What has "$\delta$ Gromov hyperbolic space" got to do with $\mathbb{H}_n$?

Question

The start of section $2$ of this paper, http://homes.cs.washington.edu/~jrl/papers/kl06-neg.pdf defines something called a ``$\delta$ Gromov hyperbolic space".

• Can someone explain what has this at all got to do with the usual $\mathbb{H}_n = SO(n,1)/SO(n)$ ?

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• To me one of the most important features of $\mathbb{H}_n$ is the scale invariance property of its metric $d( \lambda (\vec{x}_1, z_1 ), \lambda (\vec{x}_2, z_2 ) ) = d( (\vec{x}_1, z_1 ), (\vec{x}_2, z_2 ) ) = cosh^{-1} (1 + \frac{\vert \vert \vec{x}_1 - \vec{x}_2\vert \vert ^2 + (z_1-z_2)^2 }{2z_1 z_2} )$

  Does this property have an analogue in the $\delta$-Gromov hyperbolic space?

• Does these exist a notion of doing harmonic analysis on the $\delta$-Gromov hyperbolic space?

• Does the $\delta$-Gromov hyperbolic space have a $G/H$ way of thinking of it where $G$ is some (Lie) group and $H$ is a normal (Lie)-subgroup of it ?

Answer 1

First, $\mathbb{H}^n$ is a complete Riemannian manifold of constant sectional curvature $-1$.

Second, for each $\kappa<1$ there exists $\delta>0$ such that every complete Riemannian manifold $M$ whose sectional curvatures are all $\le \kappa$ is $\delta$-hyperbolic; by definition means that every geodesic triangle $T \subset M$ is $\delta$ thin, meaning that each side of $T$ is contained in the $\delta$ neighborhood of the union of the other two sides of $T$.

So the main relation between $\mathbb{H}^n$ and $\delta$-hyperbolic spaces is that the latter are generalizations of the former.

Here is perhaps the most important feature of $\mathbb{H}^n$ which generalizes to $\delta$-hyperbolic spaces $X$.

• The space $X$ has an "ideal boundary" $\partial X$, consisting of asymptotic classes of geodesic rays. The metric topology on $X$ extends to a topology on $\overline X = X \cup \partial X$ in a natural way, in particular the action on $X$ of the group of isometries of $X$ extends to an action on $\overline X$. In the special case where $\mathbb{H}^n$ is modelled by the open unit ball in $\mathbb{R}^n$ with the Poincare metric, the space at infinity is identified with the unit sphere.

Most of the features that you list do not extend to general Gromov hyperbolic spaces. The main reason is that you have listed properties or features that are either rather analytic in nature (such as the cosh formula), or are very homogeneous. The ideal of generalizing $\mathbb{H}^n$ to $\delta$-hyperbolic spaces is to capture the "large scale" properties of $\mathbb{H}^n$, at the expense of ignoring the "infinitesmal" properties of analysis and the homogeneous properties of hyperbolic spaces.

Added: To briefly sketch the proof that $\mathbb{H}^n$ is $\delta$-hyperbolic, with $\delta=1$:

• The "worst case" occurs when you let the three vertices of a triangle converge to three "ideal" points on the sphere at infinity, forming what is called an "ideal triangle".
• All ideal triangles in all dimensions are isometric to each other, so you might as well just consider one special case, the ideal triangle with ieal vertices $0,1,\infty \in \partial \mathbb{H}^2$ in the upper half plane model of $\mathbb{H}^2$. Now it's not too hard to see, using the Riemannian metric $ds^2=\frac{dx^2+dy^2}{y^2}$ that the side with ideal endpoints $0,\infty$ is contained in the $1$-neighborhood of union of the other two sides.

You can get a lot more details on this kind of stuff in a textbook of hyperbolic geometry such as Ratcliffe.