# Gromov hyperbolicity constant vs. Gromov-Hausdorff distance to a tree

Let $$X$$ be a compact, geodesic metric space which is Gromov hyperbolic with a constant $$\delta>0$$. To fix scaling, let us also assume that $$X$$ has diameter $$1$$.

To fix a definition of Gromov hyperbolicity, I will use the first definition here under Definitions using triangles''.

I understand that mostly one does not care about the precise value of $$\delta$$ because we think of this as an asymptotic notion, and that all compact metric spaces are Gromov hyperbolic for $$\delta$$ larger than their diameters. Nonetheless, in this question I would like to care about the size of $$\delta$$.

Gromov hyperbolicity is often motivated as saying that a metric space looks tree-like'' (at least at large scales), since trees are Gromov hyperbolic with $$\delta=0$$. So:

Question 1: Given $$\epsilon>0$$, is there $$\delta>0$$ such that if $$X$$ is $$\delta$$-hyperbolic, then $$X$$ is within Gromov-Hausdorff distance $$\epsilon$$ from a compact metric tree?

Here a metric tree means a geodesic metric space in which each pair of points is joined by a unique arc.

I am nervous that someone might answer the above question, if the answer is indeed yes, by an appeal to ultralimits. Although this would be fine for question 1, I would be happier with a more constructive understanding. So I also ask:

Question 2: If the answer to question 1 is yes, is there an explicit estimate for the rate of decay for $$\delta$$ given $$\epsilon$$? Equivalently, is there an upper bound, tending to zero with $$\delta$$, for the maximal Gromov-Hausdorff distance to a tree among all metric spaces $$X$$ with the above properties?

I apologize if these questions are simple or well-known. If the latter, a reference would be great.

• Hint: Round disks in hyperbolic plane will provide counter examples to Q1. Nov 20, 2020 at 0:11

Just to remove this question from the un-answered list. First of all, if $$d_{GH}(X,Y)\le \epsilon$$ then there is a $$(1,2\epsilon)$$-quasi-isometry $$X\to Y$$. See for instance

Burago, D.; Burago, Yu.; Ivanov, S., A course in metric geometry, Graduate Studies in Mathematics. 33. Providence, RI: American Mathematical Society. 2001.

If a geodesic metric space $$X$$ is $$(L,A)$$-quasi-isometric to a tree then it satisfies the following bottleneck property:

For every pair of points $$x, y\in X$$ and every mid-point $$m$$ of this pair, every path in $$X$$ connecting $$x$$ and $$y$$ passes through the open ball $$B(m, L(A/2 +D) +A)$$ centered at $$m$$.

Here $$D=D(L,A)$$ is some function which I can write for you if you really want. Incidentally, the bottleneck property also characterizes geodesic metric spaces quasi-isometric to trees. For a proof, see Theorem 4.6 in

J. Manning, Geometry of pseudocharacters. Geom. Topol. 9 (2005), 1147–1185.

On the other hand, if $$X_R$$ is the closed ball $$\bar{B}(m, R)$$ of radius $$R$$ in the hyperbolic plane and $$x, y$$ are boundary points of $$\bar{B}(m,R)$$ at the distance $$2R$$ from each other (with the mid-point $$m$$), then either arc on the boundary circle of $$\bar{B}(m,R)$$ connecting the points $$x, y$$ avoids $$B(m,R)$$. Hence, assuming that $$R>(A +D) +2A$$, the GH-distance from $$X_R$$ to any tree is $$>A$$. Here $$D=D(1,2A)$$. Of course, all the spaces $$X_R$$ are $$\delta$$-hyperbolic, where $$\delta$$ is the hyperbolicity constant of the hyperbolic plane.

Thus, your question 1 has negative answer. However, the following might be true, I am not sure:

Conjecture. There exists a separable geodesic Gromov-hyperbolic spaces $$U$$ (the "universal hyperbolic space") and a function $$\epsilon(\delta)$$ such that every compact $$\delta$$-hyperbolic hyperbolic space $$X$$ is within Gromov-Hausdorff distance $$\epsilon(\delta)$$ from a subset of $$U$$.

M. Bonk, O. Schramm, Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10 (2000), no. 2, 266–306.

A straightforward estimates show that any $$n$$-point $$\delta$$-hyperbolic space lies on distance at most $$n{\cdot}\delta$$ from a $$0$$-hyperbolic space. The following statement improves this bound to $$\log n\cdot \delta$$. I learned it from Rostislav Matveyev, who attributed it to Étienne Ghys.

Let $$d$$ be a $$\delta$$-hyperbolic metric on an $$n$$-points set $$F$$. Then there is a $$0$$-hyperbolic metric $$d'$$ on $$F$$ such that $$d \leqslant d' \leqslant d+\mathrm{const}\cdot\log n\cdot \delta.$$

To prove it, you choose a base point and define new Gromov's product $$(\ |\ )'$$ using the old one $$(\ |\ )$$ $$(x|y)'\mathrel{:=}\max \{\,\min \{\,(x_0|x_1),\dots,(x_{m-1}|x_m)\,\}\,\},$$ where the maximum is taken for all chains of points $$x=x_0,\dots, x_m=y$$. Plus you need to apply the following lemma:

For any sequence of points $$x_0,\dots,x_m$$ in a $$\delta$$-hyperbolic space, we have $$(x_0|x_m)\geqslant \min \{\,(x_0|x_1),\dots,(x_{m-1}|x_m)\,\}-\mathrm{const}\cdot\log m\cdot \delta.$$

The proof is recursive application of the inequality for Gromov's product for triples $$(x_i|x_k)\geqslant \min\{\,(x_{i}|x_j),(x_{j}|x_k)\,\}- \delta.$$ You start with $$(x_0|x_m)$$ and choose roughly the middle index each time. This way you get $$\lceil\log_2 n\rceil$$ iterations; each iteration gives you extra $$\delta$$.

Postscript. Actually, it is 6.1.B in Gromov's "Hyperbolic groups".