Quasiflats in CAT(0) complexes

QUASIFLA TS IN CA T(0) COMPLEXES MLADEN BESTVINA, BRUCE KLEINER, AND MICHAH SA GEEV Abstract. W e show that if X is a piecewise E uclidean 2 -complex with a cocompa ct isometry group, then every 2-quasiflat in X is at finite Ha usdorff distance from a subset Q which is lo cally flat outside a compact set, and asymptotica lly conical. 1. Introd uction In a n umber o f rigidit y theorems for quasi-isometries, an imp orta n t step is to determine the structure of individual quasi-flats; this is then used to restrict the b eha vior of quasi-isometries, often b y exploiting the pattern of asymptotic incidence of the quasiflats [Mos73, KL97 a, KL97b, EF97, Esk98 , BKMM12]. In this pap er, we study 2- quasiflats in CA T(0) 2- complexes, and sho w that they hav e a v ery simple asymptotic structure: Theorem 1.1. L et X b e a pr op er, pie c ewi s e Euclide an, CA T(0) 2 - c om plex with a c o c omp act isometry g r oup. T hen every 2 -quas i flat Q ⊂ X lies at fi n ite Hausdorff distanc e fr om a subse t Q ′ ⊂ X which is lo c al ly flat (i.e. lo c al ly isometric to R 2 ) outside a c omp a ct set. This result, and more refined statemen t s app earing in later sections, are applied to 2-dimensional rig ht-angled Artin gr oups in [BKS08]. The main application is to show t hat if X, X ′ are the standard CA T (0) complexes of 2-dimensional righ t-angled Artin groups, then any quas i- isometry X → X ′ b et w een them m ust map flats to within finite Haus- dorff distance of flats. The strategy for provin g Theorem 1.1 is to replace the quasiflat Q with a canonical ob ject that has more rigid structure. T o t ha t end, w e first asso ciate an elemen t [ Q ] of the lo cally finite homology group H lf 2 ( X ) , and then show t ha t t he supp ort set supp([ Q ]) of [ Q ] – the set of p oin ts x ∈ X suc h that the induced homomor phism This resea r ch was supported by NSF grant s DMS-1308 1 78 and D MS-1 40589 9. 1 2 MLADEN B ESTVI NA, BRUCE KLEINER, AN D MICHAH S AGE EV H lf 2 ( X ) → H 2 ( X , X \ { x } ) is non trivial on [ Q ] – is at b ounded Haus- dorff distance f r o m Q . The supp ort set Q ′ := supp ( [ Q ]) b ehav es muc h lik e a minimizing lo cally finite cycle, and this leads to asymptotically rigid b ehavior, in part icular asymptotic flatness. Remarks. (1) Supp ort sets we re used implicitly in [KL97b], and also in [Xie05]. (2) The pap er [KL], whic h may b e view ed as a more sophisticated v ersion of the results presen ted here, exploits similar geomet- ric ideas in asymptotic cones, to study k - quasiflats in CA T(0) spaces whic h hav e no ( k + 1)-quasiflats. (3) Man y of the results of this pap er (though not Theorem 1.1 itself ) can b e adapted to n - quasiflats in n -dimensional CA T(0) complexes. (4) One ma y use the results in this pap er to giv e a new pro of that quasi-isometries b etw een Euclidean buildings map flats t o within uniform Hausdorff distance of flats [KL97b]. This t hen leads to a (partly) differen t pro of of rigidit y of quasi-isometries b et w een Euclidean buildings. Contents 1. In tro duction 1 2. Preliminaries 2 3. Lo cally finite homology and supp ort sets 4 4. Quasi-flats in 2-complexes 12 5. Square complexes 13 References 15 2. Preliminaries 2.1. CA T( κ ) spaces. W e recall some standard facts, and fix notation. W e refer the reader to [BH9 9 , KL9 7b] fo r more detail. Our notation and con ve ntions are consisten t with [KL97b]. Let X b e a CA T ( 0 ) space. If x, y ∈ X , then xy ⊂ X denotes the geo desic segmen t with end- p oin ts x, y . If p ∈ X , we let ∠ p ( x, y ) denote the angle b et w een x and QUASIFLA TS IN CA T(0) COMPLEXES 3 y at p . This induces a pseudo-distance on X \ { p } . By collapsing sub- sets of zero diameter and completing, we obtain the space of directions Σ p X , whic h is a CA T(1) space. The quotient map yields the logarithm log p : X \ { p } → Σ p X ; it a sso ciates to x ∈ X \ { p } the direction a t p of the geo des ic segmen t px . The t a ngen t cone at p , denoted C p X , is a CA T(0) space isometric to the cone o v er Σ p X . Giv en tw o constant ( no t necess arily unit) sp eed rays γ 1 , γ 2 : [0 , ∞ ) → X , their distance is defined to b e lim t →∞ d ( γ 1 ( t ) , γ 2 ( t )) t . This defines a pseudo-distance on the set of constant sp eed ra ys in X ; the metric space obtained by collapsing zero diameter subsets is the Tits c one of X , denoted C T X . The Tits cone is isometric to the Euclidean cone ov er t he Tits b oundary ∂ T X . F or ev ery p ∈ X , there are natural logarithm maps: log p : X → C p X , log p : C T X → X , log p : X \ { p } → Σ p X , log p : ∂ T X → Σ p X . Definition 2.1. If Z is a CA T(1) space, Y ⊂ Z , and z ∈ Z , then the antip o dal set o f z in Y , is An t( z , Y ) := { y ∈ Y | d ( z , y ) = π } . Recall that b y our definition, eve ry CA T (1) space has dia meter ≤ π . If X is a CA T(0 ) complex and p, x ∈ X are distinct p oin ts, Y ⊂ Σ x X , then the antipo dal set An t(lo g x p, Y ) is the set of directions in Y whic h are tangent to extensions of the geo desic segmen t px b eyond x . 2.2. Lo cally finite homology. Let Z b e a to p ological space. W e recall tha t the k th lo cally finite (singular) c hain g roup C lf k ( Z ) is the collection of (p ossibly infinite) fo rmal sums o f singular k - simplices , suc h that for ev ery compact subset Y ⊂ Z , o nly finitely man y nonzero terms are contributed b y singular simplices whose image in tersects Y . The usual b oundary o p erator yields a well-define d ch ain complex C lf ∗ ( Z ); its homolog y is the lo c al ly finite ho m olo gy of Z . Supp ose K is a simplicial complex. Then there is a simplicial vers ion of the lo cally finite c hain complex – the lo cally finite simplicial c hain complex – defined by taking (p ossibly infinite) for ma l linear combina- tions of orien ted simplices of K , where ev ery simplex σ of K touc hes 4 MLADEN BESTVINA, BRUCE KL EINER, AND MI CHAH SAGEEV only finitely man y simplices with nonzero co effic ients . The usual pro of that simplicial homology is isomorphic to singular homology g iv es an isomorphism b etw een t he lo cally finite simplicial homology of K , and the lo cally finite ho mo lo gy of its geometric realization | K | , when K is lo cally finite [Hat02, 3.H, Exercise 6]. The supp ort set of σ ∈ H lf k ( Z ) is the subset supp( σ ) ⊂ Z consisting of the p oin ts z ∈ Z for whic h the inclusion homomorphism H lf k ( Z ) → H k ( Z , Z \ { z } ) is nonzero on σ . This is a closed subset when Z is Hausdorff. No w supp ose K is a n n -dimensional lo cally finite simplicial complex, with p olyhed ro n Z . Then the simplicial c hain g roups C lf k ( K ) v anish for k > n , and henc e H lf n ( Z ) is isomorphic to the group o f lo cally finite simpicial n -cycles Z lf n ( K ). The supp ort set of a lo cally finite simplicial n -cycle σ ∈ Z lf n ( Z ) is the union of the closed n -simplices hav ing nonzero co efficien t in σ , as fo llows from excision. 3. Locall y finite homology and suppor t se ts The k ey results in this section are the geo de sic extension prop erty of Lemma 3.1, and the asymptotic conicalit y result f or supp ort sets with quadratic area growth, in Theorem 3.11. W e remark that most of the statemen ts (and pro of s) in this section extend with minor mo difi- cations to suppo rts of n - dimensional lo cally finite homology classes in n -dimensional CA T (0) complexes. In this section X will b e a prop er, piecewis e Euclidean, CA T(0) 2- complex. 3.1. The geo desic extension prop erty and metric monotonic- it y . The fundamen tal prop ert y of supp ort sets is the extendabilit y of geo desics: Lemma 3.1. Supp ose σ ∈ H lf 2 ( X ) , and let S := supp( σ ) ⊂ X b e the supp o rt of σ . If p ∈ X , and x ∈ S , the ge o desi c se gment px m ay b e pr ol o nge d to a r a y in S : ther e i s a r ay xξ ⊂ S with fits to gether with px to fo rm a r ay pξ . Pr o of. Let γ : [0 , L ] → X b e t he unit sp eed parametrization of px , a nd let ˆ γ : I → X b e a maximal extension of γ suc h that ˆ γ ( I \ [0 , L ]) ⊂ S , where I is an interv a l con tained in [0 , ∞ ). Since S is a closed subs et of the complete space X , either I = [0 , R ] for some R < ∞ , o r I = [0 , ∞ ). QUASIFLA TS IN CA T(0) COMPLEXES 5 Supp ose I = [0 , R ] for R < ∞ , and let y := ˆ γ ( R ). Consider the closed ball B := B ( y , r ), where r is small enough that B is isometric to the r -ball in t he tangen t cone C y X . Not e that this implies t ha t S ∩ B is also a cone. Let σ = [ σ B + τ ], where σ B ∈ C lf 2 ( X ) is carried b y B (and is therefore a finite 2- c hain), τ ∈ C lf 2 ( X ) is carried b y X \ B ( y , r ), and ∂ σ B = − ∂ τ is carried b y ∂ B ∩ S . Consider the singular c hain µ obtained by coning off ∂ σ B at p . Then ∂ µ = ∂ σ B , so the con tractibility of X implies tha t µ is homologo us to σ B relativ e to ∂ µ . Th us µ + τ b elongs to the homolo g y class of σ . T herefore y lies in the car r ier of µ , for otherwise µ + τ would b e carried by X \ { y } , con tra dicting the fa ct that y ∈ supp( σ ). Th us there is a p oin t z ∈ ∂ B ∩ S suc h that the segmen t pz passes through y . Since B ∩ S is a cone, w e ha ve y z ⊂ S . This implies tha t ˆ γ is not a maximal extension, which is a con tradiction. Another w ay to a rgue the last part of the pro of is to o bserv e that σ B pro jects under log y : X \ { y } → Σ y X to a non trivial 1-cycle η in Σ y X . Therefore, there must b e a direction v ∈ Σ y S making an angle π with lo g y p , since o t herwise η would lie in the op en ball of radius π cen tered at log y p , which is con tra ctible. Then ˆ γ ma y b e extended in the direction v , whic h contradicts the maximalit y of ˆ γ .  Remark 3.2. The geo desic extension pro p ert y has a flav or similar to con v exit y , but note tha t supp ort sets need not b e con ve x. T o obtain an example, let Z b e the union of tw o disjoin t circles Y 1 , Y 2 of length 2 π with a geo desic segmen t of length < π (so Z is a “pair of glasses”), and let X b e t he Euclidean cone ov er Z . Then cone ov er Y 1 ∪ Y 2 is a supp ort set, but is not con v ex. Corollary 3.3 (Monotonicit y a nd lo we r densit y b ound) . Supp ose σ ∈ H lf 2 ( X ) and S := supp( σ ) . 1. (Metric monotonicity) F or al l 0 < r ≤ R , p ∈ X , if Φ : X → X is the ma p which c ontr a cts p oints towar d p by the f a ctor r R , then (3.4) B ( p, r ) ∩ S ⊂ Φ( B ( p, R ) ∩ S ) . 2. (Monotonicity of den sity) F or al l 0 ≤ r ≤ R , (3.5) Area( B ( p, r ) ∩ S ) r 2 ≤ Area( B ( p, R ) ∩ S ) R 2 . 3. (L ower density b ound) F or al l p ∈ S , r > 0 , (3.6) Area( B ( p, r ) ∩ S ) ≥ π r 2 , with e quality only if B ( p, r ) ∩ S is isometric to an r -b al l in R 2 . 6 MLADEN BESTVINA, BRUCE KL EINER, AND MI CHAH SAGEEV Her e Area( Y ) r e f e rs to 2 -dim e nsional Hausdorff me asur e, which is the same as L eb esgue me asur e ( c om p ute d by summing ove r the inter- se c tion s with 2 -dimens i o nal f a c es). Remark 3.7. Since t he map Φ in a ssertion 1 ha s Lipsc hitz constan t r R , the inclusion (3.4 ) can b e view ed as a m uc h stronger v ersion of t he usual monotonicit y fo rm ula for minimal submanifolds in nonp ositive ly curv ed spaces, whic h corresp onds to (3.5). Pr o of of Cor ol lary 3 .3. (3.4) fo llo ws fr o m Lemma 3.1 . Assertion 2 follows from assertion 1 and the fact that Φ has L ipschitz constan t r R . If p ∈ S , then σ determines a nonzero class Σ p σ ∈ H 1 (Σ p X ), by the comp osition H 2 ( X , X \ { p } ) ∂ − → H 1 ( X \ { p } ) log Σ p X − → H 1 (Σ p X ) . Since Σ p X is a CA T(1) g r aph, supp (Σ p σ ) contains a cycle of length at least 2 π . If r > 0 is small, then B ( p, r ) ∩ S is isometric to a cone of radius r ov er supp( Σ p σ ), and therefore ha s area at least π r 2 . No w (3.5) implies (3.6). Equalit y in (3 .6) implies that supp(Σ p σ ) is a circle of length 2 π , B ( p, r 0 ) ∩ S is isometric to a n r 0 -ball in R 2 for small r 0 > 0, and that the contraction map Φ is similarit y . This implies 3.  The coro llary implies that the ratio Area( B ( p, r ) ∩ S ) r 2 has a (p ossibly infinite) limit ¯ A as r → ∞ , whic h is clearly indep enden t of the basepo int. When it is finite w e sa y that σ has q uadr atic gr owth . In this case, Corollary 3.3 implies that (3.8) Area( B ( p, r ) ∩ S ) r 2 ≤ ¯ A for all p ∈ X , r > 0. QUASIFLA TS IN CA T(0) COMPLEXES 7 3.2. Asymptotic conicalit y . W e will use Lemma 3.1 and Corollary 3.3 to see that quadratic growth supp ort sets are asymptotically coni- cal, prov ided the CA T(0) 2- complex X satisfies a mild additional con- dition. T o see wh y an additional assumption is needed, consider a piecewise Euclidean CA T(0) 2-complex X homeomorphic to R 2 , whose singular set consists of a seque nce of cone p oints { p i } tending to infin- it y , where Σ p i X is a circle of length 2 π + θ i , and P i θ i < ∞ . Then X is the supp ort set o f the lo cally finite fundamen tal class [ X ] of t he 2-manifold X , but it is not lo cally flat outside any compact subset of X . T o exclude this kind o f b ehavior, one w ould lik e to kno w, fo r instance, that the cone angle 2 π is isolated among the set of cone angles of p oin ts in X . When dealing with g eneral CA T(0) 2-complexes, o ne needs to kno w that if p ∈ X and v ∈ Σ p X is a direction whose a n tip o dal set An t( v , supp( τ )) in a 1-cycle τ ∈ Z 1 (Σ p X ) has small diameter, then v is close to a suspension p oin t of τ . This condition will hold aut o matically if X admits a co compact group of isometries. The precise condition we need is: Definition 3.9. A family F o f CA T(1) graphs has isolate d susp ens i ons if for ev ery α > 0 there is a β > 0 suc h that if Γ ∈ F , τ ∈ Z 1 (Γ) is a 1-cycle, v ∈ Γ, and diam(An t( v , supp( τ )) < β , then supp( τ ) is a metric susp ension and v lies at distance < α f r om a p ole (i.e. suspension p oint) of supp( τ )). A CA T (0) 2-complex X has isolate d susp ens i ons if the collection of spaces of directions { Σ x X } x ∈ X has isolated susp ensions. Remark 3.10. It fo llows from a compactness argumen t that any finite collection of CA T(1) gr aphs has the isolated susp ensions prop erty . In particular, an y CA T(0) 2-complex with a co compact isometry g roup has the isolated susp ension prop erty . F or the r emainder of this section X will b e a piecewise Euclidean, prop er CA T (0) 2-complex with isolated susp ensions. Theorem 3.11. Supp ose σ ∈ H lf 2 ( X ) has quadr a tic ar e a gr owth, and S := supp( σ ) . Then for al l p ∈ X ther e is an r 0 < ∞ such that 1. If x ∈ S \ B ( p, r 0 ) , then S is lo c al ly isometric to a pr o d uct of the form R × W ne ar x , whe r e W is an i -p o d (i. e . a c on e over a fi nite set). In p articular S is lo c al ly c onvex ne ar x . 8 MLADEN BESTVINA, BRUCE KL EINER, AND MI CHAH SAGEEV 2. The map S \ B ( p, r 0 ) → [ r 0 , ∞ ) given by the distanc e function fr om p is a fibr ation with fib er home omorphic to a finite g r aph with al l vertic es of valen c e ≥ 2 . 3. S is asymptotic al ly c onic al , in the fol lowing sense. F or every p ∈ X and every ǫ > 0 , ther e is an r < ∞ such that if x ∈ S \ B ( p , r ) , then the angle (at x ) b etwe en the ge o desic se gment xp and the R -factor of some lo c al pr o duct s p litting of S is < ǫ . 4. If the ar e a gr o wth of S is Euclide an, i.e. Area( B ( p, r ) ∩ S ) π r 2 → 1 as r → ∞ , then S is a 2 -flat. Before en tering in to the pro of of this theorem, w e p oint out t ha t the pro of is driv en b y t he following observ atio n. The lo cally finite cycle σ is an area minimizing ob ject in the strong est p oss ible sense: any compact piece τ solv es the Plateau problem with b oundary condition ∂ τ (i.e. filling ∂ τ with a least area c hain); in fact, b ecause of the dimension assumption, there is only one w ay to fill ∂ τ with a c hain. Then w e adapt the standard monotonicit y form ula from minimal surface theory to see that the supp ort set is asymptotically conical. Roug hly sp eaking the idea is that the ra tio Area( B ( p, r ) ∩ supp( σ )) r 2 is nondecreasing and b ounded ab ov e, and hence ha s limit as r → ∞ . F or large r , one concludes that the monotonicit y inequalit y is nearly an equalit y , which leads to 2 of Theorem 3.11. Pr o of of The or em 3.11. W e b egin with a packing estimate. Lemma 3.12. F or al l ǫ > 0 ther e is an N such that for al l r ≥ 0 , the interse ction B ( p, r ) ∩ S do es not c ontain an ǫr - s ep ar ate d subset of c ar dinality gr e ater than N . Pr o of. T ake ǫ < 1, and supp ose the p o in ts x 1 , . . . , x k ∈ B ( p, r ) ∩ S are ǫr -separated. Then the collection n B  x i , ǫr 2  ∩ S o 1 ≤ i ≤ k is disjoint, is contained in B ( p, 2 r ) ∩ S , and b y asse rtio n 2 o f Corollar y 3.3 it ha s area at least k π ( ǫr 2 ) 2 . Th us (3.8) implies the lemma.  QUASIFLA TS IN CA T(0) COMPLEXES 9 Lemma 3.13. F or al l β > 0 ther e is an r < ∞ such that if x ∈ S \ B ( p, r ) , then (3.14) diam(An t(lo g x p , Σ x S )) < β . Pr o of. The idea is that quadratic area growth b ounds the complexit y of the suppo rt set from a b o ve , whic h implies that on sufficien tly large scales, it lo oks v ery m uch like a metric cone. On the other hand, failure of (3.14 ) implies tha t there is a pair of ra ys leav ing p whic h coincide un til x , and then branc h apart with an angle at least β ; when x is far enough from p , this will con tra dict the appro ximately conical structure of S at larg e scales. Pic k δ , µ > 0, to b e determined later. By Lemma 3.12 there is finite upp er b ound on the cardinality of an δ r -sep ar a ted subset sitting in B ( p, r ) ∩ S , where r ranges ov er [1 , ∞ ). Let N b e the maximal suc h cardinalit y , which will b e attained b y some δ r 0 -separated subset { x 1 , . . . , x N } ⊂ B ( p, r 0 ) ∩ S , fo r some r 0 . Applying Lemma 3.1, let γ 1 , . . . , γ N : [0 , ∞ ) → X b e constan t sp eed geo desics emanating from p , suc h that γ i ( r 0 ) = x i , and γ i ( t ) ∈ S for all t ∈ [ r 0 , ∞ ), 1 ≤ i ≤ N . The functions (3.15) t 7→ d ( γ i ( t ) , γ j ( t )) t are nondecreasing, and hence for all r ∈ [ r 0 , ∞ ) the collection γ 1 ( r ) , . . . , γ N ( r ) is δ r -separated, and by ma ximality , it is therefore a δ r -net in B ( p, r ) ∩ S as w ell. Using the monotonicit y (3 .15) again, w e ma y find r 1 ∈ [ r 0 , ∞ ) suc h that for all 1 ≤ i, j ≤ N , and ev ery r ∈ [ r 1 , ∞ ), (3.16) d ( γ i ( r ) , γ j ( r )) r + µ > lim t →∞ d ( γ i ( t ) , γ j ( t )) t . No w supp ose x ∈ S \ B ( p, r 1 ), and v 1 , v 2 ∈ Ant(log x p, Σ x S ) sat- isfy ∠ x ( v 1 , v 2 ) ≥ β . The idea of the rest of the pro of is to inv oke Lemma 3.1 to pro duce tw o ra ys emanating from p whic h agree un til they r each x , but then “diverge a t angle at least β ”; since b oth rays will b e w ell-approx imated by one o f the γ i ’s, their separation b e havior will contradict (3.16). Let r 2 := d ( p, x ). By Lemma 3.1 w e may prolong the segmen t px into t w o ra ys pξ 1 , pξ 2 , suc h that lo g Σ x ξ i = v i , and pξ i \ B ( p, r 2 ) ⊂ S . Let 10 MLADEN BESTVINA, BRUCE KL EINER, AND MI CHAH SAGEEV η 1 , η 2 b e the unit sp eed parametrizations of pξ 1 and pξ 2 resp ectiv ely . Applying triangle comparison, w e ma y c ho ose an r 3 ≥ r 2 suc h that (3.17) d ( η 1 ( r 3 ) , η 2 ( r 3 )) > r 3 cos β 2 . Pic k i, j suc h that d ( γ i ( r 3 ) , η 1 ( r 3 )) < δ r 3 and d ( γ j ( r 3 ) , η 2 ( r 3 )) < δ r 3 . By triangle comparison, w e hav e d ( γ i ( r 3 ) , γ j ( r 3 )) ≥ d ( η 1 ( r 3 ) , η 2 ( r 3 )) − 2 δ r 3 > r 3 cos β 2 − 2 δ r 3 while d ( γ i ( r 2 ) , γ j ( r 2 )) ≤ d ( γ i ( r 2 ) , η 1 ( r 2 )) + d ( η 1 ( r 2 ) , η 2 ( r 2 )) + d ( η 2 ( r 2 ) , γ j ( r 2 )) ≤ 2 δ r 2 , since d ( η 1 ( r 2 ) , η 2 ( r 2 )) = 0. On the o t her hand, by (3.16) µ > d ( γ i ( r 3 ) , γ j ( r 3 )) r 3 − d ( γ i ( r 2 ) , γ j ( r 2 )) r 2 ≥ cos β 2 − 4 δ. When µ + 4 δ < cos β 2 this give s a con tradiction.  The Lemma together with the definition of isolated suspensions im- plies parts 1 and 3 of Theorem 3.1 1. P art 4 follows from Lemma 3.3. T o pro v e 2 of Theorem 3.11, w e apply the definition of isolated sus- p ensions with α 0 = π 4 and let β 0 > 0 b e the corresp onding constant; then w e apply Lemma 3.13 with β = β 0 , and let r 0 b e the resulting radius. F or eac h x ∈ X \ B ( p, r 0 ), the space of directions Σ x S is a metric suspension, and the direction log x p ∈ Σ x X mak es an angle at most π 4 from a p ole of Σ x S . W e call a p oint x ∈ S \ B ( p, r 0 ) si n gular if its tangent cone is not isometric to R 2 ; th us singular po in ts in S \ B ( p, r 0 ) hav e tangen t cones of the form R × W , where W is a n i -p od with i > 2, and the set of regular p oin ts forms an op en subset whic h carries the structure of a flat Riemannian manifold. Using a partition of unit y , we ma y construct a smo oth v ector field ξ on the regular part of S \ B ( p, r 0 ) suc h that • ξ ( x ) mak es an angle at least 3 π 4 with log x p at ev ery regular p o int x . QUASIFLA TS IN CA T(0) COMPLEXES 11 • F or each singular p oin t x ∈ S \ B ( p, r 0 ) whose space of directions is the metric susp ension of an i - p od, if we decomp ose a small neigh- b orho o d B ( x, ρ ) ∩ S into a union C 1 ∪ . . . ∪ C i , where the C j ’s are Euclidean half-disks of radius ρ which in tersect along a segmen t η of length 2 ρ , then the restriction of ξ to C j extends to a smo oth v ector field ξ j on the manifold with b oundary C j , and ξ j ( y ) is a unit v ector tangen t to η = ∂ C j for ev ery y ∈ η . No w a standard Morse theory argument using a reparametrization of the flo w of ξ implies that d p : S \ B ( p, r 0 ) → [ r 0 , ∞ ) is a fibration, and that the fib er is lo cally ho meomorphic to an i -p o d near eac h p o int x ∈ S \ B ( p, r 0 ) whose space of directions is the metric suspension of an i - p o d. Here i ≥ 2 .  3.3. Asymptotic branc h p oin ts. The next result will b e used when w e consider supp ort sets asso ciated with quasiflats. Lemma 3.18. L et σ ∈ H lf 2 ( X ) b e a quadr atic gr owth class with supp ort S , pick p ∈ X , and let d p : S \ B ( p, r 0 ) → [ r 0 , ∞ ) b e the fibr ation as in 2 of The or em 3.1 1. If the fib er ha s a b r anch p oi n t, then for al l R < ∞ , the supp ort set S c o n tains an is ometric a l ly emb e dde d c o py of an R -b al l (3.19) B R := B ( z , R ) ⊂ R × W , wher e W is an infinite trip o d, and z ∈ R × W li e s on the singular line. Pr o of. Let π : Y → S \ B ( p, r 0 ) b e the univ ersal co v ering map. Since S \ B ( p, r 0 ) is homeomorphic to G × [0 , ∞ ), the cov ering map π is equiv alen t to the pro duct of t he univ ersal cov ering ˜ G → G with the iden tit y map [0 , ∞ ) → [0 , ∞ ). Since G con tains a branc h p oin t, w e may find a prop er em b edding φ : V → ˜ G of a trip o d V in to ˜ G . Consider the map ψ giv en by the comp osition V × [0 , ∞ ) − → ˜ G × [0 , ∞ ) − → G × [0 , ∞ ) ≃ S \ B ( p, r 0 ) . W e ma y put a lo cally CA T (0) metric on V × (0 , ∞ ) b y pulling back the metric fro m S \ B ( p, r 0 ). F or eac h of the three “ra ys” γ i ⊂ V whose union is V , t he metric on γ i × (0 , ∞ ) is lo cally isometric to a flat metric with g eo desic b oundary . It follow s fro m a standard argument 12 MLADEN BESTVINA, BRUCE KL EINER, AND MI CHAH SAGEEV that if y ∈ V × (0 , ∞ ) lies on the singular lo cus and ψ ( y ) lies o utside B ( p, r 0 + R ), then the R - ball in V × (0 , ∞ ) is isometric to B R as in (3.19). Since ψ is a lo cally isometric map of a CA T( 0 ) space in to a CA T(0) space, it is an isometric em b edd ing.  4. Quasi-fla ts in 2 -complexe s In this section, X will denote a piecewise flat, prop er CA T(0) 2- complex with isolated susp ensions. Theorem 4.1. L et Q ⊂ X b e an ( L, A ) -quasiflat. Then ther e is a nontrivial quadr atic gr owth, lo c al ly fin i te homolo gy class σ ∈ H lf 2 ( X ) whose supp ort set S ⊂ X is at Hausdorff distanc e at m o st D = D ( L, A ) fr om Q , with the fol lowing pr op erty. 1. F o r every p ∈ X , ther e is an r 0 ∈ [0 , ∞ ) such that S \ B ( p, r 0 ) is lo c al ly isometric to R 2 . 2. S is asymptotic al ly c onic al, i n the fol lowing sense. F or every p ∈ X and every ǫ > 0 , ther e is an r 1 ∈ [ r 0 , ∞ ) such that if x ∈ S \ B ( p, r 1 ) , then the angle at x b etwe en the ge o desic se gment xp and S i s < ǫ , and the map S \ B ( p, r 1 ) → [ r 0 , ∞ ) given by the distanc e function fr o m p is a fi b r ation w ith cir cle fib er. 3. If the ar e a gr o wth of S is Euclide an, i.e. Area( B ( p, r ) ∩ S ) π r 2 → 1 as r → ∞ , then S is a 2 -flat. Pr o of. Using a standard argumen t, we may assume without lo ss of generalit y (and at the cost of some deterioration in quasi-isometry con- stan ts whic h will b e suppressed), tha t Q is the image of a C -Lipschitz ( L, A )-quasi-isometric embedding f : R 2 → X , where C = C ( L, A ). The mapping f is prop er, and hence induces a homomorphism f ∗ : H lf 2 ( R 2 ) → H lf 2 ( X ) o f lo cally finite ho mology groups. W e define S to b e the supp o rt set of the image of the fundamen tal class of R 2 under f ∗ : (4.2) S := supp ( f ∗ ([ R 2 ])) ⊂ Im( f ) = Q. Lemma 4.3. Ther e ar e c onstants D = D ( L, A ) and a = a ( L, A ) such that such that: 1. The Hausdorff d istanc e b etwe en S and Q is at most D . 2. F o r every p ∈ X , the ar e a of B ( p, r ) ∩ S is at most a (1 + r ) 2 . QUASIFLA TS IN CA T(0) COMPLEXES 13 Pr o of. Using the uniform contractibilit y of R 2 , o ne ma y construct a prop er ma p g : Q → R 2 suc h that d ( g ◦ f , id R 2 ) is b o unded by a function of ( L, A ). In par t icular, the comp osition of prop er maps R 2 f − → Q g − → R 2 is prop erly homotopic to id R 2 . Hence ( g ◦ f ) ∗ ([ R 2 ]) = [ R 2 ], so supp(( g ◦ f ) ∗ ([ R 2 ])) = R 2 . On the other hand supp(( g ◦ f ) ∗ ([ R 2 ])) ⊂ g ( S ) , whic h implies tha t Q = Im( f ) is contained in a controlled neighborho o d of S . The last assertion follo ws from the fa ct t hat S ⊂ Q and Q has quadratic area growth, b eing t he image of a Lipschitz quasi-isometric em b edding.  Therefore Theorem 3.11 applies to S , a nd b y part 2, w e get a fibration d p : S \ B ( p, r 0 ) → [ r 0 , ∞ ) whose fib er is homeomorphic to a finite graph G all of whose v ertices ha v e v alence ≥ 2. If G had a bra nc h p o int, w e could apply Lemma 3.18, con tra dicting the fact that S is a quasi-flat. Th us S is lo cally isometric to R 2 outside B ( p, r 0 ).  5. Square complexes In this section X will b e a lo cally finite CA T(0) square complex with isolated susp ensions. Remark 5.1. It is not difficult to sho w that if F is the collection of CA T(1) graphs Γ all o f whose edges hav e length π 2 , then F has isolated suspensions. In particular, an y CA T(0) square complex has isolated suspensions. Ho we ve r, w e will not need this fact fo r our primary a p- plications, so w e omit the pro of. Theorem 5.2. L et σ ∈ H lf 2 ( X ) b e a quadr atic gr owth lo c al ly finite homolo gy class whose supp ort set S is a quasiflat. Then ther e is a finite c ol le ction { H 1 , . . . , H k } of ha lf-plane sub c o mplexes c ontaine d in S , and a finite sub c omplex W ⊂ S such that S = W ∪ ( ∪ i H i ) . 14 MLADEN BESTVINA, BRUCE KL EINER, AND MI CHAH SAGEEV Pr o of. Pic k p ∈ X and ǫ ∈ (0 , π 2 ). Let r 1 b e as in Theorem 4.1, and set Y 1 := S \ B ( p, r 1 ) . Then Y 1 is a complete flat Riemannian surface with conca ve b oundary ∂ Y 1 = S ( p, r 0 ) ∩ Y 1 . No w pic k α ∈ (0 , π 8 ), r 2 ∈ [ r 1 , ∞ ), and let Y 2 := S \ B ( p, r 2 ). Lemma 5.3. Pr ovide d r 2 is sufficiently lar ge (dep ending o n α ) , for every x ∈ Y 2 , and every semi-cir cle τ ⊂ Σ x S such that d ( τ , log x p ) > α, ther e is a subset Z ⊂ S isometric to a Euclide an half-pl a ne, such that Σ x Z = τ . Pr o of. First supp ose y ∈ Y 2 , and v ∈ Σ y S is a tangent ve ctor suc h that ∠ y ( v , log y p ) > α . Provided r 2 sin α > r 1 , there will b e a unique geo desic ra y γ v ⊂ S starting at y with direction v ; t his f o llo ws from a con tin uity argumen t, since tria ngle comparison implies that an y geo- desic segmen t with initial direction v remains outside B ( p, r 1 ). If τ ⊂ Σ x S is a semi-circle (i.e. a geo desic se gment of length π ), a nd ∠ x ( τ , log x p ) > α , then the union of the ra ys γ v , for v ∈ τ , will f o rm a subset of S isometric to a Euclidean half-pla ne.  Pr o of o f The or em 5.2 c ontinue d . W e no w assume that r 2 is larg e enough that Lemma 5.3 applies. Our next step is to construct a finite collection of half-planes in S . Consider the b oundary ∂ Y 2 . This is the fron tier o f the set K := S ∩ B ( p, r 2 ) in S . Since K is lo cally con v ex near ∂ K = ∂ Y 2 , it follows that for eac h x ∈ ∂ Y 2 , there is a w ell-defined space of directions Σ x K , whic h consists of the directions v ∈ Σ x S suc h that ∠ x ( v , log x p ) ≤ π 2 . Also, there is a normal space ν x K ⊂ Σ x S consisting of the directions v ∈ Σ x S making an angle a t least π 2 with Σ x K . When ǫ is small, the angle ∠ x (log x p, Σ x S ) is small, and hence π − ∠ x ( v , log x p ) will b e small for eve ry v ∈ ν x K . In particular, when ǫ is small, fo r eve ry v ∈ ν v K there will b e a semi-circle τ v ⊂ Σ x S suc h tha t 1. τ v mak es an a ng le at least π 8 with log x p . 2. If Z v ⊂ S is the subset obtained b y applying Lemma 5.3 to τ v , then the b oundary of Z v is parallel to one of the sides of a square P ⊂ S whic h con tains x . QUASIFLA TS IN CA T(0) COMPLEXES 15 3. The angle b etw een ∂ Z v and v is at least π 8 . W e let H v ⊂ Z v b e the largest half-plane sub complex of Z v . It follow s from prop ert y 2 that H v ma y b e obtained from Z v b y remo ving a strip of thic kness < 1 around ∂ Z v . No w let H b e the collection of all half-planes obtained this wa y , where x ranges ov er ∂ Y 2 , and v ∈ ν x K . Observ e t ha t this is a finite collection, since eac h H ∈ H has a bo undary square lying in B ( p, 1 + r 2 ), and t w o half-planes H , H ′ ∈ H sharing a b oundary square must b e the same. W e now claim that S \ ∪ H ∈H H is contained in B ( p, r 2 + sec π 8 ). T o see this note that if y ∈ Y 2 , then there is a shortest path in S from y to K . Since S is lo cally con v ex, this path will b e a geo des ic segmen t y x in X , where x ∈ ∂ Y 2 . Let v := log x y ∈ Σ x S . 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