and constants \(k,\lambda> 0\), such that \(|\lambda d_X(x,y)- d_Y(f(x), f(y)) |<k\), for all \(x,y\in X\).
Then the main result is the following. Let \(X\) be a Gromov hyperbolic geodesic metric space with bounded growth at some scale. Then there exists an integer \(n\) such that \(X\) is roughly similar to a convex subset of the hyperbolic \(n\)-space \(\mathbb{H}^n\).
The proof goes by first showing that the Gromov boundary \(\partial X\) has finite Assouad dimension. Then Assouad’s theorem implies that after rescaling \(\partial X\) has a bilipschitz embedding into \(\mathbb{R}^{n-1}\), for some \(n\). As \(\mathbb{R}^{n-1}\cup \infty\) is the boundary of \(\mathbb{H}^n\) the embedding of \(\partial X\) into \(\partial \mathbb{H}^n\) induces an embedding of \(X\) into \(\mathbb{H}^n\).
In the last section of the paper the relation of a Gromov hyperbolic space to its boundary is investigated. A characterization of the hyperbolic plane up to rough quasi-isometries is given.