Existence of quasi-arcs

a rXiv:078.824v1[math.M G]6A ug27EXISTENCE OF QUASI-ARCSJOHN M.MACKAY Abstract.We show that doubling,linearly connected metric spaces are quasi-arc connected.This gives a new and short proof of a theorem of Tukia. 1.Introduction It is a standard topological fact that a complete metric space which is locally connected,connected and locally compact is arc-wise connected.In [Tuk96]Tukia showed that an analogous geometric statement is true:if a complete metric space is linearly connected and doubling,then it is con-nected by quasi-arcs,quantitatively.(Definitions will be explained below.)In fact,he proved a stronger result:any arc in such a space may be approx-imated by a local quasi-arc in a uniform way.In this note we give a new and more direct proof of this fact.This result is of interest in studying the quasi-symmetric geometry of met-ric spaces.Such geometry arises in the study of the boundaries of hyperbolic groups;Tukia’s result was used in this context by Bonk and Kleiner [BK05]and also by the author [Mac07].(Bonk and Kleiner use Assouad’s embed-ding theorem to translate Tukia’s result from subsets of R n to the situation of doubling,linearly connected metric spaces that we will be working in directly.)Before stating the theorem precisely,we recall some definitions.A metric space (X,d )is said to be doubling if there exists a constant N such that every ball can be covered by at most N balls of half the radius.We say (X,d )is L -linearly connected for some L ≥1if for all x,y ∈X there existsa continuum J ∋x,y of diameter less than or equal to Ld (x,y ).(This is also known as bounded turning or LLC(1).)Note that a complete,doubling metric space is locally compact.Also,increasing L by an arbitrarily small amount allows us to assume J is an arc,since then we can extend J to be connected,locally connected and complete,hence arc-wise connected.See,for example,[HY61].We denote by A [x,y ]the sub-arc of an embedded arc A between x and y inA .We say that an arc A in a doubling and complete metric space is an ǫ-local λ-quasi-arc if diam(A [x,y ])≤λd (x,y )for all x,y ∈A such that d (x,y )≤ǫ.(This terminology is explained by Tukia and Vaisala’s characterization of quasi-symmetric images of the unit interval as those metric arcs that are2J.M.MACKAYdoubling and bounded turning[TV80].)We say that an arc Bǫ-follows an arc A if there exists a coarse map p:B→A such that for all x,y∈B, B[x,y]is in theǫ-neighborhood of A[p(x),p(y)];in particular p displaces points at mostǫ.(We call the map p coarse to emphasize that it is not necessarily continuous.)We can now state(the stronger version of)Tukia’s theorem precisely, and as a immediate corollary our initial statement[Tuk96,Theorem1B, Theorem1A]:Theorem1.1(Tukia).Suppose(X,d)is a L-linearly connected,N-doubling, complete metric space.For every arc A in X and everyǫ>0,there is an arc J in theǫ-neighborhood of A whichǫ-follows A and is anαǫ-localλ-quasi-arc,whereλ=λ(L,N)>0andα=α(L,N)>0.Corollary1.2(Tukia).Every pair of points in a L-linearly connected,N-doubling,complete metric space is connected by aλ-quasi-arc,whereλ=λ(L,N)>0.Our strategy for proving Theorem1.1is straightforward:firstfind a method of straightening an arc on a given scale(Proposition2.1),then secondly apply this result on a geometrically decreasing sequence of scales to get the desired local quasi-arc as a limiting object.The statement of this proposition and the resulting proof of the theorem are essentially the same as in[Tuk96],but the proof of the proposition is new and much shorter.We include a complete proof for convenience to the reader.The author thanks Mario Bonk and,in particular,his advisor Bruce Kleiner for many helpful suggestions and fruitful conversations.2.Straightening arcsGiven any arc A andǫ>0,the following proposition allows us to straight-en A on a scaleǫinside theǫ-neighborhood of A,denoted by N(A,ǫ)={x∈X:d(x,A)<ǫ}.Proposition2.1.Given a complete metric space X that is L-linearly con-nected and N-doubling,there exist constants s=s(L,N)>0and S= S(L,N)>0such that:Given any arc A⊂X,ǫ>0then there exists an arc J⊂N(A,ǫ)with endpointsǫ-close that satisfies•for all x,y∈J,if d(x,y)<sǫthen diam(J[x,y])<Sǫ;and•Jǫ-follows A.We can use this proposition to prove Theorem1.1:Proof of Theorem1.1.We may assume s<1and S>1.Letδ=s10 andǫn=ǫδn.(We will make frequent use of the inequality ∞n=1δn≤1EXISTENCE OF QUASI-ARCS3 convergent to some points a and b respectively.If a=b then let J={a} and we are done.Otherwise,we claim that the sequence J n tends(in the Hausdorfftopology)to a non-trivial local quasi-arc.First,we show that the limit J=lim H J n exists,where lim H denotes the Hausdorfflimit.As{J n}is a sequence of compact,connected sets in a bounded region of a locally compact metric space,it suffices to show that the sequence is Cauchy with respect to the Hausdorffmetric.One bound follows by construction:J n+m⊂N(J n,109ǫδn-follows J n.Since d(z i,z i+1)≤ǫδn,d(p(z i),p(z i+1))≤4ǫδn≤sǫδn−1it follows that J n⊂N(J n+m,ǫδn+Sǫδn−1),and so{J n} is Cauchy and the limit J=lim H J n exists.Moreover,J is a connected, non-trivial,compact subset of X inside N(A,ǫ).To show J is an arc it suffices to demonstrate that every point,with the exception of(at most)two,is a cut point[HY61,Theorems2-18and 2-27].We use the order on J n induced by its topology,with least element a n and greatest b n.Given x∈J we define three points 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 chosen such that d(x,x n)<109ǫδn)-neighborhoods of J n[a n,h n(x)]andJ n[t n(x),b n]respectively,and define H(x)=∪{H n(x):n≥n0}(the Head) and T(x)=∪{T n(x):n≥n0}(the Tail).By definition,H(x)and T(x)are open.Suppose y∈J and y=x.We want to show that y lies in H(x)∪T(x).To this end we bound the diameter of J n[h n(x),t n(x)]using J n−1.d(p n(h n(x)),p n(t n(x)))≤2ǫδn−1+6Sǫδn−1≤sǫδn−2,and so we have that the diameter of J n−1[p n(h n(x)),p n(t n(x))]is less than Sǫδn−2.Thus the diameter of J n[h n(x),t n(x)]is less than Sǫδn−2+ 2ǫδn−1,as J n(ǫδn−1)-follows J n−1.For some n,d(x,y)>3Sǫδn−2.Now y is109ǫδn+(S+2δ)ǫδn−2<(2109ǫδm)by definition.We project 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],2ǫδn).Now since4J.M.MACKAYd(f(x m),x n)≤d(f(x m),x m)+d(x m,x)+d(x,x n)<4ǫδn<sǫδn−1we have (even for n=m):T m(x)⊂N(J n[x n,b n],2ǫδn)∪B(x n,(S+2δ)ǫδn−1).Thus H n(x)∩T m(x)=∅implies that there exists points p and q in J n such that a n≤p≤h n(x)<x n≤q≤b n and d(p,q)<3ǫδn<sǫδn−1.(Since(S+2δ)ǫδn−1+102Sǫδn−1,H n(x)cannot meet T m(x)in the ballB(x n,(S+2δ)ǫδn−1).)But now we know that the diameter of J n[p,q]is less than Sǫδn−1,while being109ǫδn)and has diameter less than Sǫδn−1+3ǫδn≤S+3δδ2we have that J is a local quasi-arc.The proof of Proposition2.1is based on a quantitative version of the following trivial topological result:given a maximal separated r-net N in an L-linearly connected,N-doubling space X then to each x in N we can associate V x,a connected union offinitely many arcs,satisfying for all x,y∈N:(1)If d(x,y)≤2r then y∈V x.(2)diam(V x)≤5Lr(3)Either d(V x,V y)>0or V x∩V y=∅.This may be done by letting V x,for x∈N,be the union of(finitely many) arcs joining x to each y∈N with d(x,y)≤2r.By linear connectedness,we get V x⊂B(x,3Lr).The third condition is trivially satisfied for compact subsets of a metric space,but this is the condition that we will extend to the following:(3′)Either d(V x,V y)>δr or V x∩V y=∅.Lemma2.2.We can construct the sets V x satisfying(1),(2)and(3′)for δ=δ(L,N).Without loss of generality,we can rescale the metric to set r=1.We now use a Zorn’s Lemma argument and the doubling property to split N into the disjoint union of separated20L-nets N1,...,N M,where M=M(20L,N). Consider the followingClaim∆(n).For the above situation we canfind V x for all x∈∪n k=1N k satisfying(1)and(2).Moreover,if we set all other V x={x}then(3′)is satisfied withδ=δ(L,N).EXISTENCE OF QUASI-ARCS5 We will show that∆(n)holds for0≤n≤M.(Lemma2.2immediately follows from∆(M).)As N is2-separated,∆(0)certainly holds forδ≤12(5L)−M.Proof of Claim.∆(n)holds so we are given V x for all x in N i,1≤i≤n. As N n+1is10L-separated we can treat the constructions of V x for x∈N n+1independently,and for each x we can construct a V(0)x usingfinitely many arcs to join x to its2-neighbors in N,inside the ball B(x,3L).Now construct V(i)x inductively,for1≤i≤M.(Again,assume V x={x}for x∈N k,k>n+1.)V(i−1)x can be5L-close to at most one y∈N i.Ifd(V(i−1)x,V y)∈(0,12(5L)−M)for some y∈N i,i<M.(The i=M case follows from the last step of the construction.)Then,since V x=V(M)x⊃V (i)x,d(V(i)x,V y)>12(5L)−i(1−(2L)(5L)−1−(2L)(5L)−2−...−(2L)(5L)−(M−i))>11−(1/(5L)) ≥52(5L)−M,contradicting our assumption,so∆(n+1)holds.We nowfinish by using this construction to prove our proposition. Proof of Proposition2.1.By rescaling the metric,we may assume thatǫ= 20L.Let r=1in the discussion above,and let N be a maximally separated 1-net in e Lemma2.2to get{V x}x∈N satisfying(1),(2)and(3′)for δ=δ(L,N)>0.Discretize A as follows:send A to afinite sequence(x0,...,x n)where x i∈N for0≤i≤n and there exists a sequence of points y i∈A for 0≤i≤n,such that in the order on A a=y0≤...≤y n=b and A[y i,y i+1]⊂B(x i,2).Now we cut out a path without repeats along the V x’s.Let r0=0,and for j∈N+define r j inductively as max{k:V xk ∩V xr j−1=∅},untilr m=n for some m≤n.Note that for i+2≤j,V xr i ∩V xr j=∅andthus d(V xr i ,V xr j)>δ.Let us construct our arc J in segments.First,letz0=x r0.Second,for each i from0to m−1,let J i=J i[z i,z i+1]be an arcjoining z i∈V xr i to some z i+1∈V xr i+1,inside thefinite connected union of6J.M.MACKAYarcs V xr i ,where z i+1is thefirst point of J i in V xr i+1.(In the case i=m−1,join z m−1to x rm =z m.)Set J=J0∪...∪J m.This path J is an arc since each J i⊂V xr iis an arc,and if p∈J i∩J j fori<j then V xr i ∩V xr j=∅,so j=i+1and therefore p=z i+1=z j.Thisimplies that J is an arc;in fact it is the arc we want.For any y,y′∈J, y<y′,we canfind i≤j such that z i≤y<z i+1,z j≤y′<z j+1.(If y=z m,set i=m;likewise for y′.)If d(y,y′)≤δthen because y∈V xr iandy′∈V xr j we have d(V xr i,V xr j)≤δ,and so either i=j or i=j+1,givingthat J[y,y′]⊂V xr i ∪V xr jand diam(J[y,y′])is bounded above by10L.Furthermore,there is a coarse map sending y∈J[z i,z i+1)⊂J to x ri ∈N,and sending x rm to itself.We then send this x rito its corresponding y ri∈Aand call the entire composition f:J→A.The construction coarsely preserves order(up to constant10L+2<20L=ǫ).So taking s=120L10L we are done. Remark:This method of proof allows one to explicitly estimate the con-stants given in the statements of Theorem1.1and Corollary1.2,but for most applications this is not necessary.References[BK05]M.Bonk and B.Kleiner,Quasi-hyperbolic planes in hyperbolic groups,Proc.Amer.Math.Soc.133(2005),no.9,2491–2494(electronic).[HY61]J.G.Hocking and G.S.Young,Topology,Addison-Wesley,Reading,Mass.,1961. [Mac07]J.M.Mackay,Spaces with conformal dimension greater than one,Preprint(2007). [Tuk96]P.Tukia,Spaces and arcs of bounded turning,Michigan Math.J.43(1996),no.3, 559–584.[TV80]P.Tukia and J.V¨a is¨a l¨a,Quasisymmetric embeddings of metric spaces,Ann.Acad.Sci.Fenn.Ser.A I Math.5(1980),no.1,97–114.Department of Mathematics,University of Michigan,Ann Arbor MICurrent address:Department of Mathematics,Yale University,New Haven CTE-mail address:jmmackay@。