Explicit equivalences between CAT(0) hyperbolic type geodesics
aa r X i v : . [ m a t h . G T ] D ec EXPLICIT EQUIVALENCES BETWEEN CAT(0) HYPERBOLIC TYPE GEODESICS
HAROLD SULTANA
BSTRACT . We prove an explicit equivalence between various hyperbolic type properties for quasi-geodesics in CAT(0) spaces. Specifically, we prove that for X a CAT(0) space and γ ⊂ X a quasi-geodesic, the following four statements are equivalent and moreover the quantifiers in the equiva-lences are explicit: (i) γ is S-Slim, (ii) γ is M(K,L)–Morse, (iii) γ is (b,c)–contracting, and (iv) γ is C–strongly contracting. In particular, this explicit equivalence proves that for f a ( K, L ) quasi-isometry between CAT(0) spaces, and γ a C–strongly contracting (K’,L’)–quasi-geodesic, then f ( γ ) isa C ′ ( C, K, L, K ′ , L ′ ) –strongly contracting quasi-geodesic. This result is necessary for a key technicalpoint with regard to Charney’s contracting boundary for CAT(0) spaces. C ONTENTS
1. Introduction and Overview 12. Background 23. Main Theorem and Proof 4References 121. I
NTRODUCTION AND O VERVIEW
In the study of spaces of non-positive curvature, Euclidean and hyperbolic space represent the twoclassically well understood extreme ends of the spectrum. More generally, in the literature a robustapproach for studying spaces of interest is to identify particular directions, geodesics, or subspaces ofthe space in question which share features in common with one of these two prototypes. In particular,with regard to identifying hyperbolic type geodesics in spaces of interest, or geodesics which sharefeatures in common with geodesics in hyperbolic space, there are various well studied precise notionsincluding being Morse, being contracting, and being slim. Specifically, such studies have provenfruitful in analyzing right angled Artin groups [BC], Teichm¨uller space [B, BrF, BrM, BMM, Mos],the mapping class group [B], CAT(0) spaces [Sul, BD, BeF, Cha], and Out( F n ) [A] amongst others(See for instance[DMS, DS, KL, Osi, MM]).A Morse geodesic γ is defined by the property that all quasi-geodesics σ with endpoints on γ remain within a bounded distance from γ. A strongly contracting geodesic has the property thatmetric balls disjoint from the geodesic have nearest point projections onto the geodesic with uniformlybounded diameter. A geodesic is called slim if geodesic triangles with one edge along the geodesicare δ -thin. It is an elementary fact that in hyperbolic space, or more generally δ -hyperbolic spaces,all quasi-geodesics are Morse, strongly contracting, and slim. On the other hand, in product spacessuch as Euclidean spaces of dimension two and above, there are no Morse, strongly contracting, orslim quasi-geodesics. Date : November 8, 2018.2010
Mathematics Subject Classification.
Key words and phrases. hyperbolic type geodesics, contracting geodesics, morse geodesics, slim geodesics.
Building on results in [Sul], in this paper we prove that the various aforementioned hyperbolic typeproperties are equivalent and moreover the quantifiers in the equivalences are explicit.
Theorem 3.5. (Main Theorem).
Let X be a CAT(0) space and γ ⊂ X a quasi-geodesic. Then thefollowing are equivalent: (1) γ is (b,c)–contracting, (2) γ is C ′ –strongly contracting, (3) γ is M –Morse, and (4) γ is S –slimMoreover, any one of the four sets of constants { ( b, c ) , C’ , M, S } can be written in terms of any of theothers. Theorem 3.5 should be considered in the context of related theorems in [BeF, B, Cha, DMS, KL,Sul] among others. In particular, in [BeF] geodesics with property (2) are studied and in fact amongother things it is shown that for the case of γ a geodesic (2) = ⇒ (4). In [Cha] geodesics with property(2) are studied and it is shown that (2) = ⇒ (3), an explicit proof of which also appears in [A]. In[DMS] geodesics with property (3) are studied. In [Sul] building on work of the previous authors it isshown that properties (1),(2), and (3) are equivalent, although the proof relies on limiting argumentsand hence the constants of the equivalence could not be recovered.As a corollary of Theorem 3.5 we highlight the following consequence, which in fact served asmotivation for the results in this paper. Corollary 3.6.
Let X be a CAT(0) space, γ ⊂ X a C–strongly contracting (K’,L’)–quasi-geodesic,and f : X → X a ( K, L ) quasi-isometry. Then f ( γ ) is C ′ ( C, K, L, K ′ , L ′ ) –strongly contractingquasi-geodesic. In particular, Corollary 3.6 is very useful in [Cha] where it is used to show that self quasi-isometriesof CAT(0) spaces give rise to continuous maps on Charney’s contracting boundary for CAT(0) spaces.
Acknowledgements.
I want to thank Ruth Charney for motivating conversations as well as Michael Carr and ThomasKoberda for useful conversations and insights regarding arguments and ideas in this paper.2. B
ACKGROUND
Quasi-geodesics and CAT(0) spaces.Definition 2.1 (quasi-geodesic) . A (K,L) quasi-geodesic γ ⊂ X is the image of a map γ : I → X where I is a connected interval in R (possibly all of R ) such that ∀ s, t ∈ I we have the followingquasi-isometric inequality:(2.1) | s − t | K − L ≤ d X ( γ ( s ) , γ ( t )) ≤ K | s − t | + L We refer to the quasi-geodesic γ ( I ) by γ, and when the constants ( K, L ) are not relevant omit them.CAT(0) spaces are geodesic metric spaces defined by the property that triangles are no “fatter” thanthe corresponding comparison triangles in Euclidean space. In particular, using this property one canprove the following lemma, see [BH, Section II.2] for details. Lemma 2.2.
Let X be a CAT(0) space. C1: (Projections onto convex subsets). Let C be a convex subset, complete in the induced metric,then there is a well-defined distance non-increasing nearest point projection map π C : X → C. In particular, π C is continuous. We will often consider the case where C is a geodesic. XPLICIT EQUIVALENCES BETWEEN CAT(0) HYPERBOLIC TYPE GEODESICS 3
C2: (Convexity). Let c : [0 , → X and c : [0 , → X be any pair of geodesics parameterizedproportional to arc length. Then the following inequality holds for all t ∈ [0 ,
1] : d ( c ( t ) , c ( t )) ≤ (1 − t ) d ( c (0) , c (0)) + td ( c (1) , c (1)) Hyperbolic type quasi-geodesics.
In this section we define the hyperbolic types of quasi-geodesics we will consider in this paper. The following definition of Morse (quasi-)geodesics hasroots in the classical paper [Mor]:
Definition 2.3 (Morse quasi-geodesics) . A (quasi-)geodesic γ is called an M– Morse (quasi-)geodesic if for every ( K, L ) -quasi-geodesic σ with endpoints on γ, we have σ ⊂ N M ( K,L ) ( γ ) . That is, σ iswithin a bounded distance, M = M ( K, L ) , from γ, with the bound depending only on the constants K, L.
In the literature, Morse (quasi-)geodesics are sometimes referred to as stable quasi-geodesics .The following generalized notion of contracting quasi-geodesics can be found for example in [B,BrM], and is based on a slightly more general notion of (a,b,c)–contraction found in [MM] where itserves as a key ingredient in the proof of the hyperbolicity of the curve complex.
Definition 2.4 (contracting quasi-geodesics) . A (quasi-)geodesic γ is said to be (b,c)–contracting if ∃ constants < b ≤ and < c such that ∀ x, y ∈ X,d X ( x, y ) < bd X ( x, π γ ( x )) = ⇒ d X ( π γ ( x ) , π γ ( y )) < c. For the special case of a (b,c)–contracting quasi-geodesic where b can be chosen to be , the quasi-geodesic γ is called c–strongly contracting. The following elementary lemma shows that given a ( b, c ) –contracting quasi-geodesic one canincrease b to be arbitrarily close to at the expense of increasing c. Lemma 2.5. If γ is (b,c)–contracting quasi-geodesic, then for any arbitrarily small ǫ > , the quasi-geodesic γ is ( − ǫ, c ′ ( ǫ, b, c )) –contracting.Proof. Notice that if γ is ( b, c ) –contracting, then it is also ( b + b (1 − b ) , c ) –contracting. Similarly, itis also ( b + b (1 − b ) + b (1 − b ) , c ) –contracting. Iterating this process, the statement of the lemmafollows, as for < b < the sum of the geometric series P ∞ i =0 b (1 − b ) i converges to . (cid:3) Finally, the following definition of a slim quasi-geodesic is introduced in [BeF].
Definition 2.6 (slim quasi-geodesics) . A (quasi-)geodesic γ is said to be S–slim if ∃ constant S suchthat for all x ∈ X and y ∈ γ, we have: d ( π γ ( x ) , [ x, y ]) ≤ S. Note that if γ is an S–slim quasi-geodesic, then | [ x, π γ ( x )] | + | [ π γ ( x ) , y ] | − S ≤ | [ x, y ] | ≤ | [ x, π γ ( x )] | + | [ π γ ( x ) , y ] | . Moreover, if z ∈ [ x, y ] is a point such that d ( y, z ) = d ( y, π γ ( x )) (or similarly such that d ( x, z ) = d ( x, π γ ( x )) ), then d ( z, π γ ( x )) ≤ S. We conclude this section by citing a lemma relating contracting and slim geodesics.
Lemma 2.7 ([BeF] Lemma 3.5) . Let γ be a C–strongly contracting geodesic in a CAT(0) space. Then γ is (3 C + 1) -slim. HAROLD SULTAN
3. M
AIN T HEOREM AND P ROOF
Throughout this section we will assume we are in the setting of a CAT(0) metric space X. Thefollowing two elementary lemmas regarding the concatenation of geodesic segments will be useful inthe proof of Theorem 3.5.
Lemma 3.1.
For any triple of points a, b, c ∈ X, the concatenated path φ = [ a, π [ b,c ] ( a )] ∪ [ π [ b,c ] ( a ) , c ] , is a (3,0) quasi-geodesic.Proof. We must show that ∀ x, y ∈ φ, the (3,0)–quasi-isometric inequality of Equation 2.1 is satisfied.Since φ is a concatenation of two geodesic segments, without loss of generality we can assume x ∈ [ a, π [ b,c ] ( a )] , y ∈ [ π [ b,c ] ( a ) , c ] . Since x ∈ [ a, π [ b,c ] ( a )] it follows that π [ b,c ] ( x ) = π [ b,c ] ( a ) , and hence d ( x, π [ b,c ] ( a )) ≤ d ( x, y ) . Let d φ ( x, y ) denote the distance along φ between x and y. Then, the following inequality completes the proof: d ( x, y ) ≤ d φ ( x, y ) = d ( x, π [ b,c ] ( a )) + d ( π [ b,c ] ( a ) , y ) ≤ d ( x, π [ b,c ] ( a )) + (cid:0) d ( π [ b,c ] ( a ) , x ) + d ( x, y ) (cid:1) ≤ d ( x, π [ b,c ] ( a )) + d ( x, y ) ≤ d ( x, y ) (cid:3) Building on Lemma 3.1, presently we will prove a lemma which ensures that the concatenation offive geodesic segments under certain hypothesis is a quasi-geodesic with controlled quasi-constants.Let γ be a geodesic, and x, y ∈ X. Set D = d ( π γ ( x ) , π γ ( y )) . Let a, b, c be constants such d ( x, π γ ( x )) = aD, d ( x, y ) = bD, d ( y, π γ ( y )) = cD. See Figure 1. Note that by property [C1] ofLemma 2.2, b ≥ . Consider the continuous function ρ ( z ) = d ([ x, π γ ( x )] , z ) . If we restrict the func-tion ρ to the geodesic [ x, y ] , by definition ρ ( x ) = 0 , ρ ( y ) ≥ D, where the latter inequality followsfrom property [C1] of Lemma 2.2. Then by intermediate value theorem there is some point s ∈ [ x, y ] such that ρ ( s ) = D . Moreover, we can assume s ∈ [ x, y ] is the point in [ x, y ] closest to y such that ρ ( s ) = D . Similarly, we can define the continuous function ρ ( z ) = d ([ y, π γ ( y )] , z ) . If we restrictthe function ρ to the geodesic [ x, y ] , then as above the intermediate value theorem ensures there issome point t ∈ [ x, y ] such that ρ ( t ) = D , and moreover we can assume t ∈ [ x, y ] is the point in [ x, y ] closest to x such that ρ ( t ) = D . Notice that since d ( s, [ x, π γ ( x )]) = D , d ( t, [ y, π γ ( y )]) = D , and d ( x, y ) ≥ d ( π γ ( x ) , π γ ( y ) = D, it follows that s precedes t along the geodesic [ x, y ] . In fact, iffollows that d ( s, t ) ≥ D . Let r be a point in [ x, π γ ( x )] such that d ( s, r ) = D . Similarly, let u be a point in [ y, π γ ( y )] suchthat d ( t, u ) = D . Note that r, u are uniquely defined as they are nearest point projections, that is π [ x,π γ ( x )] ( s ) = r and similarly π [ y,π γ ( y )] ( t ) = u. Furthermore, by construction we similarly have that π [ s,y ] ( r ) = s and π [ x,t ] ( u ) = t. Lemma 3.2.
In the situation described above, the concatenation φ = [ π γ ( x ) , r ] ∪ [ r, s ] ∪ [ s, t ] ∪ [ t, u ] ∪ [ u, π γ ( y )] , is a ((1 + 4( a + b + c )) , -quasi-geodesic. In particular, if a + c > b and γ is M(K,L)-Morse, then D ≤ M (1 + 4( a + b + c ) , a + c − b . XPLICIT EQUIVALENCES BETWEEN CAT(0) HYPERBOLIC TYPE GEODESICS 5 bD us t<
DaD cDa+c-b2rx y πγ(x) πγ(y)D D F IGURE
1. Illustration of Lemma 3.2.
Proof.
We will show that ∀ w, z ∈ φ, that the ((1 + 4( a + b + c )) , –quasi-isometric inequality ofEquation 2.1 is satisfied. Since φ is a concatenation of geodesics, without loss of generality we canassume a, b belong to different geodesic segments within φ. Since there are 5 different geodesicsegments in φ, there are (cid:0) (cid:1) = 10 , cases to consider. By Lemma 3.1 we know that the (3,0)–quasi-isometric inequality is satisfied in the case where w and z belong to adjacent geodesic seg-ments in the concatenation. Since b ≥ it follows that a + b + c ) > , and in particular,the ((1 + 4( a + b + c )) , -quasi-isometric inequality is satisfied. To complete the proof of the firststatement of the lemma we will consider the six remaining cases and in each case verify the quasi-isometric inequality:(1) w ∈ [ π γ ( x ) , r ] , z ∈ [ s, t ] : By definition, in this case D = d ( r, s ) ≤ d ( w, z ) . Hence, d ( w, z ) ≤ d φ ( w, z ) = d ( w, r ) + | [ r, s ] | + d ( s, z ) ≤ | [ π γ ( x ) , x ] | + | [ r, s ] | + | [ s, t ] |≤ aD + D bD = D a + 4 b + 1) ≤ d ( w, z )(1 + 4 a + 4 b ) (2) w ∈ [ r, s ] , z ∈ [ t, u ] : By definition, in this case D ≤ d ( s, t ) ≤ d ( w, z ) . Hence, d ( w, z ) ≤ d φ ( w, z ) = d ( w, s ) + | [ s, t ] | + d ( t, z ) ≤ | [ r, s ] | + | [ s, t ] | + | [ t, u ] |≤ D bD + D D b ) ≤ d ( w, z )(1 + 2 b ) (3) w ∈ [ s, t ] , z ∈ [ u, π γ ( y )] : By definition, in this case D = d ( t, u ) ≤ d ( w, z ) . Hence, d ( w, z ) ≤ d φ ( w, z ) = d ( w, t ) + | [ t, u ] | + d ( u, z ) ≤ | [ s, t ] | + | [ t, u ] | + | [ π γ ( y ) , y ] |≤ bd + D cD = D b + 1 + 4 c ) ≤ d ( w, z )(1 + 4 b + 4 c ) (4) w ∈ [ π γ ( x ) , r ] , z ∈ [ t, u ] : By property [C1] of Lemma 2.2 in this case, d ( w, z ) ≥ d ( π γ ( w ) , π γ ( z )) = d ( π γ ( x ) , π γ ( z )) ≥ d ( π γ ( x ) , π γ ( y )) − d ( π γ ( y ) , π γ ( z )) = D − d ( π γ ( u ) , π γ ( z )) ≥ D − | [ u, z ] | ≥ D − | [ u, t ] | = 3 D . HAROLD SULTAN
Then, the following inequality proves the desired quasi-isometric inequality in this case: d ( w, z ) ≤ d φ ( w, z ) = d ( w, r ) + | [ r, s ] | + | [ s, t ] | + d ( t, z ) ≤ | [ π γ ( x ) , x ] | + | [ r, s ] | + | [ s, t ] | + | [ t, u ] |≤ aD + D bD + D D a + 4 b ) ≤ d ( w, z ) (2 + 4 a + 4 b )3 (5) w ∈ [ r, s ] , z ∈ [ u, π γ ( y )] : As in the previous case, by property [C1] of Lemma 2.2 in thiscase, we have that d ( w, z ) ≥ D . Hence, d ( w, z ) ≤ d φ ( w, z ) = d ( w, s ) + | [ s, t ] | + | [ t, u ] | + d ( u, z ) ≤ | [ r, s ] | + | [ s, t ] | + | [ t, u ] | + | [ π γ ( y ) , y ] |≤ D bD + D cD = D b + 4 c ) ≤ d ( w, z ) (2 + 4 b + 4 bc )3 (6) w ∈ [ π γ ( x ) , r ] , z ∈ [ u, π γ ( y )] : By property [C1] of Lemma 2.2 in this case d ( w, z ) ≥ D. Hence, d ( w, z ) ≤ d φ ( a, b ) = d ( w, r ) + | [ r, s ] | + | [ s, t ] | + | [ t, u ] | + d ( u, z ) ≤ | [ π γ ( x ) , x ] | + | [ r, s ] | + | [ s, t ] | + | [ t, u ] | + | [ π γ ( y ) , y ] |≤ aD + D bD + D cD = D ( 12 + a + b + c ) ≤ d ( w, z )( 12 + a + b + c ) For the “in particular” clause of the lemma note that if a + c > b then d ([ x, y ] , γ ) ≥ D a + c − b . Since [ s, t ] ⊂ [ x, y ] , in particular d ([ s, t ] , γ ) ≥ D a + c − b . On the other hand, [ s, t ] is a non-trivial portionof the quasi-geodesic φ and hence must stay within a neighborhood of γ controlled by the Morseconstant of γ. Specifically, [ s, t ] ⊂ N M (1+4( a + b + c ) , ( γ ) . Combining the inequalities completes theproof of the lemma. (cid:3)
The following lemma and ensuing corollary will be be useful in the proof of Theorem 3.5. Specif-ically, these results will be used to reduce arguments regarding quasi-geodesics to case of geodesics.The lemma is closely related to and should be compared with Lemma 3.8 of [BeF].
Lemma 3.3.
Let γ be an M–Morse, C–strongly contracting geodesic, and let γ ′ be a (K,L)–quasi-geodesic with endpoints on γ. Then γ ′ is C ′ ( C, M ) -strongly contracting. Similarly, let γ be an M–Morse ( b, c ) –contracting geodesic, and let γ ′ be a (K,L)–quasi-geodesic with endpoints on γ. Then γ ′ is ( b, c ′ ( c, M ) –contracting. We will prove the first statement of the lemma. The proof of the “similarly” statement is identical.Since nearest point projections onto quasi-geodesics is not uniquely determined, in the proof ofthe lemma we will use the convention that π γ ′ ( x ) represents an arbitrary element in the nearest pointprojection set of x onto γ ′ . Additionally, when measuring distances between elements in nearestpoint projection sets, such as d ( π γ ′ ( x ) , π γ ′ ( y )) , we will use the convention that the distance is thesupremum over all possible choices of elements in the nearest point projection sets. That is, d ( π γ ′ ( x ) , π γ ′ ( y )) =: sup { d ( x ′ , y ′ ) | x ′ ∈ π γ ′ ( x ) , y ′ ∈ π γ ′ ( y ) } . XPLICIT EQUIVALENCES BETWEEN CAT(0) HYPERBOLIC TYPE GEODESICS 7
Proof.
First we will prove that ∀ z ∈ X, d ( π γ ′ ( z ) , π γ ′ ( π γ ( z ))) is bounded above in terms of theconstants C, M.
Set the Morse constant M ( K, L ) = M. Then,(3.1) d ( z, π γ ( π γ ′ ( z ))) ≤ d ( z, π γ ′ ( z )) + d ( π γ ′ ( z ) , π γ ( π γ ′ ( z ))) ≤ d ( z, π γ ′ ( z )) + M ≤ d ( z, π γ ( z )) + d ( π γ ( z ) , γ ′ ) + M ≤ d ( z, π γ ( z )) + 2 M. By Lemma 2.7, the geodesic γ is (3C+1)-slim. Consider the triangle △ ( z, π γ ( z ) , π γ ( π γ ′ ( z ))) . (3C+1)–slimness in conjunction with Equation 3.1 implies that d ( π γ ( z ) , π γ ( π γ ′ ( z ))) ≤ M + 2(3 C + 1) . (3.2)Since γ ′ ⊂ N M ( γ ) , by the triangle inequality d ( π γ ( z ) , π γ ( π γ ′ ( π γ ( z )))) ≤ M. (3.3)Combining Equations 3.2 and 3.3, by the triangle inequality we have d ( π γ ( π γ ′ ( π γ ( z ))) , π γ ( π γ ′ ( z ))) ≤ M + 2(3 C + 1) . (3.4)Finally, using the Equation 3.4 in conjunction with the fact that γ ′ ⊂ N M ( γ ) and the triangle inequal-ity, it follows that ∀ z ∈ X,d ( π γ ′ ( π γ ( z )) , π γ ′ ( z )) ≤ M + 2(3 C + 1) . (3.5)Now assume we have x, y ∈ X such that d ( x, y ) < d ( x, π γ ′ ( x )) . We must show that we canbound d ( π γ ′ ( x ) , π γ ′ ( y )) from above in terms of the constants C, M.
Since d ( x, y ) < d ( x, π γ ′ ( x )) ≤ d ( x, π γ ( x )) + M, using the facts that γ is C –strongly contracting and nearest point projections ontogeodesics are distance non-increasing, we have that d ( π γ ( x ) , π γ ( y )) ≤ C + M. (3.6)As above, using the fact that γ ′ ⊂ N M ( γ ) , in conjunction with Equation 3.6 and the triangle inequal-ity, it follows that d ( π γ ′ ( π γ ( x )) , π γ ′ ( π γ ( y ))) ≤ C + 3 M. (3.7)Putting together Equations 3.5 and 3.7, the following completes the proof: d ( π γ ′ ( x ) , π γ ′ ( y )) ≤ d ( π γ ′ ( x ) , π γ ′ ( π γ ( x ))) + d ( π γ ′ ( π γ ( x )) , π γ ′ ( π γ ( y ))) + d ( π γ ′ ( π γ ( y )) , π γ ′ ( y )) ≤ (6 M + 2(3 C + 1)) + ( C + 3 M ) + (6 M + 2(3 C + 1)) = 15 M + 7 C + 4 (cid:3) As a corollary of Lemma 3.3 we have the following:
Corollary 3.4.
If it’s true that a geodesic being M-Morse implies that the geodesic is C ( M ) –stronglycontracting, then it’s also true that a quasi-geodesic being M ′ -Morse implies that the quasi-geodesicis C ′ ( M ′ ) –strongly contracting. Similarly, if it’s true that a geodesic being M-Morse implies that thegeodesic is ( b, c ) –contracting, then it’s also true that a quasi-geodesic being M ′ -Morse implies thatthe quasi-geodesic is ( b, c ′ ( M ′ )) –contracting.Proof. Once again we will prove the first statement, and the “similarly” statement follows identically.Assume that if a geodesic is M–Morse then it is also C ( M ) –strongly contracting. Let γ ′ be an M ′ –Morse quasi-geodesic. Fix x ∈ X and x ′ ∈ π γ ′ ( x ) . Let y ∈ X be such that d ( x, y ) < d ( x, x ′ ) , andfix y ′ ∈ π γ ′ ( y ) . Notice that d ( x ′ , y ′ ) ≤ d ( x ′ , x ) + d ( x, y ) + d ( y, y ′ ) ≤ d ( x ′ , x ) + d ( x ′ , x ) + 2 d ( x ′ , x ) = 4 d ( x ′ , x ) . HAROLD SULTAN
Let α x ∈ γ ′ be any point preceding x ′ such that d ( α x , x ′ ) ≥ d ( x ′ , x ) . Similarly, let β x ∈ γ ′ beany point following x ′ such that d ( x ′ , β x ) ≥ d ( x ′ , x ) [if these choices are not possible because γ ′ terminates, then set α x ( β x ) to be equal to the terminal point of γ ′ which precedes (follows) x ′ ].Since γ ′ is an M ′ –Morse quasi-geodesic and because [ α x , β x ] is a geodesic with endpoints on γ ′ , it follows that [ α x , β x ] is similarly M ′′ ( K, L ) –Morse, where the constant M ′′ ( K, L ) = M ′ (0 ,
1) + M ′ ( K, L ) . In particular, the constant M ′′ only depends on M ′ . Then, by assumption the geodesic [ α x , β x ] is C ( M ′′ ) = C ( M ′ ) –strongly contracting. By Lemma 3.3 it follows that γ ′ | [ α x ,β x ] is C ′ ( C ( M ′ ) , M ′ ) = C ′ ( M ′ ) –strongly contracting. In particular, since for all y ∈ X such that d ( x, y ) ≤ d ( x, x ′ ) , we know that π γ ′ ( y ) ⊂ γ ′ | [ α x ,β x ] , it follows that d ( π γ ′ ( x ) , π γ ′ ( y )) ≤ C ′ ( M ′ ) . Since for any starting x ∈ X we can preform this process of creating such an interval [ α x , β x ] andproceeding as above, it follows that the quasi-geodesic γ ′ is C ′ ( M ′ ) –strongly contracting. (cid:3) We are now prepared to prove the main theorem.
Theorem 3.5.
Let X be a CAT(0) space and γ ⊂ X a (K,L)–quasi-geodesic. Then the following areequivalent: (1) γ is C ′ –strongly contracting, (2) γ is (b,c)–contracting, (3) γ is M –Morse, and (4) γ is S –slim.Moreover, any one of the four sets of constants { ( b, c ) , C ′ , M, S } can be written purely in terms ofthe any of the others in conjunction with the quasi-isometry constant (K,L).Proof. By definition (1) = ⇒ (2). The fact that (2) = ⇒ (3) is a slight generalization of the well known“Morse stability lemma.” For an explicit proof see Lemma 3.3 in [Sul] (or similarly Lemma 5.13 in[A]). In order to complete the proof of the theorem we will provide an explicit proof that: (3) = ⇒ (2),(3) [+(2)] = ⇒ (1), (1)+(3) = ⇒ (4), and (4) = ⇒ (2). (3) = ⇒ (2) : By Corollary 3.4 it suffice to prove (3) = ⇒ (2) in the special case of γ a geodesic.Fix x, y ∈ X such that d ( x, y ) ≤ d ( x, π γ ( x )) . Set A = d ( x, π γ ( x )) , D = d ( π γ ( x ) , π γ ( y )) . Notethat A ≤ d ( y, π γ ( y )) ≤ A . Let ρ : [0 , → X be the geodesic parameterized proportional to arc length joining π γ ( x ) = ρ (0) and x = ρ (1) . Similarly, let ρ : [0 , → X be the geodesic parameterized proportional to arclength joining π γ ( y ) = ρ (0) and y = ρ (1) . Note that by property [C1] of Lemma 2.2 A ≥ D. Set s = DA , so s ∈ [0 , . Applying property [C2] of Lemma 2.2 to the geodesics ρ , ρ , we have that d ( ρ ( s ) , ρ ( s )) ≤ (1 − s ) d ( π γ ( x ) , π γ ( y )) + sd ( x, y ) ≤ (1 − s ) D + s A ≤ D + DA A D As in the discussion proceeding Lemma 3.2 with ρ ( s ) taking the place of x and ρ ( s ) takingthe place of y, we can construct a quasi-geodesic φ composed of the concatenation of five geodesicsegments. By construction, in this case our constants are a = 1 , b ∈ [1 , ] , and c ∈ [ , ] . ByLemma 3.2 φ is a (15,0)–quasi-geodesic. Furthermore, since a + c − b ≥ , again by Lemma 3.2it follows that D ≤ M (15 , . Hence, we have just shown that if γ is M(K,L)–Morse then it is ( , M (15 , –contracting. This completes the proof of (3) = ⇒ (2). (3) [+(2)] = ⇒ (1) : By Corollary 3.4 it suffice to prove (3) = ⇒ (1) in the special case of γ ageodesic.Fix x, y ∈ X such that d ( x, y ) < d ( x, π γ ( x )) . Set A = d ( x, π γ ( x )) , D = d ( π γ ( x ) , π γ ( y )) , and B = d ( π γ ( y ) , y ) . In order to complete the proof we must bound D. XPLICIT EQUIVALENCES BETWEEN CAT(0) HYPERBOLIC TYPE GEODESICS 9
DA xx y πγ(x) πγ(y)