Existence of quasi-arcs

EXISTENCE OF QUASI-AR CS JOHN M. MA CKA Y Abstract. W e sho w that doubling, linearly connecte d metric spaces are quasi- arc connec ted. This give s a new and sho rt pro of of a theorem of T ukia. 1. Introduction It is a standard topo logical fact that a complete metric space which is lo cally connected, connected and lo ca lly compact is arc-wise connected. T ukia [6] show ed that an analog ous geo metric statement is true: if a complete metric space is linear ly connected and doubling, then it is connected b y quasi-arcs, quantitatively . In fact, he prov ed a str o nger result: any arc in such a space may b e appr oximated b y a lo cal quasi-arc in a uniform wa y . In this note we give a new and more direct pro of of this fact. This result is of in ter est in studying the qua sisymmetric geometry of metric spaces. Such geo metry arise s in the study of the b oundar ies of hyp e rbo lic gr oups; T ukia’s result was used in this co n text by Bonk and Kleiner [1], and also by the au- thor [5]. (Bonk and Kleiner use Ass o uad’s embedding theore m to trans la te T ukia’s result fr om its or iginal context of subsets of R n int o our setting of doubling and linearly connected metric spaces.) Before stating the theo rem precisely , w e recall some definitions. A metric s pace ( X, d ) is said t o b e doubling if ther e exists a constan t N such that every ba ll can be covered by at mos t N balls of half the ra dius. Note that any co mplete, doubling metric space is prop er: all clo sed balls are compact. W e say ( X, d ) is L -line arly c onne cte d for some L ≥ 1 if for all x, y ∈ X there exists a co mpact, connected se t J ∋ x, y of diameter less than o r equal to Ld ( x, y ). (This is also known as bo unded turning or LLC(1).) W e ca n actually assume that J is an ar c , at the cost o f increa sing L by an arbitrarily small amoun t. T o see this, note that X is lo cally connec ted, a nd so the connected co mponents of an op en set are op en. Thus, for any op en neighbor ho o d U o f J , the connected comp onent o f U that co n tains J is an o p en set. W e can r eplace J inside U by an arc with the sa me endpo in ts, since any op e n, c onnected subset of a loc a lly co mpact, lo c a lly co nnected metric space is arc-wise connected [3, Cor ollary 32.36]. F or a n y x and y in an embedded arc A , we deno te by A [ x, y ] the clo sed, po ssibly trivial, subarc o f A that lies b etw een them. W e say that an a rc A in a doubling and complete metric space is an ǫ -lo c al λ -quasi-ar c if diam( A [ x, y ]) ≤ λd ( x, y ) for all x, y ∈ A such that d ( x, y ) ≤ ǫ . (This terminolo gy is explained by T ukia a nd Date : Decem ber 10, 2007. 2000 Mathematics Subje ct Classific ation. Pr imary 30C65, Secondary 54D05. Key wor ds and phr ases. Q uasi-arc, linearly connecte d, bounded turning. This r esearc h w as partially supported by NSF grant DMS-0701515. 1 2 J. M. M A CKA Y V¨ ais¨ al¨ a ’s characteriza tio n of quasisymmetric images of the unit interv al as those metric arcs that are doubling and b ounded turning [7].) One no n- standard definition will b e useful in our exp osition. W e say that an arc B ǫ -fol lows an ar c A if there ex is ts a coa rse map p : B → A , sending end- po in ts t o endp oints, s uc h that for all x , y ∈ B , B [ x, y ] is in the ǫ -neighbor ho o d of A [ p ( x ) , p ( y )]; in pa rticular, p displaces p oints at mos t ǫ . (W e call the ma p p c o arse to emphasize that it is not necessa rily c ont inuous.) The condition that B ǫ -follows A is stronger than the condition that B is con- tained in the ǫ -neighborho o d of A . It says that, coar sely , the a rc B can be obtained from the ar c A b y cutting out ‘lo o ps.’ (O f course, A co n tains no ac tua l loops , but it may have subarcs of large diameter whose endp oints are 2 ǫ -close.) W e can now state the stronger version of T ukia’s theorem precisely , and as an immediate corollar y our initial statement [6, Theorem 1B, Theorem 1A]: Theorem 1.1 (T ukia ) . Supp ose ( X, d ) is a L-line arly c onne cte d, N -doubling, c om- plete metric sp ac e. F or every ar c A in X and every ǫ > 0 , ther e is an ar c J that ǫ -fol lows A , has the same endp oints as A , and is an αǫ -lo c al λ -quasi-ar c, wher e λ = λ ( L, N ) ≥ 1 and α = α ( L, N ) > 0 . Corollary 1.2 (T ukia ) . Every p air of p oints in a L - line arly c onne cte d, N -doubling, c omplete metric sp ac e is c onne cte d by a λ -quasi-ar c, whe r e λ = λ ( L, N ) ≥ 1 . Our strategy for proving Theorem 1.1 is straightforward: find a metho d of straightening an ar c on a given scale (Prop osition 2.1), then apply this result on a geometrically decreas ing sequence of scales to get the desired lo cal quasi-arc as a limiting ob ject. The s tatemen t o f this pr op osition and the r esulting pro of of the theorem essentially follow T ukia [6], but the pro of of the pro pos ition is new and m uch sho r ter. W e include a co mplete pro of for convenience to the rea der. The author thanks Mario Bonk a nd, in particular , his advisor Br uce K leiner for many helpful sugg estions and fr uitful conv ersations. 2. Main Resul ts Given any arc A and ι > 0, the following prop osition allo ws us to stra ight en A on a scale ι inside the ι -neighbo r ho o d of A . Prop osition 2 .1. Given a c omplete metric sp ac e X that is L -line arly c onne cte d and N -doubling, ther e ex ist c onstant s s = s ( L, N ) > 0 and S = S ( L, N ) > 0 with the fol lowing pr op erty: for e ach ι > 0 and e ach ar c A ⊂ X , ther e exists an ar c J that ι -fol lows A , has t he same endp oints as A , and satisfies ( ∗ ) ∀ x, y ∈ J, d ( x, y ) < sι = ⇒ diam( J [ x, y ]) < S ι. W e will a pply this prop osition on a decreasing sequenc e of scale s to get a lo ca l quasi-ar c in the limit. The key step in proving this is given by the fo llowing lemma. Lemma 2 .2. Supp ose ( X , d ) is a L-line arly c onne ct e d, N-doubling, c omplete metric sp ac e, and let s, S, ǫ and δ b e fixe d p ositive c onstants satisfying δ ≤ min { s 4+2 S , 1 10 } . Now, if we have a se quenc e of ar cs J 1 , J 2 , . . . , J n , . . . in X , such t hat for every n ≥ 1 • J n +1 ǫδ n -fol lows J n , and • J n +1 satisfies ( ∗ ) with ι = ǫ δ n and s, S as fixe d ab ove, EXISTENCE OF QUASI-ARC S 3 then the Hausdorff limit J = lim H J n exists, and is an ǫδ 2 -lo c al 4 S +3 δ δ 2 -quasi-ar c. Mor e over, the endp oints of J n c onver ge to the endp oints of J , and J ǫ -fol lows J 1 . W e shall need some standar d de finitio ns. The (infimal) distance b etw een tw o subsets U , V ⊂ X is defined as d ( U, V ) = inf { d ( u, v ) : u ∈ U, v ∈ V } . If U = { u } , then we set d ( u, V ) = d ( U, V ). The r -neighbor ho od of U is the set N ( U, r ) = { x : d ( x, U ) < r } , and the Hausdorff dista nc e b et ween U and V , d H ( U, V ), is defined to b e the infimal r s uc h that U ⊂ N ( V , r ) and V ⊂ N ( U, r ). F or mor e information, see [2, Chapter 7]. W e will now prov e Theo rem 1.1. Pr o of of The or em 1.1. Let s and S be given by Prop osition 2 .1, and set δ = min { s 4+2 S , 1 10 } . Let J 1 = A a nd apply P rop osition 2.1 to J 1 and ι = ǫδ to ge t an a rc J 2 that ǫδ -follows J 1 . Rep eat, applying the le mma to J n and ι = ǫδ n , to get a sequence of arcs J n , where each J n +1 ǫδ n -follows J n , and satisfies ( ∗ ) with ι = ǫ δ n . W e can now apply Lemma 2.2 to find an αǫ -lo cal λ -quasi-ar c J that ǫ -follows A , where α = δ 2 and λ = 4 S +3 δ δ 2 . E very J n has the sa me endp oints as A , so J will also hav e the sa me endpoints.  The pro of of Lemma 2.2 relie s on some fairly stra ight forward estimates a nd a classical characterization o f an arc. Pr o of of L emma 2.2. F or ev ery n ≥ 1, J n +1 ǫδ n -follows J n . W e denote the ass o c i- ated coarse map b y p n +1 : J n +1 → J n . In the following, w e will make frequent use of the inequality P ∞ n =0 δ n < 11 9 . W e beg in by sho wing that the Hausdorff limit J = lim H J n exists. The collection of all compa ct subsets o f a compact metric s pace, given the Haus dorff metric, is itself a compact metric spa ce [2 , Theorem 7.3.8]. Since { J n } is a sequence o f compact sets in a b ounded region o f a prop er metric spac e , to show that the sequenc e conv erges with resp ect to the Hausdorff metric, it s uffices to show that the sequence is Cauch y . One b ound follows by construction: J n + m ⊂ N ( J n , 11 9 ǫδ n ) for all m ≥ 0. F or the second b ound, tak e J n + m and split it into subarcs of diameter at most ǫδ n , and write this a s J n + m = J n + m [ z 0 , z 1 ] ∪ · · · ∪ J n + m [ z k − 1 , z k ] for s ome z 0 , . . . , z k and some k > 0. Our coar se maps comp ose to give p : J n + m → J n , showing that J n + m 11 9 ǫδ n -follows J n . F urthermore , since d ( z i , z i +1 ) ≤ ǫδ n , we ha ve d ( p ( z i ) , p ( z i +1 )) ≤ 4 ǫδ n ≤ sǫδ n − 1 . Co m bining this with the fact that p maps endp oint s to endpoints, for n ≥ 2 w e have J n = J n [ p ( z 0 ) , p ( z 1 )] ∪ · · · ∪ J n [ p ( z k − 1 ) , p ( z k )] ⊂ N ( { p ( z 0 ) , . . . , p ( z k ) } , S ǫ δ n − 1 ) ⊂ N  J n + m , 11 9 ǫδ n + S ǫδ n − 1  . T aken together, t hese bounds give d H ( J n , J n + m ) ≤ 11 9 ǫδ n + S ǫδ n − 1 , so { J n } is Cauch y and the limit J = lim H J n exists. Moreover, J is compact (by definition) and connected (beca use each J n is connected). Now we let a n , b n denote the endp oint s of J n . Since p n ( a n ) = a n − 1 , and p n displaces p oints at most ǫδ n , the sequence { a n } is Ca uchy a nd hence co n verges to some p o in t a ∈ J . Similarly , { b n } conv erges to a p oint b ∈ J . 4 J. M. M A CKA Y There are tw o cas es to c o nsider. If a = b , then d ( a n , b n ) ≤ 2 11 9 ǫδ n ≤ sǫ δ n − 1 . Consequently , diam( J n ) ≤ S ǫδ n − 1 , J = lim H J n has diameter zero, a nd t hus J = { a } . Other wise, a 6 = b and so J is non-tr ivial. W e claim that in this cas e J is a lo cal quasi-ar c. T o sho w J is a n arc with endpo in ts a and b it suffices to demonstrate that e very po in t x ∈ J \ { a , b } is a cut p oint [4, Theor ems 2-18 and 2-2 7]. The top ology of J n induces an order on J n with least element a n and greatest b n . Given x ∈ J , we define three p oints h n ( x ), x n and t n ( x ) that satisfy a n < h n ( x ) < x n < t n ( x ) < b n , where x n is c hosen such that d ( x, x n ) ≤ 11 9 ǫδ n , and h n ( x ) and t n ( x ) are the first and last e le ments of J n at distance ( S + 1) ǫδ n − 1 from x . As long as x is not e q ual to a or b , for n greater than so me n 0 these p oints will ex ist and this definition will be v a lid. W e shall denote the 11 9 ǫδ n -neighborho o ds of J n [ a n , h n ( x )] and J n [ t n ( x ) , b n ] by H n ( x ) and T n ( x ) resp ectively , and define H ( x ) = ∪{ H n ( x ) : n ≥ n 0 } (the Head) and T ( x ) = ∪{ T n ( x ) : n ≥ n 0 } (the T ail). By definit ion, H ( x ) and T ( x ) are op en. W e claim that, in addition, they are disjo int and co ver J \ { x } , and so x is a cut po in t. Fix y ∈ J , and suppo s e y / ∈ H ( x ) ∪ T ( x ). W e want to show that y = x . T o this end, w e b ound the diameter of J n [ h n ( x ) , t n ( x )] using J n − 1 . Because d ( p n ( h n ( x )) , p n ( t n ( x ))) ≤ 2 ǫ δ n − 1 + 2( S + 1) ǫδ n − 1 ≤ sǫδ n − 2 , we know that the diameter of J n − 1 [ p n ( h n ( x )) , p n ( t n ( x ))] m ust be less than S ǫδ n − 2 . Thus the diam- eter of J n [ h n ( x ) , t n ( x )] is less than S ǫδ n − 2 + 2 ǫδ n − 1 , as J n ǫδ n − 1 -follows J n − 1 . F or ev ery n ≥ n 0 , y is 11 9 ǫδ n close to some y n ∈ J n . Since y / ∈ H ( x ) ∪ T ( x ), y n m ust lie in J n [ h n ( x ) , t n ( x )], so d ( x, y ) ≤ d ( x, J n [ h n ( x ) , t n ( x )]) + diam( J n [ h n ( x ) , t n ( x )]) + d ( y n , y ) ≤ 2 11 9 ǫδ n + ( S + 2 δ ) ǫδ n − 2 =  2 11 9 δ 2 + S + 2 δ  ǫδ n − 2 , therefore d ( x, y ) = 0 and J \ ( H ( x ) ∪ T ( x )) = { x } . W e now show that H ( x ) and T ( x ) are disjoin t. If not, then H n ( x ) ∩ T m ( x ) 6 = ∅ for some n and m . It suffices to a ssume n ≤ m . Now T m ( x ) ⊂ N ( J m [ x m , b m ] , 11 9 ǫδ m ) by definition. W e send J m to J n using f = p n +1 ◦ · · · ◦ p m : J m → J n , to get that T m ( x ) ⊂ N ( J n [ f ( x m ) , b n ] , 3 ǫ δ n ). Since d ( f ( x m ) , x n ) ≤ d ( f ( x m ) , x m ) + d ( x m , x ) + d ( x, x n ) < 4 ǫδ n < s ǫ δ n − 1 we hav e, ev en for n = m , T m ( x ) ⊂ N ( J n [ x n , b n ] , 3 ǫ δ n ) ∪ B ( x n , ( S + 3 δ ) ǫδ n − 1 ) . Since ( S + 3 δ ) ǫδ n − 1 + 11 9 ǫδ n < ( S + 1 2 ) ǫδ n − 1 , H n ( x ) cannot meet T m ( x ) in the ball B ( x n , ( S + 3 δ ) ǫδ n − 1 ). Thus H n ( x ) ∩ T m ( x ) 6 = ∅ implies that ther e exist p oints p and q in J n such that a n ≤ p ≤ h n ( x ) < x n ≤ q ≤ b n and d ( p, q ) < 3 ǫ δ n < sǫδ n − 1 . But then we know that J n [ p, q ] ha s diameter less than S ǫδ n − 1 , while containing bo th h n ( x ) and x n . This co n tradicts the definition of h n ( x ), so H ( x ) ∩ T ( x ) = ∅ . W e ha ve shown that J is an arc with endpoints a and b ; it re ma ins to show that J is a lo ca l quasi-arc with the required constants. Say w e a r e given x and y in J , with x n and y n as b efore. O ur arg umen t shows that the segments J n [ x n , y n ] co nverge to some a r c ˜ J [ x, y ], be c ause J n +1 [ x n +1 , y n +1 ] ( ǫδ n + S ǫδ n − 1 )-follows J n [ x n , y n ] fo r all n ≥ 2. This arc ˜ J [ x, y ] must lie in J , therefore ˜ J [ x, y ] must equa l J [ x, y ]. Now, supp ose that d ( x, y ) ∈ ( ǫδ n +1 , ǫδ n ] holds EXISTENCE OF QUASI-ARC S 5 for some n ≥ 2. Then d ( x n , y n ) ≤ 3 ǫδ n + ǫδ n < sǫδ n − 1 , and so the subarc J [ x, y ], which lies in N ( J n [ x n , y n ] , 11 9 ǫ ( δ n + S δ n − 1 )), has dia meter less than S ǫ δ n − 1 + 3 ǫ ( δ n + S δ n − 1 ) ≤ 4 S +3 δ δ 2 d ( x, y ), as desir ed. F urthermo re, this same argument gives that, for a ll n ≥ 2, J 11 9 ǫ ( δ n + S δ n − 1 )- follows J n , which itself 11 9 ǫδ -follows J 1 = A . T aking n sufficiently large, we hav e that J ǫ - follows A .  3. Discrete p a ths and the pr oof of Proposition 2.1 The pro o f of Prop osition 2.1 is based on a quantitativ e version o f a simple geo- metric result. Before we sta te this result, re c a ll that a maximal r -separ ated set N is a subset of X such that for all distinct x, y ∈ N we ha ve d ( x, y ) ≥ r , and for all z ∈ X there ex ists some x ∈ N with d ( z , x ) < r . Now suppo se that we ar e given a maximal r - s eparated set N in an L -linearly connected, N -doubling, complete metric s pa ce X . Then w e can find a collection of sets { V x } x ∈N so that each V x is a co nnected union of finitely man y arcs in X , and for all x, y ∈ N : (1) d ( x, y ) ≤ 2 r = ⇒ y ∈ V x . (2) diam( V x ) ≤ 5 Lr . (3) V x ∩ V y = ∅ = ⇒ d ( V x , V y ) > 0. F or x ∈ N , we can construct eac h V x by defining it to be the union of finitely many ar cs joining x to each y ∈ N with d ( x, y ) ≤ 2 r . By linear connectedness, we can ensure that diam( V x ) ≤ 4 Lr . Co nditio n (3) is triv ially s a tisfied for compact subsets of a metric space, but we will strengthen it to the following: (3 ′ ) V x ∩ V y = ∅ = ⇒ d ( V x , V y ) > δ r . Lemma 3.1. We c an c onst ruct the sets V x satisfying (1), ( 2) and (3 ′ ) for δ = δ ( L, N ) . Pr o of. Without loss of gener ality , w e can rescale the metric to set r = 1. Since X is doubling, the max im um n umber of 1 - separated p oints in a 20 L - ba ll is b ounded by a consta n t M = M (20 L, N ). W e can lab el every p oint of N with an int eger b etw een 1 and M , such that no t wo p oints have the same la b el if they are separated by a distance les s than 20 L . T o find this lab elling, consider the collection of a ll such la belling s on subs ets o f N under the natural pa rtial o rder. A Zorn’s Lemma arg umen t gives the existence of a maximal elemen t: our desired lab elling. So N is the disjoint union o f 20 L -separ ated sets N 1 , . . . , N M . Now let N ≤ n = ∪ n k =1 N k , and consider the following Claim ∆( n ) . We c an fi n d V x for al l x ∈ N ≤ n , such that for al l x, y ∈ N ≤ n (1), (2) a nd (3 ′ ) ar e satisfie d with δ = 1 2 (5 L ) − n . ∆(0) holds trivially , and Lemma 3.1 im mediately follo ws from ∆( M ), with δ = δ ( L, N ) = 1 2 (5 L ) − M . So we are do ne, mo dulo the s tatemen t that ∆( n ) = ⇒ ∆( n + 1) for n < M .  Pr o of that ∆( n ) = ⇒ ∆( n + 1 ) , for n < M . By ∆( n ), w e ha ve s e ts V x for all x in N ≤ n . 6 J. M. M A CKA Y As N n +1 is 20 L -se pa rated we can treat the constructions of V x for each x ∈ N n +1 independently . W e begin by creating a set V (0) x that is the union of finitely ma ny arcs joining x to its 2-neig h b ors in N . W e can ensur e that dia m( V (0) x ) ≤ 4 L . Now construct V ( i ) x inductively , for 1 ≤ i ≤ n . V ( i − 1) x can b e 5 L -close to at most one y ∈ N i . If d ( V ( i − 1) x , V y ) ∈ (0 , 1 2 (5 L ) − i ), then define V ( i ) x by adding to V ( i − 1) x an arc of diameter at most L (5 L ) − i joining V ( i ) x to V y . Otherwise, let V ( i ) x = V ( i − 1) x . Contin ue until i = n and set V x = V ( n ) x . Note that V x satisfies (1) and (2) by constructio n. The o nly no n-trivial case we need to chec k for (3 ′ ) is whether d ( V x , V y ) ∈ (0 , 1 2 (5 L ) − n ) for so me y ∈ N i , i < n . (The i = n case follows from th e las t step o f the construction.) Then, since V x = V ( n ) x ⊃ V ( i ) x , V ( i ) x ∩ V y 6 = ∅ , and d ( V ( i ) x , V y ) ≥ 1 2 (5 L ) − i . The construction then implies that d ( V x , V y ) ≥ 1 2 (5 L ) − i (1 − (2 L )(5 L ) − 1 − (2 L )(5 L ) − 2 − · · · − (2 L )(5 L ) − ( n − i ) ) > 1 2 (5 L ) − n (5 L )  1 − 2 / 5 1 − (1 / (5 L ))  ≥ 5 2  1 2 (5 L ) − n  , contradicting our a s sumption, so ∆( n + 1) holds.  W e now finish b y using this constructio n to prov e o ur prop ositio n. Pr o of of Pr op osition 2.1. B y r e s caling the metric, we may a ssume that ι = 20 L . If d ( a, b ) ≤ 2 0 = ι L , then join a to b by an arc o f diameter less than ι . This a r c will, trivially , satisfy our conclusion for an y S ≥ 1. Otherwise, d ( a, b ) > 20. In the hypotheses for Le mma 3.1, let r = 1 a nd let N b e a maximal 1-sepa rated set in X that contains b oth a and b . Now apply Lemma 3.1 to get { V x } x ∈N satisfying (1), (2) and (3 ′ ) for δ = δ ( L, N ) > 0. W e wan t to ‘discretize’ A by finding a corresp onding sequence o f points in N . Now, every op en ball in X meets the arc A in a collection o f disjoint, relatively op en in terv als. Since N is a maximal 1 -separated set, the collection of op en balls { B ( x, 1) : x ∈ N } covers X ; in particular, it co vers A . By the compac tness of A , we can find a finite cov er of A by connected, relatively op en interv als, each lying in some B ( x, 1), x ∈ N . Using this finite cov er, we can find p oints x i ∈ N and y i ∈ A fo r 0 ≤ i ≤ n , such that a = y 0 < · · · < y n = b in the order on A , a nd A [ y i , y i +1 ] ⊂ B ( x i , 1) for each 0 ≤ i < n . Since a, b ∈ N , we ha ve that x 0 = a and x n = b . The sequence ( x 0 , . . . , x n ) is a discrete path in N that corr esp onds na turally to A . W e now find a subseque nc e ( x r j ) of ( x i ) suc h that the corresp onding sequence of sets ( V x r j ) forms a ‘path’ witho ut repeats. Let r 0 = 0 , and for j ∈ N + define r j inductively a s r j = max { k : V x k ∩ V x r j − 1 6 = ∅} , until r m = n for s o me m ≤ n . The int eger r j is well defined since d ( y ( r j − 1 +1) , x k ) ≤ 1 for k = r j − 1 and k = r j − 1 + 1, so V x ( r j − 1 +1) ∩ V x r j − 1 6 = ∅ . Note that if i + 2 ≤ j then V x r i ∩ V x r j = ∅ , and thus d ( V x r i , V x r j ) > δ . Let us co nstruct our a r c J in segments. First, let z 0 = x r 0 . Seco nd, fo r ea c h i from 0 to m − 1, let J i = J i [ z i , z i +1 ] b e an arc in V x r i that joins z i ∈ V x r i to some z i +1 ∈ V x r i +1 , where z i +1 is the fir st p oint of J i to meet V x r i +1 . (In the case i = m − 1, join z m − 1 to x r m = z m .) Set J = J 0 ∪ · · · ∪ J m . EXISTENCE OF QUASI-ARC S 7 This path J is an arc s inc e each J i ⊂ V x r i is an arc, and if there exists a p oint p ∈ J i ∩ J j for some i < j , then j = i + 1 and p = z i +1 = z j . This is true b ecause V x r i ∩ V x r j 6 = ∅ implies that j = i + 1, and the definition of z i +1 implies that J i ∩ V x r i +1 = { z i +1 } . Any finite se q uence of ar cs that meet only at consecutive endpo in ts is also an ar c, so we ha ve that J is an ar c. In fact, J satis fie s ( ∗ ). F o r any y , y ′ ∈ J , y < y ′ , we can find i ≤ j such t hat z i ≤ y < z i +1 , z j ≤ y ′ < z j +1 . (If y = z m , se t i = m ; likewise for y ′ .) If d ( y , y ′ ) ≤ δ then, because y ∈ V x r i and y ′ ∈ V x r j , we hav e d ( V x r i , V x r j ) ≤ δ , so either j = i or j = i + 1. This gives that J [ y , y ′ ] ⊂ V x r i ∪ V x r j , and so diam( J [ y , y ′ ]) is b ounded ab ov e by 10 L . F urthermo re, J ι -follows A . There is a coa rse map f : J → A defined by the following comp osition: firs t ma p J to N by sending y ∈ J [ z i , z i +1 ) ⊂ J to x r i ∈ N , and sending x r m to itself. Seco nd, map ea ch x r i to the cor resp onding y r i in A . T aking arbitr a ry y < y ′ in J as b efore, w e see that J [ y , y ′ ] ⊂ J [ z i , z j +1 ] ⊂ N ( { x r i , . . . , x r j } , 5 L ) ⊂ N ( { y r i , . . . , y r j } , 5 L + 1 ) ⊂ N ( A [ y r i , y r j ] , 5 L + 1 ) ⊂ N ( A [ f ( y ) , f ( y ′ )] , ι ) . W e let s = 1 20 L δ and S = 1 20 L 10 L , and hav e proven the Prop os itio n.  R emark: This metho d of pr oo f allows one to e x plicitly estimate the constants given in the statements of Theor em 1.1 and Co rollary 1.2, but for mos t applications this is not necessa r y . References 1. M . Bonk and B. Kleiner, Quasi-hyp erb olic planes in hyp erb olic gr oups , Proc. Amer. Math. So c. 133 (2005), no. 9, 2491–2494 (elect ronic). MR 2146190 (2005m:20098) 2. D. Burago, Y. Burago, and S. Iv anov, A c ourse in metric ge ometry , Graduate Studies i n Mathematics, vo l. 33, American Mathematical Societ y , Providenc e, RI, 2001. MR 1835418 (2002e:530 53) 3. H . F. Cullen, Intr o duction to ge ner al t op olo gy , D. C. Heath an d Co., Boston, Mass., 1968. MR 02214 55 (36 #4507) 4. J. G. Ho c king and G. S. Y oung, T op olo gy , second ed., Dov er Publications Inc., New Y ork, 1988. MR 10168 14 (90h:54001) 5. J. M. Mac k ay , Sp ac e s with co nformal dimension gr e ate r tha n o ne , Preprin t (2007), arXiv:07 11.0417 . 6. P . T ukia, Sp ac es and ar cs of b ounde d turning , M ic higan Math. J. 4 3 (1996), no. 3, 559–584. MR 14205 92 (98a:30028) 7. P . T ukia and J. V¨ ais¨ al¨ a, Quasisymmetric emb e ddings of metric sp ac es , Ann. Acad. Sci. F enn. Ser. A I Math. 5 (1980), no. 1, 97–114. MR 595180 (82g:30038) Dep ar tment of Ma thema tics, University of Michigan, Ann Arbor, Michigan 48109- 1109 Curr ent addr ess : Departmen t of M athematics, Y ale University , New Hav en, Connecticut 06520-8283 E-mail addr ess : jmmackay@umich. edu