Uniformity from Gromov hyperbolicity

Transcription

1 Uniformity from Gromov hyperbolicity David Herrron, Nageswari Shanmugalingam, and Xiangdong Xie. December 3, 2006 Abstract We show that, in a metric space X with annular convexity, uniform domains are precisely those Gromov hyperbolic domains whose quasiconformal structure on the Gromov boundary agrees with that on the metric boundary in X. As an application we show that quasimöbius maps between geodesic spaces with annular convexity preserve uniform domains. These results are quantitative. Introduction In this paper we study a connection between Gromov hyperbolic spaces and uniform domains. We characterize uniform domains among Gromov hyperbolic domains in metric spaces with annular convexity in terms of the quasiconformal structure of the Gromov boundary. Let (X, d) be a metric space and C a 2 a constant. We say (X, d) is C a -annular convex if for all x X, all r > 0 and every pair of points y, z B(x, r)\b(x, r/2) there is a path γ joining y and z satisfying: () the length of γ is at most C a d(y, z), (2) the path γ does not intersect the ball B(x, r/c a ). Examples of metric spaces with annular convexity include Banach spaces and Carnot groups, as well as metric measure spaces equipped with a doubling measure that supports a (, p)- Poincaré inequality (see [K]). Let (X, d) be a proper metric space (that is, closed and bounded subsets are compact), and Ω X a rectifiably connected open subset (every pair of points in Ω can be joined by a rectifiable path in Ω) with boundary Ω. We say Ω is a Gromov hyperbolic domain if Ω is Gromov hyperbolic with respect to the quasihyperbolic metric k Ω on Ω. Given a bounded Gromov hyperbolic domain Ω, we obtain the Gromov compactification Ω = Ω Ω of (Ω, k Ω ), where Ω is the Gromov boundary of Ω. The metric closure of Ω in (X, d) is denoted Ω; since X is proper and Ω is bounded, Ω is compact. In general, the identity map f : (Ω, k Ω ) (Ω, d) does not extend to a continuous map from Ω to Ω, and even if f does extend, the extension may not be injective. However, if Ω is a uniform domain, N. S. was partially supported by NSF grant DMS

2 then f extends to a homeomorphism from Ω to Ω, and the restriction of the extension to the Gromov boundary is a quasimöbius map with respect to the visual metric on Ω (see [BHK]). The main result (Theorem 9.) of this paper is that uniform domains are the only Gromov hyperbolic domains in an annular convex proper metric space with the above property. As an application of the main result we demonstrate that quasimöbius maps preserve uniform domains (Theorem 0.): if Ω is a domain in an annular convex proper metric space and Ω is quasimöbius equivalent to a uniform domain in some metric space, then Ω is also uniform. In the setting of Euclidean spaces and spheres Theorem 9. has been proven by Bonk, Heinonen, and Koskela [BHK], and Väisälä [V] proved this theorem for domains in Banach spaces. The proof in [BHK] makes use of the notion of moduli of curve families, and therefore does not extend to metric spaces that have no nice measure. The proof in [V] uses only metric properties. Our proof follows the general outline of the arguments found there, but it contains the following two new ingredients. In [V] the theorem was first proved for unbounded domains in Banach spaces, and then inversions in Banach spaces were used to reduce the study of bounded domains to the study of unbounded domains. To follow this strategy, we use a notion of inversion in general metric spaces, see Section 4 or [BHX] for more details. The second crucial property used in [V] is the fact that spheres in Banach spaces are 2-quasiconvex. Our replacement for this property is the annular convexity property of the underlying metric space X. Points in Ω can be classified as annulus points or arc points (see Section 7). A consequence of the annular convexity property is that each arc point lies on an anchor. This was first shown in [BHK] in the Euclidean setting. Under the assumption of annular convexity we establish a slightly weaker version of these facts sufficient for the proof of Theorem 9.. It is not clear whether Theorem 9. still holds if the metric space is not annular convex. As in [V] we interpret the cross ratio in the Gromov boundary with respect to a visual metric d y,ɛ in terms of distances between certain geodesics (see Section 5). Let (Y, d) be a proper geodesic Gromov hyperbolic space, Q = (ξ, ξ 2, ξ 3, ξ 4 ) a quadruple of distinct points in the Gromov boundary Y. Fix any geodesic [ξ i, ξ j ] ( i, j 4) joining ξ i and ξ j. The cross ratio of Q with respect to d y,ɛ, denoted cr(q, d y,ε ), satisfies { e ɛd([ξ,ξ 4 ],[ξ 2,ξ 3 ]) if d([ξ, ξ 4 ], [ξ 2, ξ 3 ]) d([ξ, ξ 3 ], [ξ 2, ξ 4 ]), cr(q, d y,ε ) e ɛd([ξ,ξ 3 ],[ξ 2,ξ 4 ]) otherwise. This interpretation of cross ratio is quite convenient in studying the quasiconformal structure of the Gromov boundary, and allows us to simplify some of the arguments found in [V]. Notation: In this note (X, d) denotes a metric space, B(x, r) = {y X : d(x, y) < r} is the open ball with center x X and radius r > 0, and S(x, r) = {y X : d(y, x) = r} is the sphere of radius r and center x. The image of a path α : [a, b] X is denoted α, and l d (α) denotes the d-length of α (we simply use l(α) if the metric d in question is clear). We use α : x y to indicate a path α : [a, b] X with α(a) = x and α(b) = y. If a path α : [a, b] X is an embedding and x, y α, then both α xy and α[x, y] denote the subpath of α from x to y; we prefer the second notation when a path already has a subscript, say, 2

3 α. If A X and r > 0, then N r (A) = {y X : d(y, x) r for some x A} denotes the closed r-neighborhood of A. For two bounded subsets A, B X, HD d (A, B) = inf{r > 0 : B N r (A) and A N r (B)} denotes the Hausdorff distance between A and B; if the metric d in question is clear, we use HD(A, B). Given two real numbers a, b, we denote the smaller of these by a b. By c = c(δ, η, C a ) we mean c depends only on δ, η and C a. We say that a metric space is a geodesic space if every pair of points x, y in that space can be joined by a geodesic, that is, a path with length d(x, y). A metric space is said to be c-quasiconvex for some c if each pair of points x, y in the space can be joined by a curve of length no more than c d(x, y). Thus, a geodesic space is -quasiconvex. It shouldbe noted that if Ω is a non-empty open subset of a c-quasiconvex space, then for every x Ω there exists r x > 0 such that all y, z B(x, r x ) can be joined by a curve in Ω with length at most c d(y, z). 2 Quasihyperbolic metrics and uniform domains In this section we recall some basic facts about quasihyperbolic metrics. While we do not give proofs for most of these facts, we do provide citations the reader can refer to for them. Let Ω be an open subset of a metric space (X, d). We say Ω is rectifiably connected if each pair of points x, y Ω can be joined by a rectifiable path γ Ω. The boundary Ω of Ω is the set Ω\Ω. Let (X, d) be a proper metric space, and Ω X an open rectifiably connected subset of X with Ω. For x Ω, we denote δ Ω (x) = d(x, X \ Ω). The quasihyperbolic metric k Ω on Ω is defined as follows: for x, y Ω, k Ω (x, y) := inf γ ds(z) δ Ω (z), where the infimum is over all rectifiable curves γ in Ω joining x and y, and ds denotes the arc length element along γ. It can be shown that k Ω indeed is a metric. The length metric l Ω on Ω is given by l Ω (x, y) = inf γ l d (γ) for x, y Ω, where the infimum is over paths in Ω joining x and y. Proposition 2. (Proposition 2.8 of [BHK]). If the identity map f : (Ω, d) (Ω, l Ω ) is a homeomorphism, then f : (Ω, d) (Ω, k Ω ) is also a homeomorphism with (Ω, k Ω ) a proper geodesic space. It should be noted that if X is a c-quasiconvex space and Ω is an open connected set (that is, a domain), then id : (Ω, d) (Ω, l Ω ) is indeed a homeomorphism. Thus, in this paper the assumption in Proposition 2. is always satisfied and hence the space (Ω, k Ω ) is a proper geodesic space. Lemma 2.2. If x, y Ω and α : x y is a rectifiable arc in Ω, then l d (α) ( e l k Ω (α) ) δ Ω (x). 3

4 Proof. Without loss of generality, let α be parametrized by d-arclength. Then ld (α) ld (α) ( l kω (α) = 0 δ Ω (α(t)) dt dt = log + l ) d(α). 0 δ Ω (x) + t δ Ω (x) Let x, y Ω and α be a quasihyperbolic geodesic from x to y. Then l kω (α) = k Ω (x, y) and l d (α) d(x, y). Since we may assume δ Ω (x) δ Ω (y), the proof of Lemma 2.2 shows that the following holds for all x, y Ω: ( ) d(x, y) k Ω (x, y) log +. δ Ω (x) δ Ω (y) Since δ Ω (y) δ Ω (x) + d(x, y), we have k Ω (x, y) log δ Ω(y) δ Ω (x). () Lemma 2.3 (Lemma 2.3 of [BHK]). Let (X, d) be a proper metric space and Ω X a rectifiably connected open subset with Ω. If γ : [0, ] Ω is a curve that satisfies min{l d (γ [0,t] ), l d (γ [t,] )} A δ Ω (γ(t)) for all t [0, ], then with x = γ(0) and y = γ(), ( l kω (γ) 4A log + l d (γ) δ Ω (x) δ Ω (y) The following is a modification to our setting of Lemma 3.5 of [V]. Since we replace the 2-quasiconvexity of spheres with the annular convexity property, and hence the estimates we obtain here are necessarily weaker than those in [V]. Lemma 2.4. Suppose (X, d) is C a -annular convex for some C a 2. Let α : x y be a quasihyperbolic geodesic in Ω, a Ω, and t > 0. (i) If B(a, 6C 2 a t) \ B(a, e 4C3 a t/2) Ω and x, y Ω \ B(a, 8Ca t), then α Ω \ B(a, e 4C3 a t). (ii) If B(a, 8C a t) \ B(a, t/c a ) Ω and x, y Ω B(a, 4t), then α Ω B(a, 8e 4C3 a t). Proof. We first prove (i). Suppose that α B(a, e 4C3 a t). Then α must intersect B(a, 8C a t) as well, and so we can choose z, z 2 α S(a, 8C a t) such that the subcurve α z z 2 of α satisfies both α z z 2 B(a, 8C a t) and α z z 2 B(a, e 4C3 a t). As X is annular convex, there is a path γ : z z 2 with l d (γ) C a d(z, z 2 ) C a 2(8C a t) = 6C 2 at and γ B(a, 8C 2 at) \ B(a, 8t) Ω. Hence by the hypothesis of (i), for every w γ we have ). δ Ω (w) min{8c 2 at, 8t e 4C3 a t/2} = 8t e 4C 3 a t/2 4t. 4

5 Therefore, l kω (γ) = hence we see that k Ω (z, z 2 ) 4C 2 a. Since α z z 2 γ δ Ω (w) dw 4t l d(γ) 4t 6C2 at = 4C 2 a, is a quasihyperbolic geodesic, we have l kω (α z z 2 ) = k Ω (z, z 2 ) 4C 2 a. (2) By assumption, there is a point z α z z 2 B(a, e 4C3 a t). By Lemma 2.2, [( l kω (α z z 2 ) = l kω (α z z) + l kω (α zz2 ) log + l ) ( d(α z z) + l d(α zz2 ) δ Ω (z) δ Ω (z) ( log + l ) d(α z z 2 ). δ Ω (z) However, as l d (α z z 2 ) 8C a t e 4C3 a t and δω (z) e 4C3 a t, we see that ( ) l kω (α z z 2 ) log + 8C at e 4C3 a t ( ) = log 8C a e 4C3 a 4C 3 e 4C3 a t a. (3) Combining inequalities (2) and (3) we obtain 4Ca 2 4Ca, 3 a contradiction because C a 2. Thus the curve α cannot intersect the ball B(a, e 4C3 a t). Now we prove (ii). To do so, suppose that α S(a, 8e 4C3 a t). Then clearly α intersects the sphere S(a, 4t), and so there are points w, w 2 α S(a, 4t) satisfying α w w 2 S(a, 8e 4C3 a t) and αw w 2 B(a, 4t) =. By the annular convexity of X, there is a rectifiable curve γ joining w and w 2 in the annulus B(a, 4C a t) \ B(a, 4t/C a ) Ω with l d (γ) C a d(w, w 2 ) 8C a t. For every z γ, Therefore, δ Ω (z) min { 8C a t 4C a t, 4t t } C a C a = 3t C a. k Ω (w, w 2 ) l kω (γ) C a 3t l d(γ) C a 3t 8C at = 8 3 C2 a. Since α is a quasihyperbolic geodesic, we see that l kω (α w w 2 ) = k Ω (w, w 2 ) 8 3 C2 a. (4) Meanwhile, by Lemma 2.2 and by the facts that l d (α w w 2 ) 8e 4C3 a t 4t 4e 4Ca 3 t and δ Ω (w ) 4t, ( l kω (α w w 2 ) log + l ) ( ) d(α w w 2 ) log + 4e4C3 a t ( ) log e 4C3 a = 4C 3 δ Ω (w ) 4t a. By inequality (4), we now get the contradiction 4C 3 a 8 3 C2 a as C a 2. This concludes the proof of this lemma. 5 )]

6 Given c, a path γ : [0, ] Ω is called a c-uniform curve if () l d (γ) c d(γ(0), γ()); (2) c δ Ω (z) min{l d (γ [0, t]), l d (γ [t, ])} for all t [0, ]. An open subset Ω X with Ω is called a c-uniform domain for some c if every two points x, y Ω can be joined by a c-uniform curve. If Ω is equipped with more than one metric, then to specify the metric with respect to which Ω is uniform we say that (Ω, d) is a uniform domain. Lemma 2.5. Let x Ω and x 2 Ω such that δ Ω (x ) d(x, x 2 ), and γ a geodesic in X with respect to the metric d connecting x and x 2. Then γ\{x 2 } Ω, γ is a -uniform curve, and furthermore, δ Ω (x) l d (γ x2 x) = d(x, x 2 ) for all x γ. Proof. By assumption, x 2 B(x, δ Ω (x )) Ω. Let γ : [0, d(x, x 2 )] X be the arc-length parametrization of γ with γ(0) = x 2, and γ(d(x, x 2 )) = x. Then, for every z γ \ {x 2 } we have d(x, z) < d(x, x 2 ), and hence z B(x, δ Ω (x )) Ω. Now, for t (0, d(x, x 2 )], we have d(γ(t), x ) = d(x, x 2 ) t, whence, δ Ω (γ(t)) δ Ω (x ) d(x, γ(t)) = δ Ω (x ) d(x, x 2 ) + t = t + [δ Ω (x ) d(x, x 2 )] t. Proposition 2.6. Let X be a proper geodesic metric space and Ω X an open subset with Ω. Suppose x 0 Ω. Then there exists a point b Ω and a curve γ : x 0 b such that (i) l d (γ) = δ Ω (x 0 ), (ii) γ B(x 0, δ Ω (x 0 )), (iii) γ \ {b} Ω, (iv) for every x γ \ {b} we have l d (γ bx ) = δ Ω (x), (v) γ is a geodesic with respect to the metric d, (vi) γ is a quasihyperbolic geodesic in Ω. Proof. Since X is proper, we can choose b Ω such that δ Ω (x 0 ) = d(x 0, b), and we set γ to be a geodesic in X with respect to the metric d joining x 0 and b. Then by Lemma 2.5 we see that the conditions (i) through (v) are satisfied by γ. To see (vi), let γ : [0, δ Ω (x 0 )] X be the arclength parametrization of γ with respect to the metric d, with γ(0) = b and γ(δ Ω (x 0 )) = x 0. Let 0 < t < t 2 δ Ω (x 0 ). Since δ Ω (γ(t)) = t, inequality () implies that k Ω (γ(t ), γ(t 2 )) log(t 2 /t ). On the other hand, k Ω (γ(t ), γ(t 2 )) l kω (γ [t,t 2 ]) = Hence γ is a quasihyperbolic geodesic in Ω. t2 t δ Ω (γ(t)) dt = 6 t2 t t dt = log ( t2 t ).

7 Given two rectifiable curves α, β in a metric space (Y, d), we say a map f : α β is a length map with respect to the metric d if for all x, y α we have l d (β f(x)f(y) ) = l d (α xy ). Lemma 2.7 (Lemma 3.3 of [V]). If α and β are curves in (Ω, k Ω ) with l kω (α) l kω (β), and f : α β is a length map (with respect to k Ω ) with k Ω (f(x), x) c whenever x α, then e c l d (α) l d (f( α )) e c l d (α). 3 Gromov hyperbolic spaces In this section we review some basic facts about Gromov hyperbolic spaces. See [BHK], [CDP], [GdlH], [V], and references therein for more details. Let (Y, d) be a proper geodesic space and δ 0 a constant. We say (Y, d) is δ-hyperbolic if geodesic triangles in Y are δ-thin, that is, for any x, y, z Y, any geodesics γ : x y, γ 2 : y z, γ 3 : z x, we have γ 3 N δ ( γ γ 2 ). A space (Y, d) is Gromov hyperbolic if it is δ-hyperbolic for some δ 0. Let w Y be a (fixed) base point. The Gromov product of x, y Y based at w is: (x y) w = (d(x, w) + d(y, w) d(x, y)). 2 If Y is δ-hyperbolic, a sequence of points {y i } is said to goes to infinity if lim i,j (y i y j ) w = for some (or any) base point w Y. Two sequences {x i } and {y i } going to infinity are equivalent if lim i,j (x i y j ) w =. The Gromov boundary d Y of Y is the set of equivalence classes of sequences going to infinity. If the metric d in question is clear, we simply denote the Gromov boundary by Y, and set Y = Y Y. If ξ Y and a sequence of points {x i } represent ξ we write {x i } ξ. Interested reader may refer to Chapter 7 of [GdlH] for more details. If γ : [0, ) Y is a geodesic (ray), then one sees easily from the definition that {γ(t)} goes to infinity as t and hence represent some ξ Y. In this case we say γ(0) and ξ are the endpoints of γ. Similarly, for any complete geodesic γ : R Y there are ξ +, ξ Y such that {γ(t)} ξ + as t and {γ(t)} ξ as t. We say ξ + and ξ are the endpoints of γ. A proper geodesic δ-hyperbolic space has the visibility property: given any two distinct points a, b Y, there is a geodesic γ with a and b as endpoints (Proposition 2. in Chapter 2 of [CDP]). Geodesic triangles in Y are 24δ-thin (Proposition 2.2 in Chapter 2 of [CDP]). From this it can be seen that if α, β, and γ are the three sides of a geodesic triangle, then there is some x γ satisfying d(x, α ) 24δ and d(x, β ) 24δ. Let w Y be a base point. The Gromov product of two points ξ, η Y is defined as follows: (ξ η) w = sup lim inf (x i y j ) w, i,j where the supremum is taken over all sequences {x i } ξ, {y i } η. One can show that (ξ η) w 2δ lim inf i,j (x i y j ) w (ξ η) w for all w Y, all ξ, η Y and all sequences 7

8 {x i } ξ, {y i } η; see Chapter 7 of [GdlH]. Similarly, the Gromov product of x Y and η Y is defined to be (x η) w = sup lim inf i (x y i) w, where the supremum is taken over all sequences {y i } η. We define a topology on Y by specifying when a sequence of points x i Y converges to a point ξ Y : if ξ Y, then x i ξ means d(ξ, x i ) 0 as i ; and if ξ Y, then x i ξ means (ξ x i ) w for some (equivalently all) w Y as i. In this topology, Y is compact and Y is a dense open subset. The induced topology on Y agrees with the metric topology on Y. Given ɛ > 0, w Y and ξ, η Y, let ρ w,ɛ (ξ, η) = e ɛ(ξ η) w. Proposition 3. (Proposition 0 of [GdlH], Chapter 7). Let ɛ 0 (δ) = min{, }. Then 5δ for any δ-hyperbolic metric space Y, any base point w Y, and any 0 < ɛ ɛ 0, there is a metric d w,ɛ on Y such that for all ξ, η Y, 2 ρ w,ɛ(ξ, η) d w,ɛ (ξ, η) ρ w,ɛ (ξ, η). A metric d w,ɛ satisfying the conclusion of Proposition 3. is called a visual metric. Definition 3.2. Let L and A 0. A (not necessarily continuous) map γ : I Y on an interval I is an (L, A)-quasigeodesic if for all t, t 2 I we have L t 2 t A d(γ(t ), γ(t 2 )) L t 2 t + A. Note that an (L, A)-quasigeodesic is a geodesic if and only if L = and A = 0. An important property of Gromov hyperbolic spaces is the stability of quasigeodesics. It says that quasigeodesics are close to geodesics (see also [V3]): Lemma 3.3 (Theorem.2 and Theorem 3. of [CDP], Chapter 3). Given any δ 0, L, and A 0, there is a constant M = M(δ, L, A) such that whenever Y is a proper geodesic δ-hyperbolic space, the following conditions hold: () If α : [a, b] Y and α : [a, b ] Y are two (L, A)-quasigeodesics with α(a) = α (a ) and α(b) = α (b ), then the Hausdorff distance HD( α, α ) M; (2) If α : R Y is an (L, A)-quasigeodesic, then there exists a geodesic α : R Y such that HD( α, α ) M. Lemma 3.3 (2) implies that every quasigeodesic α : R Y has two endpoints ξ +, ξ in Y. Since two complete geodesics with the same endpoints in a δ-hyperbolic space has Hausdorff distance at most 2δ from each other, by replacing M with 2δ + 2M, we have that HD( α, α ) M for any two (L, A)-quasigeodesics with the same endpoints. We also recall the following two results. 8

9 Theorem 3.4 (Chapter 8 of [CDP]). Let (Y, d) be a δ-hyperbolic space, y 0 Y, and Y 0 = {y 0, y,, y n } be a set of n + points in Y Y. Let X denote the union of geodesics [y 0, y i ] connecting y 0 and y i, i n, and we choose a positive integer k such that 2n 2 k +. Then there exists a simplicial tree, denoted T (X), and a continuous map u : X T (X) which satisfies the following properties: () For each i, the restriction of u to the geodesic [y 0, y i ] is an isometry; (2) For every x and y in X we have d(x, y) 2kδ d(u(x), u(y)) d(x, y). Lemma 3.5 (Lemma 2.7 of [V3]). Suppose (Y, d) is δ-hyperbolic and α : a b, α 2 : a 2 b 2 are two geodesics with l(α ) l(α 2 ). If d(a, a 2 ) µ, d(b, α 2 ) µ for some µ 0, and f : α α 2 is the length map with f(a ) = f(a 2 ), then d(f(x), x) 8δ + 5µ for all x α. 4 Inversions in metric spaces In this section we recall the notion of inversions in metric spaces and collect related facts useful in this paper. See [BHX] for more details. Let (X, d) be a metric space and Q = (x, x 2, x 3, x 4 ) a quadruple of distinct points in X. The cross ratio of Q with respect to d is the number cr(q, d) = d(x, x 3 ) d(x 2, x 4 ) d(x, x 4 ) d(x 2, x 3 ). Let η : [0, ) [0, ) be a homeomorphism. A homeomorphism f : (X, d ) (Y, d 2 ) between two metric spaces is called an η-quasimöbius map if for each quadruple of distinct points Q = (x, x 2, x 3, x 4 ) in X, cr(f(q), d 2 ) η(cr(q, d )), where f(q) = (f(x ), f(x 2 ), f(x 3 ), f(x 4 )). We say a homeomorphism f : (X, d ) (Y, d 2 ) is quasimöbius if it is η-quasimöbius map for some η. A homeomorphism f : (X, d ) (Y, d 2 ) between two metric spaces is called an η-quasisymmetric map if for all triples of distinct points (x, x 2, x 3 ) in X, d 2 (f(x ), f(x 2 )) d 2 (f(x ), f(x 3 )) η ( ) d (x, x 2 ). d (x, x 3 ) We say a homeomorphism f : (X, d ) (Y, d 2 ) is quasisymmetric if it is η-quasisymmetric for some η. A quasisymmetric homeomorphism is quasimöbius, but a quasimöbius homeomorphism may not be quasisymmetric. However, a quasimöbius homeomorphism between bounded metric spaces is quasisymmetric. Let (X, d) be a proper metric space and p X. Set I p (X) = X\{p} if X is bounded and I p (X) = (X\{p}) { } if X is unbounded, where is a point not in X. We now define a metric d p on I p (X). 9

10 Consider the function f p : I p (X) I p (X) [0, ) given by For x, y I p (X), we define d(x,y) if x, y X\{p}, d(x,p) d(y,p) f p (x, y) = if y = and x X\{p}, d(x,p) 0 if x = = y. k d p (x, y) := inf f p (x i, x i+ ), where the infimum is taken over all finite sequences of points x 0,, x k I p (X) with x 0 = x and x k = y. Theorem 4. ([BHX]). The following holds for all x, y I p (X): (2.) i=0 4 f p(x, y) d p (x, y) f p (x, y). In particular, d p is a metric on I p (X) and the identity map f : (X\{p}, d) (X\{p}, d p ) is an η-quasimöbius homeomorphism with η(t) = 6t. Furthermore: () If (X, d) is c-quasiconvex and c-annular convex, then (I p (X), d p ) is c -quasiconvex and c -annular convex with c depending only on c; (2) Let Ω (X, d) be a c-quasiconvex open subset of X with at least two boundary points, and p Ω. Denote by k Ω,p the quasihyperbolic metric on Ω induced by the metric d p. If d 0 := diam(ω, d) < and d 0 := diam( Ω, d) > 0, then the restriction of the identity map f Ω : (Ω, k Ω ) (Ω, k Ω,p ) is M-bilipschitz with M = max{200c, 6c d 0 /d 0}; (3) Let Ω X be an open subset with p Ω. If (Ω, d p ) is c -uniform and (X, d) is both c 2 -quasiconvex and c 2 -annular convex, then (Ω, d) is c-uniform with c = c(c, c 2 ); (4) If d 0 > 0, p Ω, and (Ω, d) is c-uniform, then (Ω, d p ) is c -uniform with c = c (c). Under the assumptions of Theorem 4. (2), (Ω, k Ω,p ) is δ -hyperbolic for some δ whenever (Ω, k Ω ) is δ-hyperbolic. In general, one cannot control δ in terms of δ and c alone. Proposition 4.2. Let (X, d) be a c-quasiconvex and c-annular convex proper metric space, Ω X a bounded rectifiably connected open subset of X such that d 0 = diam(ω, d) is finite and d 0 = diam( Ω, d) is positive. If (Ω, k Ω ) is δ-hyperbolic and p Ω, then (Ω, k Ω,p ) is δ -hyperbolic with δ depending only on c and δ. To prove this proposition we need the following preliminary results. Let L and A 0. A map f : X Y between two metric spaces is an (L, A)- quasiisometry if the following two conditions are satisfied: () d X (x, x 2 )/L A d Y (f(x ), f(x 2 )) Ld X (x, x 2 ) + A holds for all x, x 2 X; 0

11 (2) For each y Y, there is some x X with d Y (f(x), y) A. By definition an L-bilipschitz map is an (L, 0) quasiisometry, and an (L, A)-quasigeodesic is an (L, A)-quasiisometry onto it s image. Observe that we do not require a quasiisometry to be continuous. It is well-known that if f : Y Y 2 is an (L, A)-quasiisometry between geodesic spaces and Y is δ-hyperbolic, then Y 2 is δ -hyperbolic with δ = δ (δ, L, A). Proposition 4.2 does not follow from this fact since the bilipschitz constant of the identity f : (Ω, k Ω ) (Ω, k Ω,p ) depends on the ratio d 0 /d 0. In fact, from Lemma 4.4 below we see that diam(k, k Ω,p ) 7c while it can be seen that diam(k, k Ω ) as d 0 /d 0. Now suppose X is a c-quasiconvex and c-annular convex proper metric space, Ω X is a rectifiably connected open subset of X with and 0 < d 0 < d 0 <, and p Ω. Then 20c 2 the boundary of Ω in (X, d p ) is p Ω := Ω \ {p}, and the boundary of Ω in the induced quasihyperbolic metric k Ω,p is denoted pω. For x Ω, let δ p (x) = d p (x, p Ω). However, B = B(p, 0c 2 d 0) will denote the ball with respect to the original metric d. Let K = Ω\B and S = {x Ω : d(x, p) = 0c 2 d 0}. By the definition of d 0 and the assumption that d 0 is finite, observe that K is a compact subset of Ω in both metrics. Let D = diam(s, k Ω ), D 2 = diam(s, k Ω,p ) and d 2 = diam(k, k Ω,p ). Since S K, we always have D 2 d 2. Lemma 4.3 (Lemma 3.9 of [BHX]). There is a constant L depending only on c such that for all x, y Ω\K we have k Ω,p (x, y) L k Ω (x, y) + D 2 and k Ω (x, y) L k Ω,p (x, y) + D. Recall that we assume X to be both c-quasiconvex and c-annular convex. The length of a path γ I p (X) with respect to the d p -metric shall be denoted l p (γ). Lemma 4.4. The inequalities D 4c 2 and d 2 7c hold. Proof. Let x, y S. Since X is c-annular convex, there is a path γ in X joining x and y such that γ B(p, 0c 3 d 0) \ B(p, 0cd 0) and l(γ) c d(x, y). Note that δ Ω (z) 5cd 0 for all z γ and γ Ω. Now k(x, y) γ δ Ω (z) dz γ 5cd 0 dz = l(γ) cd(x, y) 20c 2 d 5cd 0 5cd 0 5d 0 = 4c 2. 0 Now we prove the second inequality. We first prove that whenever r 0c 2 d 0, for every x, y (B(p, 2r) \ B(p, r)) Ω, k Ω,p (x, y) 32c3 d 0. (5) r Assume r 0c 2 d 0 and let x, y (B(p, 2r) \ B(p, r)) Ω. Since X is c-annular convex, there is a path γ B(p, 2cr) \ B(p, r/c) connecting x and y such that l d (γ) c d(x, y). Note γ Ω. For any z, z 2 γ we have by Theorem 4., d p (z, z 2 ) d(z, z 2 ) d(z, p) d(z 2, p) d(z, z 2 ) = c2 d(z, z 2 ). (r/c) 2 r 2 It follows that l p (γ) c2 l d (γ) c2 c d(x, y) r 2 r 2 c3 4r r 2 = 4c 3 /r.

12 On the other hand, as r 0c 2 d 0 and γ B(p, 2cr)\B(p, r/c), we have d(z, w) d(z, p)/2 for all z γ and w p Ω. Hence by by Theorem 4. again, for any w p Ω and z γ, d p (z, w) d(z, w) 4d(z, p)d(w, p) 8d(w, p). 8d 0 It follows that δ p (z) for all z γ. Consequently 8d 0 k Ω,p (x, y) δ p (z) ds p 8d 0l p (γ) 8d 0 4c 3 /r = 32c3 d 0, r γ where ds p is the d p -arc length element along γ. Set r 0 = 0c 2 d 0 and let n be the integer such that 2 n r 0 < d 0 2 n r 0. Then K n+ i= Ω (B(p, 2i r 0 ) \ B(p, 2 i r 0 )). The inequality (5) now implies n 32c 3 d 0 d c3 d 0 < 7c. 2 i r 0 r 0 i=0 Let α : I Ω be a geodesic with respect to the metric k Ω,p with I a closed (not necessarily compact) interval such that the endpoints of α do not lie in K. We define a map α : I (Ω, k Ω ) as follows. Let f : (Ω, k Ω ) (Ω, k Ω,p ) be the identity map. If α K =, then we let α = f α. If α K, then let t = inf α (K) and t 2 = sup α (K); observe that α(t ), α(t 2 ) S. Since diam(s, k Ω,p ) = D 2, we have t 2 t D 2. Let α (t) = f (α(t)) if t < t or t > t 2, and α (t) = f (α(t )) if t [t, t 2 ]. Similarly, given any geodesic β : I (Ω, k Ω ) whose endpoints do not lie in K, we can define a map β : I (Ω, k Ω,p ). Lemma 4.5. The maps α, β are (L, A)-quasigeodesics with respect to both metrics k Ω and k Ω,p, with L the constant in Lemma 4.3 and L, A depend only on c. Proof. We only prove the claim for α, as the proof for β is similar. We use Lemma 4.3. Let s, t I. First assume s, t I\[t, t 2 ]. Then α (s) = α(s), α (t) = α(t) and hence s t L D 2 L = k Ω,p(α(s), α(t)) D 2 L k Ω (α (s), α (t)) L k Ω,p (α(s), α(t)) + D = L s t + D. Next assume s, t [t, t 2 ]. Then s t t 2 t D 2 and α (s) = α (t). We therefore see that the above chain of inequalities is again satisfied. Finally assume s [t, t 2 ] and t / [t, t 2 ]. Then and k Ω (α (s), α (t)) = k Ω (α (t ), α (t)) L t t + D L( t s + s t ) + D L s t + LD 2 + D, k Ω (α (s), α (t)) = k Ω (α (t ), α (t)) t t L Now the lemma follows from Lemma 4.4. D 2 L s t L s t D 2 L L 2 s t L D 2 L D 2 L.

13 Proof of Proposition 4.2. The result follows from Theorem 4. (2) if d 0 20c 2 d 0. Hence without loss of generality, d 0 > 20c 2 d 0. The goal is to prove that all geodesic triangles in (Ω, k Ω,p ) are δ -thin for some δ = δ (δ, c). Let x, x 2, x 3 Ω and setting x 4 := x, let α i (i =, 2, 3) be a geodesic in (Ω, k Ω,p ) joining x i and x i+. We want some δ = δ (δ, c) such that α N δ ( α 2 α 3 ). We consider several cases. Case : x, x 2, x 3 / K. By Lemma 4.5 we obtain an (L, A)-quasigeodesic α i in (Ω, k Ω ) connecting x i and x i+. Fixing a geodesic β i in (Ω, k Ω ) joining x i and x i+, note by Lemma 3.3 that HD kω ( β i, α i ) c with c = c (δ, L, A) = c (δ, c). Let x α, and fix y α \K such that k Ω,p (x, y) D 2. Necessarily y α. Since HD kω ( β, α ) c, there is some y β with k Ω (y, y ) c. By the δ-hyperbolicity of (Ω, k Ω ) there is some y 2 β 2 β 3 with k Ω (y, y 2 ) δ. We may assume y 2 β 2. The fact HD kω ( β 2, α 2 ) c implies that there is some y 3 α 2 with k Ω (y 2, y 3 ) c. Triangle inequality implies k Ω (y, y 3 ) 2c + δ. By Lemma 4.3 we have k Ω,p (x, y 3 ) k Ω,p (x, y) + k Ω,p (y, y 3 ) D 2 + L k Ω (y, y 3 ) + D 2 2D 2 + L(2c + δ). Since y 3 α 2 α 2, we have shown x N δ ( α 2 α 3 ) with δ = 2D 2 + L(2c + δ). Case 2: x, x 2 K. Since diam(k, k Ω,p ) = d 2, we have α N d2 ({x 2 }) N d2 ( α 2 α 3 ). Case 3: x 3 K and exactly one of x, x 2 lies in K, say x K and x 2 / K. Let x be the first point on α (oriented from x 2 to x ) that lies in K and x 3 the first point on α 2 (oriented from x 2 to x 3 ) that lies in K. Let γ be a geodesic in (Ω, k Ω,p ) connecting x and x 3. Now by Case, α N d2 ( α [x 2, x ] ) N d2 +δ ( α 2 [x 2, x 3] γ ) N 2d2 +δ ( α 2 [x 2, x 3] ) N 2d2 +δ ( α 2 ). Case 4: x 3 / K and exactly one of x, x 2 lies in K, say x K and x 2 / K. Let x 3 be the first point on α 3 (oriented from x 3 to x ) that lies in K. Let γ be a geodesic in (Ω, k Ω,p ) joining x 2 and x 3. Again Case implies that γ N δ ( α 3 [x 3, x 3] α 2 ) (strictly speaking we need x 3 K in order to apply Case here, but by the choice of x 3 we may employ a limiting argument together with Case to get the desired inclusion), and an application of Case 3 yields α N 2d2 +δ ( α 3 [x, x 3] γ ). It follows that α N 2d2 +2δ ( α 2 α 3 ). Case 5: x, x 2 / K and x 3 K. Let x be the first point on α 3 (oriented from x to x 3 ) that lies in K. Let γ be a geodesic in (Ω, k Ω,p ) connecting x 2 and x. Case implies that α N δ ( α 3 [x, x ] γ ), and Case 3 implies that γ N 2d2 +δ ( α 2 ). It follows that α N 2d2 +2δ ( α 2 α 3 ). We shall also need the following construction of Bonk Kleiner [BK]. Let (X, d) be an unbounded metric space and p X. Let S p (X) = X { }, where is a point not in X. We define a function s p : S p (X) S p (X) [0, ) as follows: d(x,y) if x, y X, [+d(x,p)][+d(y,p)] s p (x, y) = if x X and y =, +d(x,p) 0 if x = = y. 3

14 In analogy to the construction of the metric d p for the inversion, we construct the metric d p on S p (X) by the formula d p (x, y) := inf Σ k i=0 s p(x i, x i+ ), where the infimum is taken over all finite sequences of points x 0,, x k S p (X) with x 0 = x and x k = y. It was shown in [BK] that d p is a metric on S p (X) with 4 s p(x, y) d p (x, y) s p (x, y) for x, y S p (X). Furthermore, the identity map f : (X, d) (X, d p ) is an η-quasimöbius homeomorphism with η(t) = 6t. Theorem 4.6 ([BHX]). Let (X, d) be an unbounded proper metric space, Ω X a rectifiably connected open subset of X, and p Ω. We denote by k Ω,p the quasihyperbolic metric on Ω induced by the metric d p. Suppose diam(ω, d) =. () If Ω is locally c-quasiconvex, then the identity map f : (Ω, k Ω ) (Ω, k Ω,p ) is M- bilipschitz with M depending only on c; (2) If (Ω, d) is c-uniform, then (Ω, d p ) is c -uniform with c depending only on c; (3) If (X, d) is c-quasiconvex and c-annular convex, then (S p (X), d p ) is c -quasiconvex and c -annular convex with c depending only on c; (4) If (Ω, d p ) is c-uniform, then (Ω, d) is c -uniform with c depending only on c. 5 Boundary maps of quasi-isometries Under the assumptions of Theorem 4. (2), (Ω, k Ω,p ) is Gromov hyperbolic and the extension of the identity map (Ω, k Ω ) (Ω, k Ω,p ) is η-quasimöbius for some η whenever (Ω, k Ω ) is Gromov hyperbolic. In general, there is no control on η. In this section we provide a control on η in the case (X, d) is annular convex (Proposition 5.6). It is well-known that if f : Y Y 2 is an (L, A)-quasiisometry between geodesic spaces and Y is δ-hyperbolic, then the natural boundary map ( Y, d y,ɛ) ( Y 2, d y2,ɛ) of f is η-quasimöbius with η depending only on L, A and δ, see Proposition 5.0. Proposition 5.6 does not follow from this general result since the bilipschitz constant of the identity map f : (Ω, k Ω ) (Ω, k Ω,p ) depends on the ratio d 0 /d 0, where d 0 := diam(ω, d) and d 0 := diam( Ω, d). See also the remark after Proposition 4.2. We first study the cross ratio on the Gromov boundary of a Gromov hyperbolic space (Corollary 5.2). Let (Y, d) be a proper geodesic δ-hyperbolic space and Q = (ξ, ξ 2, ξ 3, ξ 4 ) a quadruple of distinct points in Y. The signed distance sd(q) of Q is the number sd(q) = inf{d([ξ, ξ 4 ], [ξ 2, ξ 3 ]) d([ξ, ξ 3 ], [ξ 2, ξ 4 ])}, where the infimum is taken over all geodesics [ξ i, ξ j ] joining ξ i and ξ j. It follows from Theorem 3.4 and the definition of δ-hyperbolicity that the Hausdorff distance between two 4

15 infinite geodesics with the same endpoints is at most 2δ. Hence for all geodesics [ξ i, ξ j ] joining ξ i and ξ j, sd(q) d([ξ, ξ 4 ], [ξ 2, ξ 3 ]) d([ξ, ξ 3 ], [ξ 2, ξ 4 ]) sd(q) + 8δ. For w Y, the cross difference of Q based at w is: cd w (Q) = (ξ ξ 4 ) w + (ξ 2 ξ 3 ) w (ξ ξ 3 ) w (ξ 2 ξ 4 ) w. Note that for all quadruples Q, each w Y, and every 0 < ɛ ɛ 0 (δ), e ɛcd w(q) /4 cr(q, d w,ɛ ) 4e ɛcd w(q). Moreover, if Y is a tree, then sd(q) = cd w (Q) for all w Y and all Q. The following result shows that in a general δ-hyperbolic geodesic space, sd(q) and cd w (Q) differ by at most a fixed multiple of δ. From Theorem 3.4 and the definition of δ-hyperbolicity it follows that geodesic triangles in Y Y are 24 δ-thin. Lemma 5.. Let (Y, d) be a δ-hyperbolic space, w Y, and Q = (ξ, ξ 2, ξ 3, ξ 4 ) a quadruple of distinct points in Y. Then cd w (Q) sd(q) 430δ. Proof. Fix w Y and we choose geodesic rays [w, ξ i ], i 4, geodesics [ξ i, ξ j ], and let X = i [w, ξ i]. By Theorem 3.4 there is a tree T (X) and a map u : X T (X) with the properties stated in Theorem 3.4. Let w = u(w) and ξ i T (X) be such that u [w,ξi ] is an isometry onto [w, ξ i]. Let x ij T (X) be the unique point with [w, x ij] = [w, ξ i] [w, ξ j] (x ij = x ji), and let x ij wξ i be such that u(x ij ) = x ij (x ij may not equal x ji ). We can find sequences x k [w, ξ i ] converging to ξ i and y k [w, ξ j ] converging to ξ j. Since T (X) is a tree and hence (u(x k ) u(y l )) w = d(w, x ij), the properties of u from Theorem 3.4 imply that d(w, x ij) 6δ (x k y l ) w d(w, x ij). Using (ξ i ξ j ) w 2δ lim inf k,l (x k y l ) w (ξ i ξ j ) w, we obtain It follows that with Q = (ξ, ξ 2, ξ 3, ξ 4), d(w, x ij) 3δ (ξ i ξ j ) w d(w, x ij) + 2δ. cd w (Q ) 0δ cd w (Q) cd w (Q ) + 0δ, (6) We next show that HD([ξ i, ξ j ], [x ij, ξ i ] [x ji, ξ j ]) 00δ. To this end, let y ij [x ij, ξ i ] be such that d(x ij, y ij ) = 25δ. The properties of u imply that d(y ij, [w, ξ j ]) 25δ. Since the triangle [w, ξ i ] [w, ξ j ] [ξ i, ξ j ] is 24δ-thin, there is some point z ij [ξ i, ξ j ] such that d(y ij, z ij ) 24δ. Consequently, the fact that [y ij, z ij ] [y ij, ξ i ] [z ij, ξ i ] is 24δ-thin implies that HD([y ij, ξ i ], [z ij, ξ i ]) 48δ. Similarly HD([y ji, ξ j ], [z ji, ξ j ]) 48δ. Since d(x ij, x ji ) 6δ, the triangle inequality implies that d(z ij, z ji ) 04δ. It follows that HD([ξ i, ξ j ], [x ij, ξ i ] [x ji, ξ j ]) 48δ + 52δ = 00δ. (7) For {i, j, k, l} = {, 2, 3, 4}, we choose p ij [ξ i, ξ j ] and p kl [ξ k, ξ l ] with d(p ij, p kl ) = d([ξ i, ξ j ], [ξ k, ξ l ]). 5

16 Inequality (7) implies that there are q ij [x ij, ξ i ] [x ji, ξ j ] and q kl [x kl, ξ k ] [x lk, ξ l ] such that d(p ij, p kl ) d(q ij, q kl ) 200δ. By the properties of u, d(q ij, q kl ) 6δ d(u(q ij ), u(q kl )) d(q ij, q kl ). Since u(q ij ) [ξ i, ξ j] and u(q kl ) [ξ k, ξ l ], we have d(u(q ij), u(q kl )) d([ξ i, ξ j], [ξ k, ξ l ]). Combining the above inequalities we obtain d([ξ i, ξ j ], [ξ k, ξ l ]) d([ξ i, ξ j], [ξ k, ξ l]) 200δ. (8) On the other hand, there exist points r ij [ξ i, ξ j] and r kl [ξ k, ξ l ] such that d(r ij, r kl) = d([ξ i, ξ j], [ξ k, ξ l]). Observe that u([x ij, ξ i ] [x ji, ξ j ]) = [ξ i, ξ j]. Hence there exists r ij [x ij, ξ i ] [x ji, ξ j ] with u(r ij ) = r ij. Similarly there is a point r kl [x kl, ξ k ] [x lk, ξ l ] with u(r kl ) = r kl. The properties of u gives d(r ij, r kl ) 6δ d(r ij, r kl ) d(r ij, r kl ). By inequality (7), there is a point w ij [ξ i, ξ j ] with d(w ij, r ij ) 00δ. Similarly there is some w kl [ξ k, ξ l ] with d(w kl, r kl ) 00δ. Thus, d([ξ i, ξ j ], [ξ k, ξ l ]) d(w ij, w kl ) d(r ij, r kl ) + 200δ whence by inequality (8), Also recall that d(r ij, r kl) + 206δ = d([ξ i, ξ j], [ξ k, ξ l]) + 206δ, d([ξ i, ξ j ], [ξ k, ξ l ]) d([ξ i, ξ j], [ξ k, ξ l]) 206 δ. (9) sd(q) 8δ d([ξ, ξ 4 ], [ξ 2, ξ 3 ]) d([ξ, ξ 3 ], [ξ 2, ξ 4 ]) sd(q) + 8δ. It follows that sd(q) sd(q ) 420δ. Since T (X) is a tree, we have cd w (Q ) = sd(q ). Therefore by inequality (6), cd w (Q) sd(q) 430δ. Since T (X) is a tree, at least one of d([ξ, ξ 4], [ξ 2, ξ 3]), d([ξ, ξ 3], [ξ 2, ξ 4]) is 0. Hence it follows from inequality (9) that { } min d([ξ, ξ 4 ], [ξ 2, ξ 3 ]), d([ξ, ξ 3 ], [ξ 2, ξ 4 ]) 206δ. (0) Corollary 5.2. Let (Y, d) be a proper geodesic δ-hyperbolic space, and set c 0 = 4e 86. Then for each w Y, all 0 < ɛ ɛ 0 (δ), and all quadruple Q of distinct points in Y, e ɛ sd(q) c 0 cr(q, d w,ɛ ) c 0 e ɛ sd(q). Proof. Recall ɛ 0 (δ) = min{, }. The corollary now follows from Lemma 5. and the inequality 5δ 4 eɛ cdw(q) cr(q, d w,ɛ ) 4 e ɛ cdw(q). 6

17 Corollary 5.3. Let c > 0, (Y, d) a proper geodesic δ-hyperbolic space, x Y, 0 < ɛ ɛ 0 (δ), and Q = (ξ, ξ 2, ξ 3, ξ 4 ) a quadruple of distinct points in Y. () If cr(q, d x,ɛ ) c, then for all geodesics [ξ i, ξ j ] joining ξ i and ξ j for i, j 4, d([ξ, ξ 4 ], [ξ 2, ξ 3 ]) c = c (c, ɛ, δ). (2) If d([ξ, ξ 4 ], [ξ 2, ξ 3 ]) c for some geodesics [ξ i, ξ j ] joining ξ i and ξ j, i, j 4, then cr(q, d x,ɛ ) c = c (c). Proof. We first prove (). By hypothesis cr(q, d x,ɛ ) c. Therefore, by Corollary 5.2 we have sd(q) log(c 0 c)/ɛ. Let [ξ i, ξ j ] be a geodesic with endpoints ξ i and ξ j for i, j 4. With the aid of inequality (0) the claim follows from the fact (see the discussion following the definition of sd(q)) that d([ξ, ξ 4 ], [ξ 2, ξ 3 ]) d([ξ, ξ 3 ], [ξ 2, ξ 4 ]) sd(q) + 8δ. Now we prove (2). Suppose that d([ξ, ξ 4 ], [ξ 2, ξ 3 ]) c. Then sd(q) c. Since ɛ, by Corollary 5.2 we have cr(q, d x,ɛ ) c 0 e ɛc c 0 e c. Corollary 5.4. Let (Y, d) be δ-hyperbolic and w, w 2 Y. Then for any 0 < ɛ ɛ 0 (δ), the identity map ( Y, d w,ɛ) ( Y, d w2,ɛ) is η-quasimöbius with η(t) = 6e 72 t = c 2 0 t. Proof. By Corollary 5.2, cr(q, d w2,ε) c 0 e ε sd(q) c 2 0 cr(q, d w,ε). Corollary 5.5. Let (Y, d) be δ-hyperbolic and 0 < ɛ, ɛ 2 ɛ 0 (δ). Then for any w Y, the identity map ( Y, d w,ɛ ) ( Y, d w,ɛ2 ) is η-quasimöbius with η(t) = 4 + ɛ 2 ɛ t ɛ 2 ɛ. Proof. let t = cr(q, d w,ɛ ) for any quadruple Q of distinct points from Y. Then cr(q, d w,ɛ2 ) 4e ɛ 2cd w (Q) = 4(e ɛ cd w (Q) ) ɛ 2 ɛ 4(4t) ɛ 2 ɛ = η(t). We now bring the construction of inversion back into the picture. By Proposition 4.2 (Ω, k Ω,p ) is δ -hyperbolic with δ depending only on c and δ. Theorem 4. implies that the identity map f : (Ω, k Ω ) (Ω, k Ω,p ) is bilipschitz, hence it has a natural extension f : ( Ω, d w,ɛ ) ( pω, d w,ɛ) to the associated boundaries for all w, w Ω, and this extension is a homeomorphism. Proposition 5.6. Suppose (X, d) is a c-quasiconvex and c-annular convex proper metric space, Ω X a domain in X, and p Ω. Suppose also that d 0 := diam(ω, d) < and d 0 := diam( Ω, d) > 0. If (Ω, k Ω ) is δ-hyperbolic, then f : ( Ω, d w,ɛ ) ( pω, d w,ɛ ) is η-quasimöbius with η depending only on δ and c. 7

18 By an abuse of notation, for ξ pω we denote f(ξ) also by ξ. Let K be the compact set given in Section 4 if d 0 > 20c 2 d 0, and K = if d 0 20c 2 d 0. The proof of Proposition 5.6 is achieved by combining Lemmas 5.7 through 5.9. Lemma 5.7. There exists a constant A = A (δ, c) such that L k Ω([ξ i, ξ j ], ]ξ k, ξ l ]) A k Ω,p ( [ξ i, ξ j ], [ξ k, ξ l ]) L k Ω ([ξ i, ξ j ], [ξ k ξ l ]) + A for every quadruple Q = (ξ, ξ 2, ξ 3, ξ 4 ) of distinct points in Ω, geodesics [ξ i, ξ j ] in (Ω, k Ω ) joining ξ i and ξ j, and geodesics [ξ i, ξ j ] in (Ω, k Ω,p ) joining ξ i and ξ j. Here L is the constant given by Lemma 4.3. Proof. Let q ij [ξ i, ξ j ] and q kl [ξ k, ξ l ] such that k Ω (q ij, q kl ) = k Ω ([ξ i, ξ j ], [ξ k, ξ l ]). Since diam(s, k Ω ) = D and if the geodesics [ξ i, ξ j ], [ξ k, ξ l ] intersect K then they both pass through S, there exist points w ij [ξ i, ξ j ]\K and w kl [ξ k, ξ l ]\K such that k Ω (q ij, w ij ) D and k Ω (q kl, w kl ) D (if either of the two geodesics, say [ξ i, ξ j ], does not intersect K, then we can choose w ij = q ij ). Given the geodesics β ij = [ξ i, ξ j ], consider the maps β ij given in the paragraph before Lemma 4.5. By Lemma 4.5 both β ij and β kl are seen to be (L, A)- quasigeodesics in (Ω, k Ω,p ), with L and A depending solely on c. By Theorem 4., the Gromov boundary points that are endpoints of β ij given by Lemma 3.3 are the same as ξ i = f(ξ i ) and ξ j = f(ξ j ). From Lemma 3.3 and the fact that geodesic triangles in (Ω pω, k Ω,p ) are 24δ thin it follows that HD kω,p (β ij, [ξ i, ξ j ]) b and HD kω,p (β kl, [ξ k, ξ l ]) b, where b = b (δ, L, A) = b (δ, c). Hence we find two points z ij [ξ i, ξ j ] and z kl [ξ k, ξ l ] with k Ω,p (w ij, z ij ), k Ω,p (w kl, z kl ) b. Now we have k Ω,p ( [ξ i, ξ j ], [ξ k, ξ l ]) k Ω,p (z ij, z kl ) k Ω,p (w ij, w kl ) + 2b L k Ω (w ij, w kl ) + D 2 + 2b L {k Ω (q ij, q kl ) + 2D } + D 2 + 2b = L k Ω ([ξ i, ξ j ], [ξ k, ξ l ]) + (D 2 + 2b + 2LD ). The second inequality can be proven in a similar manner. Let Q = ( f(ξ ), f(ξ 2 ), f(ξ 3 ), f(ξ 4 )) for each quadruple Q = (ξ, ξ 2, ξ 3, ξ 4 ) of distinct points in Ω. Lemma 5.8. There exists a constant b 2 = b 2 (δ, c) with the property that for every quadruple Q = (ξ, ξ 2, ξ 3, ξ 4 ) of distinct points in Ω, () if sd(q) 0, then sd(q ) L sd(q) + b 2 ; (2) if sd(q) 0, then sd(q ) sd(q)/l + b 2. Proof. We first prove (). Assume that sd(q) 0. Recall sd(q) 8δ k Ω ([ξ, ξ 4 ], [ξ 2, ξ 3 ]) k Ω ([ξ, ξ 3 ], [ξ 2, ξ 4 ]) sd(q) + 8δ. 8

19 Since by inequality (0) { } min k Ω ([ξ, ξ 4 ], [ξ 2, ξ 3 ]), k Ω ([ξ, ξ 3 ], [ξ 2, ξ 4 ]) 206δ, we have k Ω ([ξ, ξ 3 ], [ξ 2, ξ 4 ]) 24δ. Now Lemma 5.7 implies that sd(q ) k Ω,p ( [ξ, ξ 4 ], [ξ 2, ξ 3 ]) k Ω,p ( [ξ, ξ 3 ], [ξ 2, ξ 4 ]) + 8δ k Ω,p ( [ξ, ξ 4 ], [ξ 2, ξ 3 ]) + 8δ L k Ω ([ξ, ξ 4 ], [ξ 2, ξ 3 ]) + A + 8δ = L (sd(q) + k Ω ([ξ, ξ 3 ], [ξ 2, ξ 4 ]) + 8δ) + A + 8δ L sd(q) L δ + A + 8δ. The proof of (2) is similar to that of (). If sd(q) 0, then k Ω ([ξ, ξ 4 ], [ξ 2, ξ 3 ]) 24δ, and therefore Lemma 5.7 implies sd(q ) k Ω,p ( [ξ, ξ 4 ], [ξ 2, ξ 3 ]) k Ω,p ( [ξ, ξ 3 ], [ξ 2, ξ 4 ]) + 8δ L k Ω ([ξ, ξ 4 ], [ξ 2, ξ 3 ]) + A L k Ω([ξ, ξ 3 ], [ξ 2, ξ 4 ]) + A + 8δ 24 L δ + 2A + 8δ + L {k Ω([ξ, ξ 4 ], [ξ 2, ξ 3 ]) k Ω ([ξ, ξ 3 ], [ξ 2, ξ 4 ])} 24 L δ + 2A + 8δ + (sd(q) + 8δ) L = L sd(q) + 24Lδ + 2A + 8δ + 8δ/L. Let w, w Ω and 0 < ɛ ɛ 0 (δ), ɛ 0 (δ ). For a quadruple Q = (ξ, ξ 2, ξ 3, ξ 4 ) of distinct points in Ω, set cr(q) = cr(q, d w,ɛ ) and cr(q ) = cr(q, d w,ɛ ). Lemma 5.9. The map f : ( Ω, d w,ɛ ) ( pω, d w,ɛ) is η-quasimöbius for some η = η(δ, c). Proof. Let Q = (ξ, ξ 2, ξ 3, ξ 4 ). Recall that ε ε 0 (δ). If sd(q) 0, then by Lemma 5.8, sd(q ) L sd(q) + b 2. Let c 0 = 4e 86. It follows from Corollary 5.2 that cr(q ) c 0 e ε sd(q ) c 0 e ε (L sd(q)+b 2) = c 0 e εb 2 (e εsd(q) ) L c 0 e εb 2 (c 0 cr(q)) L If sd(q) 0, then sd(q ) sd(q)/l + b 2. It follows that = c L+ 0 e εb 2 (cr(q)) L c L+ 0 e b 2 (cr(q)) L. cr(q ) c 0 e ε sd(q ) c 0 e ε (sd(q)/l+b 2) = c 0 e εb 2 (e εsd(q) ) L Note that cr(q) 0 as sd(q). c0 e εb 2 (c 0 cr(q)) L = c + L 0 e εb 2 (cr(q)) L c + L 0 e b 2 (cr(q)) L. 9

20 The proof of Proposition 5.6 can easily be generalized to show the following: Proposition 5.0. Let f : X Y be an (L, A)-quasiisometry between two proper geodesic metric spaces. If X is δ-hyperbolic, then Y is δ -hyperbolic with δ = δ (δ, L, A) and the natural map f : ( X, d x,ε ) ( Y, d y,ε ) with x X, y Y, is η-quasimöbius with η = η(l, A, δ). 6 Necessity In this section we prove that a uniform domain is Gromov hyperbolic with respect to the quasihyperbolic metric and that the natural map exists and is quasimöbius; this result is quantitative. We first explain the notion of natural map. Let (X, d) be a proper metric space, X the one point compactification X { } of X if X is unbounded, and let X = X if X is bounded. Let Q = (x, x 2, x 3, x 4 ) be a quadruple of distinct points in X. The cross ratio cr(q, d) is defined as in Section 4 if all x i X, and if one of the x i is, then cr(q, d) is obtained from the usual definition by canceling the terms involving. For example, if x =, then cr(q, d) = d(x, x 3 )d(x 2, x 4 ) d(x, x 4 )d(x 2, x 3 ) = d(x 2, x 4 ) d(x 2, x 3 ). If Ω X is a rectifiably connected open subset with Ω, let Ω be the boundary of Ω in X. Suppose (Ω, k Ω ) is Gromov hyperbolic. If the identity map (Ω, k Ω ) (Ω, d) has a continuous extension from the Gromov closure Ω = Ω Ω of (Ω, k) into X, then the restriction of this extension to the Gromov boundary, Ω Ω, is called a natural map of Ω. Since Ω is dense in the Gromov closure, the natural map is unique if it exists. We note that if (X, d) is unbounded, then for any p X the metric space (S p (X), ˆd p ) is homeomorphic to the one point compactification X. So by a natural map φ : Ω Ω we mean the continuous extension to the Gromov boundary of the identity map (Ω, k Ω ) (Ω, ˆd p ). Since the identity map (X, d) (X, ˆd p ) is η 0 -quasimöbius with η 0 (t) = 6t, a natural map ( Ω, d x,ε ) ( Ω, d) exists and is η-quasimöbius if and only if the natural map ( Ω, d x,ε ) ( Ω, ˆd p ) exists and is η -quasimöbius, with η and η depending only on each other. Suppose there is a metric space (Y, ρ) and Ω Y a bounded rectifiably connected open subset with ρ Ω (the metric boundary of Ω in (Y, ρ)) containing at least two points, and p ρ Ω such that X = I p (Y ) with d = d ρ ; that is, the metric space (X, d) is the inversion of (Y, ρ) at p. Then Ω is unbounded in (X, d). We note that X and Y are homeomorphic. If (Ω, k Ω ) is Gromov hyperbolic and φ : Ω Ω is a natural map, then by composing φ with the identification of X and Y, we obtain another natural map φ : Ω ρ Ω. Since the identity map (Y \ {p}, ρ) (X, d) is η 0 -quasimöbius, we see that a natural map φ : ( Ω, d x,ε ) ( Ω, d) is η-quasimöbius if and only if the natural map ( Ω, d x,ε ) ( ρ Ω, ρ) is η -quasimöbius, with η and η depending only on each other. Theorem 6. (Theorem 3.6 of [BHK]). Let (X, d) be a proper metric space and Ω X a c-uniform domain. Then (Ω, k Ω ) is a geodesic δ-hyperbolic space with δ = δ(c). If Ω is 20

21 bounded, then for each w Ω and all 0 < ɛ ɛ 0 (δ) the natural map φ : ( Ω, d w,ɛ ) ( Ω, d) exists and is η-quasimöbius with η = η(c, ɛ). If we choose ɛ = ɛ 0 (δ) = ɛ 0 (c) for the visual metric d w,ɛ, then the homeomorphism η in Theorem 6. depends only on c. Theorem 6.2. Let X be a proper metric space and Ω X a c-uniform domain. There exists a constant ɛ (c) > 0 such that for every w Ω and 0 < ɛ ɛ (c), the natural map φ : ( Ω, d w,ɛ ) ( Ω, d) exists and is η-quasimöbius with η = η(c, ɛ). Proof. By Theorem 6., it only remains to consider the case when Ω is unbounded. Suppose that Ω is an unbounded c-uniform domain. Fix p Ω and consider the compact metric space (S p (X), ˆd p ). By Theorem 4.6 (2), (Ω, ˆd p ) is c -uniform with c = c (c). Let k Ω,p be the quasihyperbolic metric on Ω (S p (X), ˆd p ). By Theorem 6., (Ω, k Ω,p ) is δ -hyperbolic with δ = δ (c ) = δ (c), and therefore for any w Ω and 0 < ɛ ɛ 0 (δ ), the natural map φ : ( kω,p Ω, d w,ɛ) (ˆ p Ω, ˆd p ) exists and is η -quasimöbius with η = η (c, ɛ) = η (c, ɛ). On the other hand, Theorem 4.6 () implies that the identity map f : (Ω, k Ω ) (Ω, k Ω,p ) is M-bilipschitz with M = M(c). By Proposition 5.0, for any w Ω and any ɛ satisfying 0 < ɛ ɛ (c) := min{ɛ 0 (δ), ɛ 0 (δ )}, the boundary map f : ( Ω, d w,ɛ ) ( uω, d w,ɛ) is η 2 -quasimöbius with η 2 = η 2 (δ, M) = η 2 (c). Hence there is an η-quasimöbius natural map with η = η η 2. φ = φ f : ( Ω, d w,ɛ ) ( Ω, ˆd p ) Again if we choose ɛ = ɛ (c), then the homeomorphism η in Theorem 6.2 depends only on c. 7 Annulus points, arc points and starlikeness In this section we recall the notion of annulus points and arc points, and show that each arc point lies on an anchor (Lemma 7.3) and that domains with large boundaries are starlike (Theorem 7.4). The following definitions are from Chapter 7 of [BHK]. Let (X, d) be a proper geodesic space and Ω X a rectifiably connected open subset with Ω. Definition 7.. Let 0 < λ /2. A point x Ω is said to be a λ-annulus point if there is a point a Ω with δ Ω (x) = d(x, a) such that B(a, δ Ω (x)/λ) \ B(a, λδ Ω (x)) Ω. If x Ω is not a λ-annulus point, then it is said to be a λ-arc point. Definition 7.2. Let x 0 Ω and c. A path γ : a b (with a, b Ω) in Ω is a c-anchor of x 0 if (i) x 0 γ, (ii) l d (γ) c d(a, b), (iii) for every x γ ax0 we have l d (γ ax ) c δ Ω (x), 2

22 (iv) for every x γ x0 b we have l d (γ xb ) c δ Ω (x), (v) γ Ω = {a, b}, (vi) γ is a continuous (c, c)-quasigeodesic in (Ω, k Ω ): l kω (γ xy ) c k Ω (x, y) + c for all x, y γ \{a, b}. The following is an analog of the anchor lemma 3.8 of [V]. Lemma 7.3. Suppose (X, d) is a C a -annular convex geodesic space. If 0 < λ /(2C 2 a), then every λ-arc point x 0 Ω has a c-anchor with c = c(λ, C a ). Proof. In this proof, C and C denote constants that depend only on λ and C a, and their values may change from one occurance to another as they represent all such constants occuring in this proof that we do not need to keep track of. Let a Ω such that δ Ω (x 0 ) = d(x 0, a). Since x 0 is a λ-arc point, there is a point y / Ω such that λδ Ω (x 0 ) d(a, y) < δ Ω (x 0 )/λ. Let γ be a geodesic (with respect to the metric d) connecting a to x 0. We break up the construction of the anchor into two cases. x 0 β y γ β 0 w z a y Figure : The two cases in Lemma 7.3. Case : d(a, y) δ Ω (x 0 ). Then let β 0 be a geodesic with respect to the metric d joining y to a; β 0 intersects the sphere S(a, δ Ω (x 0 )) at exactly one point w. By the annular convexity of X there is a rectifiable curve β joining x 0 and w in the annulus B(a, C a δ Ω (x 0 ))\B(a, δ Ω (x 0 )/C a ) with l d (β ) C a d(x 0, w). Let β 0 be the segment of β 0 with endpoints y and w, and set γ = β 0 β γ to be the concatenation of the three paths β 0, β, and γ. Case 2: d(a, y) < δ Ω (x 0 ). In this case let z γ be the unique point with d(z, a) = d(a, y). Let β 0 be the subcurve of γ between x 0 and z oriented from x 0 to z, and by the annular convexity, let β be a rectifiable curve in the annulus B(a, C a d(y, a))\b(a, d(y, a)/c a ) joining z and y with l d (β ) C a d(z, y). Now set γ = β β 0 γ. Once such a a curve γ has been constructed from the above cases, we modify this curve further. Since y / Ω and δ Ω (x 0 ) > λ δ Ω (x 0 ), there is a point x Ω ( β β 0 ) at 3C a 22