A metric between quasi-isometric trees

It is known that PQ-symmetric maps on the boundary characterize the quasi-isometry type of visual hyperbolic spaces, in particular, of geodesically complete \br-trees. We define a map on pairs of PQ-symmetric ultrametric spaces which characterizes the branching of the space. We also show that, when the ultrametric spaces are the corresponding end spaces, this map defines a metric between rooted geodesically complete simplicial trees with minimal vertex degree 3 in the same quasi-isometry class. Moreover, this metric measures how far are the trees from being rooted isometric.


Introduction
The study of quasi-isometries between trees and the induced maps on their end spaces has a voluminous literature. This is often set in the more general context of hyperbolic metric spaces and their boundaries. See Bonk and Schramm [2], Buyalo and Schroeder [4], Ghys and de la Harpe [6], and Paulin [14] to name a few.
For a quasi-isometry f : X → Y between Gromov hyperbolic, almost geodesic metric spaces, Bonk and Schramm define [2, Proposition 6.3] the induced map ∂f : ∂X → ∂Y between the boundaries at infinity and prove [2, Theorem 6.5] that ∂f is PQ-symmetric with respect to any metrics on ∂X and ∂Y in their canonical gauges. Moreover, they prove that a PQ-symmetric map between bounded metric spaces can be extended to a map between their hyperbolic cones and obtain that a PQ-symmetric map between the boundaries at infinity of Gromov hyperbolic, almost geodesic metric spaces, implies a quasi-isometry equivalence between the spaces. In the special case that X and Y are R-trees, ∂X = end(X, v), ∂Y = end(Y, w) and the end space metrics are in the canonical gauges for any choice of roots.
Another source for the result that quasi-isometries between R-trees induce PQsymmetric homeomorphisms on their ultrametric end spaces, is Buyalo and Schroeder [4,Theorem 5.2.17]. They work with Gromov hyperbolic, geodesic metric spaces and with visual boundaries on their boundaries. When specialized to R-trees, these boundaries are the ultrametric end spaces. For the converse in this approach see [12].
In [10] we defined bounded distortion equivalences and we proved that bounded distortion equivalences characterize power PQ-symmetric homeomorphisms between certain classes of bounded, complete, uniformly perfect, ultrametric spaces which we called pseudo-doubling. This class includes those ultrametric spaces arising up to similarity as the end spaces of bushy trees.
Bounded distortion equivalences can be seen from a geometrical point of view as a coarse version of quasiconformal homeomorphisms (see for example [1], [7] and [8] for a geometric approach to quasiconformal maps) where instead of looking at the distortion of the spheres with the radius tending to 0 we consider the distortion of all of them.
Of central importance in the theory of quasiconformal mappings are Teichmüller spaces. The Teichmüller space is the set of Riemannian surfaces of a given quasiconformal type and the Teichmüller metric measures how far are the spaces from being conformal equivalent. There is also an extensive literature on this, see for example [1] and [8]. The question we deal with in this paper is to see if something similar to a Teichmüller metric can be defined with these bounded distortion equivalences playing the role of the quasiconformal homeomorphisms.
Here we consider the set of ultrametric spaces of a given PQ-symmetry type. What is obtained is not a metric in the general framework of pseudo-doubling ultrametric spaces because it fails to hold the triangle inequality, nevertheless, adding a condition on the metrics, it is enough to characterize what we call here the branching of the space which is a natural concept when looking at the ultrametric space as boundary of a tree. Theorem 1.1. Let (U, d), (U ′ , d ′ ) be ultrametric spaces. If the metrics d, d ′ are pseudo-discrete, then ̺(U, U ′ ) = 0 if and only if (U, d) and (U ′ , d ′ ) have the same branching.
As we mentioned before, if the ultrametric spaces are end spaces of R-trees, then the PQ-symmetry type of the boundary corresponds to the quasi-isometry type of the trees. In the case in which the trees have minimal vertex degree 3, the defined map ̺ is a metric measuring how far are the trees from being rooted isometric. This means that in the category T ≥3 the branching is enough to characterize isometry type, the quasi-isometry type and the "distance" between quasi-isometric objects.

Preliminaries on trees, end spaces, and ultrametrics
In this section, we recall the definitions of the trees and their end spaces that are relevant to this paper. We also describe a well-known correspondence between trees and ultrametric spaces. See Feȋnberg [5] for an early result along these lines and Hughes [9] for additional background.   Let F, G ∈ end(T, v).
(1) The Gromov product at infinity is In this article a map is a function that need not be continuous.
x ∈ X and all t ≥ 0. The function η is called the control function of f . Definition 2.8. A quasi-symmetric map is said to be power quasi-symmetric or PQ-symmetric, if its control function is of the form

Bounded distortion equivalences between ultrametric spaces.
Let us recall some definitions as stated in [10].
Consider (U, d), (U ′ , d ′ ) two bounded distortion equivalent ultrametric spaces. Let K U,U ′ be the greatest lower bound for K such that there exist a homeomorphism f : given U a bounded distortion equivalence class of ultrametric spaces, let us define ̺ : U × U → R + such that If there exists such a bijection we say that the ultrametric spaces have the same branching.
Remark 3.5. Note that this defines an equivalence relation.
Definition 3.6. Let (X, d) be a metric space. We say that d is pseudo-discrete if there is some δ > 1 such that for every non-empty sphere S(x, r), and any y such that r δ < d(y, x) < r · δ then d(x, y) = r. Proof. If the branching is the same then there is a homeomorphism h such that the spheres are preserved, this is, D h (x, r) = 1 and D h −1 (h(x),r) = 1 for any x ∈ U and any r > 0. Suppose that both ultrametric spaces are pseudo-discrete with the same constant δ > 1 and let ̺(U, U ′ ) = 0 which is equivalent to saying that K U,U ′ = 1. Then, for any ε > 1 there exists a homeomorphism h ε : by the properties of the ultrametric and, as we just proved, taking ǫ < δ, Let us recall the following definition from [10] Definition 3.8. A metric space is pseudo-doubling if for every C > 1 there exist N ∈ N such that: if 0 < r < R with R/r = C and x ∈ X, then there are at most N balls B such that B(x, r) ⊆ B ⊆ B(x, R).  For end(T, v): It is immediate to check that there is no sphere distorted by h −1 n and the unique spheres distorted by h n are S 1 (F 0 ) and S 1 (H 0 ) in end(T, v) where D hn (F 0 , 1) = e 1 n = D hn (G 0 , 1). Therefore, for any K > 1, there exists some n ∈ N such that e 1 n < K and K (T,v),(T ′ ,w) = 1 which implies that ̺((T, v), (T ′ , w)) = 0.
Clearly, the ramification is not the same. From the cardinality together with the bounded distortion condition we know that h(S 1 (F i )) corresponds to S 1 (h(F ′ i )) for any i, and in (T, v) we have
If there exist such a bijection, we say that (T, v) and (T ′ , w) have the same branching. Now, from Proposition 3.7, we have: In, particular, the end space metric of a rooted geodesically complete simplicial tree is pseudo-discrete with δ = e. This is not true in general for rooted geodesically complete R-trees as we saw in Example 3.10.
Definition 4.5. An R-tree T is bushy if there is a constant K > 0, called a bushy constant, such that for any point x ∈ T there is a point y ∈ T such that d(x, y) < K and T \{y} has at least 3 unbounded components.
The following result is from [10]. If such a map exists, we say that (T, v) and (T ′ , w) are rooted isometric, which defines an equivalence relation.
Let us consider, from now on, the category T of rooted isometry classes of rooted geodesically complete simplicial bushy trees.
Given (S, x) ∈ T , let [(S, x)] be the class of rooted geodesically complete R-trees quasi-isometric to (S, x) (i.e. whose end space is bounded distortion equivalent to end(S, x)).
The extended version, allowing the image of ̺ to be ∞ together with Theorem 4.6, implies the following. Let us recall that, from [2] (see also [12]), as a particular case we have that Let us consider the restriction of the category T to rooted geodesically complete simplicial trees with valence at least 3 at each vertex, T ≥3 . Let [(R, z)] ⊂ T ≥3 be the class of trees quasi-isometric to (R, z) (i.e. whose end space is bounded distortion equivalent to end(R, z)). Proof. The if part is obvious.
Suppose that ̺((T, v), (T ′ , w)) = 0. Since the trees are simplicial D g (F, r) ∈ {e k | k = 0, 1, ...} for any homeomorphism g and any radius r > 0. Therefore, there is a homeomorphism h : end(T, v) → end(T ′ , w) such that D h (F, r) = 1 for any F ∈ end(T, v) and any r > 0. Let us prove that h is an isometry by induction on the Gromov product.
Let us recall the following corollary in [9].  Note that this distance depends on the ramification of the tree and not on its isometry type. Therefore, the root plays an important role and it is immediate to check that from the same tree with two different roots (T, v), (T, w) it may happen that ̺((T, v), (T, w)) > 0. Nevertheless, it can be bounded above by a constant depending on the distance between the roots since ̺((T, v), (T, w)) ≤ ln (1 + d(v, w)). The symmetric property follows immediately from the definition. Hence, it suffices to check the triangle inequality. Let and ̺((T ′ , w), (T ′′ , x)) = d 2 = ln(1 + 2ln(K (T ′ ,w),(T ′′ ,x) )).
To simplify the notation through the proof let us denote K (T,v),(T ′ ,w) as K 1 and K (T ′ ,w),(T ′′ ,x) as K 2 . Since K 1 and K 2 are greatest lower bounds, there is a homeomorphism . It suffices to observe that for any F ∈ end(T, v) and G, H ∈ S(F, ε), there exists some ε ′ > 0 such that h 1 (G), h 1 (H) ∈ S(h 1 (F ), ε ′ ) and, as- Case 2. If K 1 > 1 then K 1 ≥ e and lnK 1 ≥ 1.
Let F i be the set of branches F ′ ∈ end(T ′ , w) such that (h(F )|F ′ ) w = t 0 + i with i = 0, t 1 − t 0 . For any F ′ 1 , F ′ 2 ∈ F i , the bounded distortion condition with respect to h 1 (F ) implies that ) ∈ F i for any i, which means that either (h 1 (F ), h −1 2 (F ′′ j0 )) w < t 0 or (h 1 (F ), h −1 2 (F ′′ j0 )) w > t 1 . In both cases, the bounded distortion condition is not hold and (1) leads to a contradiction.
Since ̺((T, v), (T ′′ , x)) = ln(1 + 2ln(K 3 )) it follows that Given a rooted simplicial tree (T, v) and a vertex x ∈ T , let us denote the order of x by ord(x) and let us call the vertices whose geodesic to the root v contains x, descendants of x. If k ∈ N, by desc k (x) we denote the set of descentants of x at a distance k whith the canonical metric on the tree (edges having length 1). Given x ∈ (T, v), we denote by T x = {y ∈ T | x ∈ [v, y]} is also a tree. If (T, v) is geodesically complete, so it is (T x , x) and there is a canonical injection j : end(T x , x) → end(T, v). Abusing of the notation we may consider end(T x , x) ⊂ end(T, v). Proof. Consider any bijection h : end(T, v) → end(T ′ w). Let x 1 , ..., x m be m different vertices in desc 1 (x) and let F i = end(T xi , x i ) ⊂ end(T, v) for i = 1, m. There is a unique vertex y ∈ T ′ such that h(F ) ∈ end(T ′ y , y) and |wy| is maximal. Then, there are three branches F i1 , F i2 , F i3 such that [h(F i1 )|h(F i2 )] = [w, y], [h(F i1 )|h(F i3 )] = [w, y] and (h(F i2 )|h(F i3 )) w ≥ |wy|+D. Therefore, D h (F 1 , e −||x|| ) ≥ D and ̺((T, v), (T ′ w)) ≥ ln(1 + 2D).