Movimientos y actividades en pro del agua

2. NORMAS, JURISPRUDENCIA Y DOCTRINA INTERNA PARA LA PRESERVACIÓN DEL AGUA

2.7. Movimientos y actividades en pro del agua

Definition 3.3.9. Let (X1, d1) and (X2, d2) be metric spaces. A map Φ : X1→X2is aquasi-isometric embedding if for everyx, y∈X1

d2(Φ(x),Φ(y))≍+,×d1(x, y).

A quasi-isometric embedding Φ is called a quasi-isometry if its image Φ(X1) is

cobounded inX2, that is, if there existsR >0 such that Φ(X1) isR-dense in X2,

meaning that minx∈X2d(x,Φ(X1)) ≤R. In this case, the spaces X1 and X2 are

said to bequasi-isometric.

Theorem 3.3.10 ([39, Theorem III.H.1.9]). Any geodesic metric space which can be quasi-isometrically embedded into a geodesic hyperbolic metric space is also a hyperbolic metric space.

Remark 3.3.11. Theorem 3.3.10 is not true if the hypothesis of geodesicity is dropped. For example,Ris quasi-isometric to [0,_∞)_{× {}0_{} ∪ {}0_{} ×}[0,_∞)_⊆R2, but the former is hyperbolic and the latter is not.

There are many more examples of hyperbolic metric spaces which we will not discuss; cf. the list in_§1.1.2.

3.4. The boundary of a hyperbolic metric space

In this section we define the Gromov boundary of a hyperbolic metric space

X. The construction will depend on a distinguished pointo∈X, but the resulting space will be independent of which point is chosen. If X is an R-tree, then the boundary ofX will turn out to be the set of infinite branches throughX, i.e. the set of all isometric embeddingsπ : [0,∞)→X sending 0 to o, where o ∈X is a distinguished fixed point. IfX is an algebraic hyperbolic space, then the boundary ofX will turn out to be isomorphic to the space∂X defined in Chapter 2.

To motivate the definition of the boundary, suppose thatX is anR-tree. An infinite branch through X can be approximated by finite branches which agree on longer and longer segments. Suppose that ([o, xn])∞1 is a sequence of geodesic

segments. For eachn, m_∈N, the length of the intersection of [o, xn] and [o, xm] is equal tod(o, C(o, xn, xm)), which in turn is equal tohxn|xmio. Thus, the sequence ([o, xn])∞1 converges to an infinite geodesic if and only if

(3.4.1) _hxn|xmio−−→ n,m ∞.

(Cf. Figure 3.4.1.) The formula (3.4.1) is reminiscent of the definition of a Cauchy sequence. This intuition will be made explicit in Section 3.6, where we will introduce a metametric onX with the property that a sequence inX satisfies (3.4.1) if and only if it is Cauchy with respect to this metametric.

o x1 x2 x3 x5 . . .

Figure 3.4.1. A Gromov sequence in anR-tree

Definition 3.4.1. A sequence (xn)∞1 in X for which (3.4.1) holds is called a

Gromov sequence. Two Gromov sequences (xn)∞1 and (yn)∞1 are calledequivalent

if hxn|ynio−→ n ∞, or equivalently if hxn|ymio−−→ n,m ∞.

In this case, we write (xn)∞1 ∼(yn)∞1 . It is readily verified using Gromov’s inequal-

ity that∼is an equivalence relation on the set of Gromov sequences inX. We will denote the class of sequences equivalent to a given sequence (xn)∞1 by [(xn)∞1 ].

Definition3.4.2. TheGromov boundary ofX is the set of Gromov sequences modulo equivalence. It is denoted ∂X. The Gromov closure orbordification ofX

is the disjoint union bordX :=X∪∂X.

Remark3.4.3. IfX is an algebraic hyperbolic space, then this notation causes some ambiguity, since it is not clear whether∂X represents the Gromov boundary of X, or rather the topological boundary of X as in Chapter 2. This ambiguity will be resolved in _§3.5.1 below when it is shown that the two bordifications are isomorphic.

Remark 3.4.4. In the literature, the ideal boundary of a hyperbolic metric space is often taken to be the set of equivalence classes of geodesic rays under asymptotic equivalence, rather than the set of equivalence classes of Gromov sequences (e.g. [39, p.427]). IfX is proper and geodesic, then these two notions are equivalent [39, Lemma III.H.3.13], but in general they may be different.

Remark 3.4.5. By (d) of Proposition 3.3.3, the concepts of Gromov sequence and equivalence do not depend on the basepoint o. In particular, the Gromov boundary∂X is independent ofo.

3.4. THE BOUNDARY OF A HYPERBOLIC METRIC SPACE 35

3.4.1. Extending the Gromov product to the boundary. We now wish to extend the Gromov product and Busemann function to the boundary “by continuity”. Fix ξ, η ∈ ∂X and z ∈ X. Ideally, we would like to define hξ|ηiz to be

(3.4.2) lim

n,m→∞hxn|ymiz,

where (xn)∞1 ∈ ξ and (ym)∞1 ∈ η. (The definition would then have to be shown

independent of which sequences were chosen.) The naive definition (3.4.2) does not work, because the limit (3.4.2) does not necessarily exist:

Example3.4.6. Let

X ={x∈R2:x2∈[0,1]}

be interpreted as a subspace of R2 _{with the} _L1 _{metric. Then} _X _{is a hyperbolic}

metric space, since it contains the cobounded hyperbolic metric spaceR_{× {}0_}. Its Gromov boundary consists of two points−∞ and +∞, which are the limits of x asx1approaches−∞or +∞, respectively. Lety= (0,1) andz= (1,0). Then for

allx_∈X,_hx_|y_iz=x2. In particular, we can find a sequencexn→+∞such that limn→∞hxn|yiz does not exist.

Fortunately, the limit (3.4.2) “exists up to a constant”:

Lemma 3.4.7. Let (xn)∞1 and(ym)∞1 be Gromov sequences, and fix y, z ∈X.

Then lim inf n,m→∞hxn|ymiz≍+lim sup_n,m_→∞hxn|ymiz (3.4.3) lim inf n→∞hxn|yiz≍+lim sup_n_→∞ hxn|yiz, (3.4.4)

with equality if X is strongly hyperbolic.

Note that except for the statement about strongly hyperbolic spaces, this lemma is simply [172, Lemma 5.6].

Proof of Lemma 3.4.7. Fixn1, n2, m1, m2∈N. By Gromov’s inequality hxn1|ym1iz&+min(hxn2|ym2iz,hxn1|xn2iz,hym1|ym2iz).

Taking the liminf overn1, m1and the limsup over n2, m2 gives

demonstrating (3.4.3). On the other hand, suppose thatX is strongly hyperbolic. Then by (3.3.6) we have exp −hxn1|ym1iz ≤exp −hxn2|ym2iz +exp −hxn1|xn2iz +exp −hym1|ym2iz ; taking the limsup overn1, m1 and the liminf overn2, m2 gives

exp ₋lim inf

n,m→∞hxn|ymiz

≤exp ₋lim sup

n,m→∞hxn|ymiz

+ + exp − lim inf

n1,n2→∞h

xn1|xn2iz

+ + exp ₋ lim inf

m1,m2→∞h

ym1|ym2iz

= exp ₋lim sup

n,m→∞hxn|ymiz

(since (xn)∞1 and (ym)∞1 are Gromov)

demonstrating equality in (3.4.3). The proof of (3.4.4) is similar and will be omitted. Remark3.4.8. Many of the statements in this monograph concerning strongly hyperbolic metric spaces are in fact valid for all hyperbolic metric spaces satisfying the conclusion of Lemma 3.4.7.

Now that we know that it does not matter too much whether we replace the limit in (3.4.2) by a liminf or a limsup, we make the following definition without fear:

Definition 3.4.9. Forξ, η_∈∂X andy, z_∈X, let

As a corollary of Lemma 3.4.7, we have the following:

Lemma3.4.10. Fix ξ, η∈∂X andy, z∈X. For all(xn)∞1 ∈ξand(ym)∞1 ∈η

we have hxn|ymiz−−−−→ n,m,+ hξ|ηiz (3.4.8) hxn|yiz−−−−→ n,+ hξ|yiz (3.4.9) Bxn(y, z)−−−−→ n,+ Bξ(y, z), (3.4.10)

3.4. THE BOUNDARY OF A HYPERBOLIC METRIC SPACE 37

Note that except for the statement about strongly hyperbolic spaces, this lemma is simply [172, Lemma 5.11].

Proof of Lemma 3.4.10. Say we are given (x(ni))∞1 ∈ξ and (y (i) m)∞1 ∈ η for eachi= 1,2. Let xn=    x(1)_n/₂ neven x(2)₍_n₊₁₎_/₂ nodd ,

and defineymsimilarly. It may be verified using Gromov’s inequality that (xn)∞1 ∈ ξand (ym)∞1 ∈η. Applying Lemma 3.4.7, we have

(2)

m )∞1 ∈η gives (3.4.8). A similar

argument gives (3.4.9). Finally, (3.4.10) follows from (3.4.9), (3.4.7), and (j) of Proposition 3.3.3.

IfX is strongly hyperbolic, then all error terms are equal to zero, demonstrating

that the limits converge exactly.

Remark3.4.11. In the sequel, the statement that “ifXis strongly hyperbolic, then all error terms are zero” will typically be omitted from our proofs.

A simple but useful consequence of Lemma 3.4.10 is the following:

Corollary3.4.12. The formulas of Proposition 3.3.3 together with Gromov’s inequality hold for points on the boundary as well, if the equations and inequalities there are replaced by additive asymptotics. IfX is strongly hyperbolic, then we may keep the original formulas without adding an error term.

Proof. For each identity, choose a Gromov sequence representing each element of the boundary which appears in the formula. Replace each occurrence of this element in the formula by the general term of the chosen sequence. This yields a sequence of formulas, each of which is known to be true. Take a subsequence on which each term in these formulas converges. Taking the limit along this subsequence again yields a true formula, and by Lemma 3.4.10 we may replace each limit term by the term which it stood for, with only bounded error in doing so,

and no error if X is strongly hyperbolic. Thus the formula holds as an additive asymptotic, and holds exactly ifX is strongly hyperbolic. Remark 3.4.13. In fact, (a), (c), (d), and (e) of Proposition 3.3.3 hold in bordX in the usual sense, i.e. as exact formulas without additive constants.

Proof. These are the identities where there is at most one Gromov product on each side of the formula. For each element of the boundary, we may simply replace each occurence of that element with the general term of an arbitrary Gromov sequence, take the liminf, and then take the infimum over all Gromov sequences.

Observation 3.4.14. hx|yiz=∞if and only ifx=y∈∂X.

Proof. This follows directly from (3.4.5) and (3.4.6).

3.4.2. A topology on bordX. One can endow the bordification bordX =

X∪∂X with a topological structureT _{as follows: Given}_S_⊆_bord_X_{, write}_S_∈T

(i.e. callS open) if

(I) S∩X is open, and

(II) For eachξ_∈S_∩∂X there existst_≥0 such thatNt(ξ)⊆S, where

Nt(ξ) :=Nt,o(ξ) :={y∈bordX :hy|ξio> t}

Remark3.4.15. The topologyT _{may equivalently be defined to be the unique}

topology on bordX satisfying:

(I) T _↿_X _{is compatible with the metric}_d_{, and}

(II) For eachξ∈∂X, the collection

(3.4.11) _{Nt(ξ) :t≥0} is a neighborhood base forT _at_ξ_.

Remark 3.4.16. It follows from Lemma 3.4.23 below that the sets Nt(ξ) are open in the topologyT_.

Remark 3.4.17. By (d) of Proposition 3.3.3 (cf. Remark 3.4.13), we have

Nt,x(ξ)⊇Nt+d(x,y),y(ξ) for allx, y∈X,ξ∈∂X, andt≥0. Thus the topologyT is independent of the basepointo.

The topologyT _{is quite nice. In fact, we have the following:}

Proposition 3.4.18. The topological space (bordX,T₎_{is completely metriz-}

able. If X is proper and geodesic, thenbordX (and thus also ∂X) is compact. If

X is separable, thenbordX (and thus also ∂X) is separable.

Remark 3.4.19. If X is proper and geodesic, then Proposition 3.4.18 is [39, Exercise III.H.3.18(4)].

3.4. THE BOUNDARY OF A HYPERBOLIC METRIC SPACE 39

Proof of Proposition 3.4.18. We delay the proof of the complete metriz- ability of bordX until Section 3.6, where we will introduce a class of compatible complete metrics on bordX which are important from a geometric point of view, the so-calledvisual metrics.

Since X is dense in bordX, the separability of X implies the separability of bordX. Moreover, since bordX is metrizable (as we will show in Section 3.6), the separability of bordX implies the separability of∂X.

Finally, assume thatXis proper and geodesic; we claim that bordXis compact. Let (xn)∞1 be a sequence in X. If (xn)∞1 contains a bounded subsequence, then

sinceX is proper it contains a convergent subsequence. Thus we assume that (xn)∞1

contains no bounded subsequence, i.e. _kxnk → ∞. For eachn_∈Nandt_≥0 let

xn,t= [o, xn]t∧kxnk, 3

where [o, xn] is any geodesic connectingoandxn. SinceX is proper, there exists a sequence (nk)∞1 such that for eacht≥0, the sequence (xnk,t)

∞

1 is convergent, say xnk,t−→

k xt.

It is readily verified that the mapt_7→xtis an isometric embedding from [0,∞) to

X. Thus there exists a point ξ∈∂X such thatxt→ξ. We claim thatxnk →ξ.

Indeed, for eacht_≥0, lim sup k→∞ D(xnk, xnk,t)≍×lim sup k→∞ D(xnk,t, xt)≍×lim sup k→∞ D(xt, ξ)≍× b−t,

and so the triangle inequality gives lim sup

k→∞

D(xnk, ξ).×b

−t_.

Lettingt_{→ ∞}shows thatxnk →ξ.

Observation3.4.20. A sequence (xn)∞1 in bordXconverges to a pointξ∈∂X

if and only if

(3.4.12) _hxn|ξio−→ n ∞.

Observation 3.4.21. A sequence (xn)∞1 inX converges to a pointξ∈∂X if

and only if (xn)∞1 is a Gromov sequence and (xn)∞1 ∈ξ.

We now investigate the continuity properties of the Gromov product and Buse- mann function.

Lemma 3.4.22 (Near-continuity of the Gromov product and Busemann function). The maps (x, y, z)_{7→ h}x_|y_iz and(x, z, w)7→ Bx(z, w) are nearly continuous

in the following sense: Suppose that (xn)∞1 and (yn)∞1 are sequences in bordX

which converge to points xn → x∈ bordX and yn →y ∈ bordX. Suppose that (zn)∞1 and (wn)∞1 are sequences inX which converge to points zn → z ∈ X and

wn→w∈X. Then hxn|ynizn−−→ n,+ hx|yiz (3.4.13) Bxn(zn, wn)−−→ n,+ Bx(z, w), (3.4.14) with₋_→ n ifX is strongly hyperbolic.

Proof. In the proof of (3.4.13), there are three cases:

Case 1: x, y_∈X. In this case, (3.4.13) follows directly from (d) and (e) of Propo- sition 3.3.3.

Case 2: x, y_∈∂X. In this case, for eachn_∈N, choosex_bn ∈X such that either (1) xbn=xn (ifxn∈X), or

(2) _hbxn|xniz≥n(if xn ∈∂X).

Choosey_bnsimilarly. Clearly,bxn →xandbyn →y. By Observation 3.4.21, (bxn)∞1 ∈xand (byn)∞1 ∈y. Thus by Lemma 3.4.10,

(3.4.15) _hbxn|byniz−−→ n,+ hx|yiz.

Now by Gromov’s inequality and (e) of Proposition 3.3.3, either

hbxn|byniz≍+hxn|ynizn or

(1)

hbxn|byniz&+n,

(2)

with which asymptotic is true depending onn. But fornsufficiently large, (3.4.15) ensures that the (2) fails, so (1) holds.

Case 3: x_∈X,y_∈∂X, or vice-versa. In this case, a straightforward combination of the above arguments demonstrates (3.4.13).

Finally, note that (3.4.14) is an immediate consequence of (3.4.13), (3.4.7), and (j)

of Proposition 3.3.3.

Although Lemma 3.4.22 is generally sufficient for applications, we include the following lemma which reassures us that the Gromov product does behave some- what regularly even on an “exact” level.

Lemma 3.4.23. The function (x, y, z) 7→ hx|yiz is lower semicontinuous on bordX×bordX×X.

Proof. Since bordX is metrizable, it is enough to show that if xn → x,

yn →y, andzn→z, then

lim inf

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