aa r X i v : . [ m a t h . M G ] A p r PROJECTIONS AND RELATIVE HYPERBOLICITY
ALESSANDRO SISTO
Abstract.
We give an alternative definition of relative hyperbolicitybased on properties of closest-point projections on peripheral subgroups.We also derive a distance formula for relatively hyperbolic groups, sim-ilar to the one for mapping class groups.
Introduction
The main aim of this paper is to introduce a new characterization of rel-atively hyperbolic groups in terms of projections on left cosets of peripheralsubgroups. The properties we will consider are similar to those appeared in[ ? , ? ] and are used in [ ? ] in a more general setting. The characterization wewill give is similar to the characterization of tree-graded spaces given in [ ? ],the link being provided by asymptotic cones in view of results in [ ? ]. Ourcharacterization only involves the geometry of the Cayley graph, alongsidethe ones given in [ ? ] and [ ? ]. Also, the statement deals with the more gen-eral setting of metric relative hyperbolicity (i.e. asymptotic tree-gradednesswith the established terminology).We defer the exact statement to Section 2, see Definitions 2.1, 2.11 andTheorem 2.14.We will use projections also to provide an analogue for relatively hyper-bolic groups of the distance formula for mapping class groups [ ? ].Let G be a relatively hyperbolic group and let P be the collection of allleft cosets of peripheral subgroups. For P ∈ P , let π P be a closest pointprojection map onto P . Denote by ˆ G the coned-off graph of G , that is tosay the metric graph obtained from a Cayley graph of G by adding an edgeconnecting each pair of (distinct) vertices contained in the same left coset ofperipheral subgroup. Let (cid:8)(cid:8) x (cid:9)(cid:9) L denote x if x > L , and 0 otherwise. Wewrite A ≈ λ,µ B if A/λ − µ ≤ B ≤ λA + µ . Theorem 0.1 (Distance formula for relatively hyperbolic groups) . Thereexists L so that for each L ≥ L there exist λ, µ so that the following holds.If x, y ∈ G then (1) d ( x, y ) ≈ λ,µ X P ∈P (cid:8)(cid:8) d ( π P ( x ) , π P ( y )) (cid:9)(cid:9) L + d ˆ G ( x, y ) . This formula will be used in [ ? ] to study quasi-isometric embeddings ofrelatively hyperbolic groups in products of trees. It is useful for applica-tions that projections admit alternative descriptions, see Lemma 1.15. In subsection 3.1 we will give a sample application of the distance formula andshow that a quasi-isometric embedding between relatively hyperbolic groupscoarsely preserving left cosets of peripheral subgroups gives a quasi-isometricembedding of the corresponding coned-off graphs (the reader may wish tocompare this result with [ ? , Theorem 10.1]). Acknowledgment.
The author would like to thank Cornelia Drut¸u, RobertoFrigerio and John MacKay for helpful discussions and comments.1.
Background on relatively hyperbolic groups
Definition 1.1.
A geodesic complete metric space X is tree-graded withrespect to a collection P of closed geodesic subsets of X (called pieces ) ifthe following properties are satisfied:( T ) two different pieces intersect in at most one point,( T ) each geodesic simple triangle is contained in one piece.Tree-graded spaces can be characterized in terms of closest-point projec-tions on the pieces. Let us denote by X a complete geodesic metric spaceand by P a collection of subsets of X . Consider the following properties. Definition 1.2.
A family of maps Π = { π P : X → P } P ∈P will be called projection system for P if, for each P ∈ P ,( P
1) for each r ∈ P , z ∈ X , d ( r, z ) = d ( r, π P ( z )) + d ( π P ( z ) , z ),( P π P is locally constant outside P ,( P
3) for each Q ∈ P with P = Q , we have that π P ( Q ) is a point. Definition 1.3.
A geodesic is P− transverse if it intersects each P ∈ P inat most one point. A geodesic triangle in X is P− transverse if each side is P− transverse. P is transverse-free if each P− transverse geodesic triangle is a tripod. Theorem 1.4. [ ? ] Let X be a complete geodesic metric space and let P acollection of subsets of X . Then X is tree-graded with respect to P if andonly if P is transverse-free and there exists a projection system for P . The following properties have also been considered in [ ? ]. Properties ( P )and ( P ) are equivalent to ( P ′ ) and ( P ′ ). Lemma 1.5.
Properties ( P and ( P can be substituted by: ( P ′ for each P ∈ P and x ∈ P , π P ( x ) = x , ( P ′ for each P ∈ P and for each z , z ∈ X such that π P ( z ) = π P ( z ) , d ( z , z ) = d ( z , π P ( z )) + d ( π P ( z ) , π P ( z )) + d ( π P ( z ) , z ) . The reader unfamiliar with asymptotic cones is referred to [ ? ]. Convention 1.6.
Throughout the paper we fix a non-principal ultrafilter µ on N . We will denote ultralimits by µ − lim and the asymptotic cone of X with respect to (the ultrafilter µ ,) the sequence of basepoints ( p n ) and thesequence of scaling factor ( r n ) by C ( X, ( p n ) , ( r n )). ROJECTIONS AND RELATIVE HYPERBOLICITY 3
Definition 1.7. [ ? ] The geodesic metric space X is asymptotically tree-graded with respect to the collection of subsets P if all its asymptotic cones,with respect to the fixed ultrafilter, are tree-graded with respect to thecollection of the ultralimits of elements of P . Definition 1.8.
The finitely generated group G is hyperbolic relative to itssubgroups H , . . . , H n , called peripheral subgroups , if its Cayley graphs areasymptotically tree-graded with respect to the collection of all left cosets ofthe H i ’s.Let X be asymptotically tree-graded with respect to P . We report belowsome useful lemmas from [ ? ] that will be used later.When A is a subset of the metric space X , the notation N d ( A ) will de-note the closed neighborhood of radius d around A , i.e. N d ( A ) = { x ∈ X | d ( x, A ) ≤ d } . Lemma 1.9. [ ? , Theorem 4.1 − ( α )] If γ is a geodesic connecting x to y ,and d ( x, P ) , d ( y, P ) ≤ d ( x, y ) / for some P ∈ P , then γ ∩ N M ( P ) = ∅ . Lemma 1.10. [ ? , Lemma 4.7] For each H ≥ there exists B such that diam ( N H ( P ) ∩ N H ( Q )) ≤ B for each P, Q ∈ P with P = Q . We will also need that each P ∈ P is quasi-convex , in the following sense. Lemma 1.11. [ ? , Lemma 4.3] There exists t such that for each L ≥ eachgeodesic connecting x, y ∈ N L ( P ) is contained in N tL ( P ) . If G is hyperbolic relative to H , . . . , H n , its coned-off graph, denoted ˆ G ,is obtained from a Cayley graph of G by adding edges connecting verticeslying in the same left coset of peripheral subgroup.By [ ? ], ˆ G is hyperbolic and the following property holds. Proposition 1.12 (BCP property) . Let α, β be geodesics in ˆ G , for G rel-atively hyperbolic, and let P be the collection of all left cosets of peripheralsubgroups of G . There exists c with the following property.(1) If α contains an edge connecting vertices of some P ∈ P but β doesnot, then such vertices are at distance at most c in G .(2) If α and β contain edges [ p α , q α ] , [ p β , q β ] (respectively) connectingvertices of some P ∈ P , then d G ( p α , p β ) , d G ( q α , q β ) ≤ c . Geodesics and projections.Convention 1.13.
In this subsection X is an asymptotically tree-gradedspace with respect to a collection of subsets P . Sometimes we will restrictto X a Cayley graph of a relatively hyperbolic group, and in that case P will always be the collection of left cosets of peripheral subgroups.The following definition is taken from [ ? ] (Definition 4.9). Definition 1.14. If x ∈ X and P ∈ P define the almost projection π P ( x )to be the subset of P of points whose distance from x is less than d ( x, P )+ 1. ROJECTIONS AND RELATIVE HYPERBOLICITY 4
The following lemma gives two alternative characterizations of the maps π P . Lemma 1.15. (1) If α is a continuous ( K, C ) − quasi-geodesic connecting x to P ∈ P then for each D ≥ D = D ( K, C ) there exists M so that the firstpoint in α ∩ N D ( P ) is at distance at most M from π P ( x ) .(2) [Bounded Geodesic Image] If X is the Cayley graph of G , there exists M so that if ˆ γ is a geodesic in ˆ G connecting x ∈ G to P ∈ P thenthe first point in ˆ γ ∩ P is at distance at most M from π P ( x ) .Proof. (1) The saturation of a geodesic is the union of the geodesic and all P ∈ P whose µ − neighborhood intersects the geodesic (for some appropri-ately chosen µ ). By [ ? , Lemma 4.25] there exists R = R ( K, C ) so that if γ is a geodesic and the ( K, C ) − quasi-geodesic α connects points in the satu-ration Sat ( γ ) of γ , then α is contained in the R − neighborhood of Sat ( γ ).We can apply this when α is as in our statement and γ is a geodesicfrom x to π P ( x ). Let D ≥ µ, R and let p be the first point in α ∩ N D ( P ).There are two cases to consider. If p ∈ N R ( γ ) then we are done as diam ( γ ∩ N D + R ( P )) ≤ D + R and p, π P ( x ) ∈ N R ( γ ) ∩ N D ( P ). Otherwise there exists P ′ = P so that P ′ ⊆ Sat ( γ ) and p ∈ N R ( P ′ ). By [ ? , Lemma 4.28] thereexists B = B ( D ) so that N D ( P ) ∩ N R ( P ′ ) ⊆ N B ( γ ). As noticed earlier diam ( γ ∩ N B + D ( P )) ≤ B + D and π P ( x ) , p ∈ N B ( γ ) ∩ N D ( P ), so we aredone.2) Let ˆ γ be a geodesic in ˆ G connecting x to π P ( x ) and denote by p thefirst point in ˆ γ ∩ P , and let ˆ γ be any geodesic from x to P intersecting P only in its endpoint q . By adding an edge to ˆ γ connecting q to π P ( x ) weare in a situation where we can apply the BCP property to get a uniformbound on d ( p, q ). So, it is enough to prove the statement for ˆ γ = ˆ γ .By [ ? , Lemma 8.8], we can bound by some constant, say B , the distancefrom p to a geodesic γ in G from x to π P ( x ). As in the first part, wehave p, π P ( x ) ∈ N B ( γ ) ∩ N D ( P ), a set whose diameter can be bounded by B + D . (cid:3) The lift of a geodesic in ˆ G is a path in G obtained by substituting edgeslabeled by an element of some H i and possibly the endpoints with a geodesicin the corresponding left coset. The following is a consequence of [ ? , Lemma8.8] (or of the distance formula and the second part of Lemma 1.15, but [ ? ,Lemma 8.8] is used in the proof). Proposition 1.16 (Hierarchy paths for relatively hyperbolic groups) . Thereexist λ, µ so that if α is a geodesic in ˆ G then its lifts are ( λ, µ ) − quasi-geodesics. Lemma 1.17.
There exists L so that if d ( π P ( x ) , π P ( y )) ≥ L for some P ∈ P then ROJECTIONS AND RELATIVE HYPERBOLICITY 5 (1) all ( K, C ) − quasi-geodesics connecting x to y intersect B R ( π P ( x )) and B R ( π P ( y )) , where R = R ( K, C ) ,(2) all geodesics in ˆ G connecting x to y contain an edge in P , when X is a Cayley graph of G .Proof. In view of Lemma 1.15 − (1), in order to show 1) we just have to showthat any quasi-geodesic α as in the statement intersects a neighborhood of P of uniformly bounded radius. We can suppose that γ is continuous. Let p be a point on α minimizing the distance from P , and let γ be a geodesicfrom p to P of length d ( p, P ). The point p splits α in two halves α , α , andit is easy to show that the concatenation β i of α i and γ is a quasi-geodesicwith uniformly bounded constants: Lemma 1.18.
Let δ be a geodesic connecting q to p and let δ be a ( K, C ) − quasi-geodesic starting at p . Suppose that d ( q, p ) = d ( q, δ ) . Then the concate-nation δ of δ and δ is a ( K ′ , C ′ ) − quasi-geodesic, for K ′ , C ′ depending on K, C only.Proof.
It is clear that the said concatenation is coarsely lipschitz. Let I = I ∪ I be the domain of δ , where I , I are (translations of) the domainsof δ , δ . We will denote by t the intersection of I and I so that δ ( t ) = δ ( t ) = δ ( t ) = p . Let t , t ∈ I and set x i = δ ( t i ). We can assume t i ∈ I i , the other cases being either symmetric or trivial. Suppose first d ( x , p ) = | t − t | ≤ | t − t | / (2 K ) − C/
2. In this case d ( x , p ) ≤ d ( x , p ) / d ( p, x ) ≤ d ( p, x ) + d ( x , x ) ≤ d ( p, x ) / d ( x , x ) and hence d ( p, x ) ≤ d ( x , x ). Then | t − t | = | t − t | + | t − t | ≤ d ( p, x ) / ≤ d ( x , x ) . On the other hand, if | t − t | ≥ | t − t | / K − C/ | t − t | ≤ (2 K + 1) d ( x , p ) + KC ≤ (2 K + 1) d ( x , x ) + KC, as d ( x , p ) ≤ d ( x , x ). (cid:3) Again by Lemma 1.15 − (1) we can uniformly bound the distance betweenthe projections of x and y on P if d ( p, P ) > D = D ( K, C ), so that β i ∩ N D ( P ) = γ ∩ N D ( P ), so that for L large enough we must have α ∩ N D ( P ) = ∅ as required.Let ˆ γ be a geodesic in ˆ G . Part 1) applies in particular to lifts ˆ γ , so thatthe conclusion follows applying the BCP property to a sub-geodesic of ˆ γ connecting points close to π P ( x ) to π P ( y ) and the geodesic in ˆ G consistingof a single edge connecting π P ( x ) to π P ( y ). (cid:3) Alternative definition of relative hyperbolicity
In this section we state the analogue of the alternative definition of tree-graded spaces that can be found in [ ? ]. Throughout the section let X be ageodesic metric space and let P be a collection of subsets of X . ROJECTIONS AND RELATIVE HYPERBOLICITY 6
We will need the coarse versions of the definitions of projection systemand being transverse-free, as defined in [ ? ]. Definition 2.1.
A family of maps Π = { π P : X → P } P ∈P will be called almost-projection system for P if there exist C ≥ P ∈ P ,( AP
1) for each x ∈ X , p ∈ P , d ( x, p ) ≥ d ( x, π P ( x )) + d ( π P ( x ) , p ) − C ,( AP
2) for each x ∈ X with d ( x, P ) = d , diam ( π P ( B d ( x ))) ≤ C ,( AP
3) for each P = Q ∈ P , diam ( π P ( Q )) ≤ C . Remark 2.2.
For each x ∈ X and P ∈ P , by ( AP
1) with p = π P ( x ) wehave d ( x, π P ( x )) ≤ d ( x, P ) + C .2.1. Technical lemmas.
First of all, let us prove some basic lemmas. Oneof the aims will be to prove that properties ( AP
1) and ( AP
2) are equivalentto coarse versions of properties ( P ′
1) and ( P ′
2) that will be formulated later.Consider an almost-projection system for P and let C be large enough sothat ( AP
1) and ( AP
2) hold. Let us start by proving that projections arecoarsely contractive, in 2 different senses.
Lemma 2.3. (1) Consider some k ≥ and a path γ connecting x to y such that d ( x, P ) ≥ kC for each x ∈ γ . Then d ( π P ( x ) , π P ( y )) ≤ l ( γ ) /k + C .(2) d ( π P ( x ) , π P ( y )) ≤ d ( x, y ) + 6 C .Proof. (1) : Consider a partition of γ in subpaths γ i = [ x i , y i ] of length kC and one subpath γ ′ = [ x ′ , y ′ ] of length at most kC . By property ( AP
2) wehave d ( π P ( x i ) , π P ( y i )) ≤ C = d ( x i , y i ) /k and d ( π P ( x ′ ) , π P ( y ′ )) ≤ C , so d ( π P ( x ) , π P ( y )) ≤ X d ( π P ( x i ) , π P ( y i )) + d ( π P ( x ′ ) , π P ( y ′ )) ≤ X d ( x i , y i ) /k + d ( x ′ , y ′ ) /k + C ≤ l ( γ ) /k + C. (2) : Consider a geodesic γ connecting x to y . If γ ∩ N C ( P ) = ∅ wecan apply the first point. Otherwise, let γ ′ = [ x, x ′ ] (resp. γ ′′ = [ y ′ , y ]) be a(possibly trivial) subgeodesic such that γ ′ ∩ N C ( P ) = x ′ (resp. γ ′′ ∩ N C ( P ) = y ′ ). Applying the previous point to γ ′ and γ ′′ and Remark 2.2 we get d ( π P ( x ) , π P ( y )) ≤ d ( π P ( x ) , π P ( x ′ ))+ d ( π P ( x ′ ) , x ′ )+ d ( x ′ , y ′ )+ d ( y ′ , π P ( y ′ ))+ d ( π P ( y ′ ) , π P ( y )) ≤ ( d ( x, x ′ ) + C ) + 2 C + d ( x ′ , y ′ ) + 2 C + ( d ( y ′ , y ) + C ) = d ( x, y ) + 6 C, as required. (cid:3) Lemma 2.4.
For each r and c ≥ we have that each (1 , c ) − quasi-geodesic γ from x ∈ X to y ∈ N r ( P ) , for some P ∈ P , intersects B ρ ( π P ( x )) , where ρ = 2 r + 6 C + 5 c . Moreover, any point y ′ on γ such that d ( x, P ) − c ≤ d ( x, y ′ ) ≤ d ( x, P ) belongs to B ρ ( π P ( x )) . ROJECTIONS AND RELATIVE HYPERBOLICITY 7
Proof.
Note that y ′ as in the statement exists if and only if d ( x, y ) ≥ d ( x, P ) − c . Suppose d ( x, y ) < d ( x, P ) − c . In this case d ( π P ( x ) , π P ( y )) ≤ C by ( AP d ( y, π P ( x )) ≤ r + 2 C (we used Remark 2.2).Let us now consider the other case. Let y ′ ∈ γ be such that d ( x, P ) − c ≤ d ( x, y ′ ) ≤ d ( x, P ) and let γ ′ be the sub-quasi-geodesic of γ from x to y ′ . As d ( y, π P ( y )) ≤ r + C and d ( π P ( y ′ ) , π P ( x )) ≤ C , we have, using ( AP
1) in thesecond inequality, d ( y ′ , y ) ≥ d ( y ′ , π P ( y )) − r − C ≥ d ( y ′ , π P ( y ′ )) + d ( π P ( y ′ ) , π P ( y )) − r − C ≥ d ( y ′ , π P ( x )) + d ( π P ( x ) , π P ( y )) − r − C. Also, d ( x, y ) ≤ d ( x, π P ( x )) + d ( π P ( x ) , π P ( y )) + r + C. As d ( x, y ) ≥ d ( x, y ′ ) + d ( y ′ , y ) − c (as these points lie on a (1 , c ) − quasi-geodesic) and d ( x, y ′ ) ≥ d ( x, P ) − c , we obtain[ d ( y ′ , π P ( x )) + d ( π P ( x ) , π P ( y )) − r − C ] + d ( x, P ) ≤ d ( y ′ , y ) + d ( y ′ , x ) + 2 c ≤ d ( x, y ) + 5 c ≤ d ( x, π P ( x ))+ d ( π P ( x ) , π P ( y ))+ r + C +5 c ≤ d ( x, P )+ d ( π P ( x ) , π P ( y ))+ r +2 C +5 c. Therefore, d ( y ′ , π P ( x )) ≤ r + 6 C + 5 c. (cid:3) The following can be thought as another coarse version of property ( P Lemma 2.5.
Consider a geodesic γ starting from x and some P ∈ P suchthat γ ∩ N r ( P ) = ∅ , for some r ≥ C . Let y be the first point on γ in N r ( P ) .Then d ( y, π P ( x )) ≤ r + 22 C .Proof. If d ( x, y ) ≤ d ( x, P ), we have d ( π P ( x ) , π P ( y )) ≤ C by ( AP d ( y, π P ( x )) ≤ r + 2 C (we used Remark 2.2). Suppose that this is not thecase and let y ′ be as in the previous lemma. Consider a geodesic γ ′ = [ y, y ′ ].By d ( y, π P ( y )) ≤ r + C , d ( y ′ , π P ( y ′ )) ≤ r + 7 C (because of Remark 2.2),Lemma 2.3 − (1) with k = 2 (recall that r ≥ C and notice that γ ′ ∩ N r ( P ) = { y } ), we have d ( y, y ′ ) ≤ d ( y, π P ( y ))+ d ( π P ( y ) , π P ( y ′ ))+ d ( π P ( y ′ ) , y ′ ) ≤ r +8 C + d ( y, y ′ ) / . So, d ( y, y ′ ) ≤ r + 16 C and d ( y, π P ( x )) ≤ d ( y, y ′ ) + d ( y ′ , π P ( x )) ≤ r +22 C . (cid:3) Corollary 2.6.
Consider a geodesic γ from x to y and some P ∈ P suchthat γ ∩ N r ( P ) = { y } , for some r ≥ C . Then l ( γ ) ≤ d ( x, P ) + 8 r + 23 C and π P ( γ ) ⊆ B r +30 C ( π P ( x )) .Proof. Using the previous lemma, l ( γ ) = d ( x, y ) ≤ d ( x, π P ( x ))+ d ( π P ( x ) , y ) ≤ d ( x, P ) + C + (8 r + 22 C ). The second part is an easy consequence of thisfact, using ( AP
2) and Lemma 2.3 − (2). (cid:3) ROJECTIONS AND RELATIVE HYPERBOLICITY 8
Corollary 2.7.
Let γ be a geodesic from x to x . Then diam ( γ ∩ N r ( P )) ≤ d ( π P ( x ) , π P ( x )) + 18 r + 62 C for each r ≥ C and P ∈ P .Proof. Let x ′ , x ′ be the first and last point in γ ∩ N r ( P ). By Corollary 2.6,we have d ( π P ( x i ) , π P ( x ′ i )) ≤ r + 30 C . So, d ( π P ( x ) , π P ( x )) ≥ d ( x ′ , x ′ ) − r + 30 C ) − r + C ) = d ( x ′ , x ′ ) − r − C . As d ( x ′ , x ′ ) = diam ( γ ∩ N r ( P )), this is what we wanted. (cid:3) We will consider the following coarse analogs of properties ( P ′
1) and( P ′ AP ′
1) There exists C ≥ x ∈ X , d ( x, π P ( x )) ≤ d ( x, P ) + C .( AP ′
2) There exists C ≥ x , x ∈ X such that d ( π P ( x ) , π P ( x )) ≥ C , we have d ( x , x ) ≥ d ( x , π P ( x )) + d ( π P ( x ) , π P ( x )) + d ( π P ( x ) , x ) − C. Lemma 2.8. ( AP
1) + ( AP ⇐⇒ ( AP ′
1) + ( AP ′ . Definition 2.9.
We will say that C is a projection constant if the properties( AP , ( AP , ( AP ′ , ( AP ′
2) hold with constant C . Proof. ⇐ : Fix C large enough so that ( AP ′ , ( AP ′
2) hold. Property ( AP d ( π P ( x ) , x ) is large, and in this case it follows from( AP ′
2) setting x = x and x = π P ( x ) = p and keeping into account d ( π P ( p ) , p ) ≤ C . Let us show property ( AP d ( π P ( x ) , π P ( x ′ )) >C implies d ( x, x ′ ) > d ( x, P ) − C . We want to exploit this fact. Set d = d ( x, P ). Note that if x ′ ∈ B ( x, d ), then there exists x ′′ ∈ B d − C such that d ( x ′ , x ′′ ) ≤ C and one of of the following 2 cases holds: • x ′ ∈ N C ( P ), or • d ( x ′′ , P ) ≥ C .In the first case either d ( π P ( x ′ ) , π P ( x ′′ )) < C or d ( x ′ , π P ( x ′ )) + d ( π P ( x ′ ) , π P ( x ′′ )) + d ( π P ( x ′′ ) , x ′′ ) − C ≤ d ( x ′ , x ′′ ) ≤ C, and so d ( π P ( x ′ ) , π P ( x ′′ )) ≤ C . In the second case d ( x ′ , x ′′ ) ≤ d ( x ′ , P ) − C ,and so d ( π P ( x ′ ) , π P ( x ′′ )) ≤ C .These considerations yield diam ( π P ( B d ( x ))) ≤ C . ⇒ : We already remarked that ( AP ′
1) holds. Let
C > AP
1) and ( AP
2) hold. We will prove the following, which implies( AP ′
2) setting c = 0 and which will be useful later. Lemma 2.10. If d ( π P ( x ) , π P ( x )) ≥ C + 8 c + 1 , for some c ≥ and P ∈ P , then any (1 , c ) − quasi-geodesic γ from x to x intersects N C ( P ) and B C +5 c ( π P ( x i )) .Proof. Once we show that γ ∩ N C ( P ) = ∅ , we can apply Lemma 2.4 toobtain B C +5 c ( π P ( x i )) ∩ γ = ∅ Set d i = d ( x i , P ). We have that B d ( x ) ∩ B d ( x ) = ∅ , for otherwisewe would have d ( π P ( x ) , π P ( x )) ≤ C . Let z i be a point on γ such that ROJECTIONS AND RELATIVE HYPERBOLICITY 9 d i − c ≤ d ( x i , z i ) ≤ d i . Suppose by contradiction that [ z , z ] ∩ N C ( P ) = ∅ . Then d ( π P ( z ) , π P ( z )) ≤ d ( z , z ) / C by Lemma 2.3 − (1), and inparticular d ( z , z ) / ≥ C + 8 c + 1 (notice that d ( π P ( z ) , π P ( x i )) ≤ C ). So, d ( x , x ) ≤ d ( x , π P ( x )) + d ( π P ( x ) , π P ( z )) + d ( π P ( z ) , π P ( z ))+ d ( π P ( z ) , π P ( x )) + d ( π P ( x ) , x ) ≤ ( d ( x , P ) + C ) + C + ( d ( z , z ) / C ) + C + ( d ( x , P ) + C ) ≤ d ( x , z ) + d ( z , z ) + d ( z , x ) + 5 C + 4 c − d ( z , z ) / ≤ ( d ( x , x ) + 4 c ) + 5 C + 4 c − d ( z , z ) / < d ( x , x ) , a contradiction. Therefore [ z , z ] ∩ N C ( P ) = ∅ and in particular γ ∩ N C ( P ) = ∅ , as required. (cid:3)(cid:3) Main result.Definition 2.11.
A (1 , c ) − quasi-geodesic triangle ∆ is P− almost-transversewith constants K, D if, for each P ∈ P and each side γ of ∆, diam ( N K ( P ) ∩ γ ) ≤ D . P is asymptotically transverse-free if there exist λ, σ such that for each D ≥ K ≥ σ the following holds. If ∆ is a geodesic triangle which is P− almost-transverse with constants K, D , then ∆ is λD − thin.Recall that a triangle is δ − thin if any point on one of its sides is atdistance at most δ from the union of the other two sides.The definition of being asymptotically transverse-free only involves ge-odesic triangles. But, as we will see, if there exists an almost-projectionsystem for P , then we can deduce something about (1 , c ) − quasi-geodesictriangles as well. Definition 2.12. P is strongly asymptotically transverse-free if there exist λ, σ such that for each c, D ≥ K ≥ σc the following holds. If ∆ is a(1 , c ) − quasi-geodesic triangle which is P− almost-transverse with constants K, D , then ∆ is λ ( D + c ) − thin. Lemma 2.13. If P is asymptotically transverse-free and there exists analmost-projection system for P , then P is strongly asymptotically transverse-free.Proof. Let C be a projection constant for P and let λ , σ be the constantssuch that P is asymptotically transverse-free with those constants. We willshow that P is strongly asymptotically transverse-free for σ = 10 C + 5.Let ∆ be a (1 , c ) − quasi-geodesic triangle, for c ≥
1, which is P− almost-transverse with constants K ≥ σc, D ≥
1, and let { γ i } be its sides.Consider x, y ∈ γ i . We want to prove that any geodesic γ from x to y is P− almost-transverse with “well-behaved” constants. Let us start byproving that d ( π P ( x ) , π P ( y )) ≤ D + 20 C + 10 c + 1 for each P ∈ P . In fact,if that was not the case, by Lemma 2.10 we would have that γ i intersects ROJECTIONS AND RELATIVE HYPERBOLICITY 10 B C +5 c ( π P ( x )), B C +5 c ( π P ( x )), so diam ( γ i ∩ N C +5 c ( P )) ≥ D + 1 (acontradiction as σc ≥ C +5 c ). By Corollary 2.7 (we can assume σ ≥ C ),we have diam ( γ ∩ N σ ( P )) ≤ D + 18 σ + 82 C + 10 c + 1 for each P ∈ P .By the fact that P is asymptotically transverse-free, we obtain that eachgeodesic triangle whose vertices lie on γ i is λ ′ − thin, for λ ′ = λ ( D + 18 σ +82 C + 10 c + 1). This is all that is needed to apply verbatim the proof of [ ? ,Theorem III.H.1.7] (which roughly states that in a hyperbolic space quasi-geodesics are at finite Hausdorff distance from geodesics). The constantsappearing in the proof are explicitly determined in terms of the hyperbolicityconstant δ ( λ ′ plays the role of δ ) and the quasi-geodesics constants λ, ǫ (inour case λ = 1, ǫ = c ), and one can easily check that the bound on theHausdorff distance can be chosen to be linear in δ + ǫ , when fixing λ = 1 (and,say, for δ, ǫ ≥ D + c )from the sides of a triangle whose thinness constant is linear in ( D + c ), sowe are done. (cid:3) Theorem 2.14.
The geodesic metric space X is asymptotically tree-gradedwith respect to the collection of subsets P if and only if P is asymptoticallytransverse-free and there exists an almost-projection system for P .Proof. ⇐ : Consider an asymptotic cone Y = C ( X, ( p n ) , ( r n )) of X andconsider the collection P ′ of ultralimits of elements of P in Y . It is quiteclear that elements of P ′ are geodesic, by the assumptions on P . Also, it isvery easy to see that an almost projection system for P induces a projectionsystem for P ′ .Let us prove that P ′ is transverse-free. Consider a geodesic triangle ∆in Y . We would like to say that its sides are ultralimits of geodesics in X .This is not the case, but, as shown in the following lemma, it is not too farfrom being true. Lemma 2.15.
Any geodesic γ : [0 , l ] → Y is the ultralimit of a sequence ( γ n ) of (1 , c n ) − quasi-geodesics, where µ − lim c n /r n = 0 .Proof. By [ ? , Lemma 9.4], γ is a ultralimit of lipschitz paths γ n . Let c n bethe least real number so that γ n is a (1 , c n ) − quasi-geodesic. As the ultralimitof ( γ n ) is a geodesic, it is readily seen that µ − lim c n /r n = 0. (cid:3) Using this lemma, we obtain that ∆, the geodesic triangle we are consider-ing, is the ultralimit of some triangles ∆ n of X whose sides are (1 , c n ) − quasi-geodesics and µ − lim c n /r n = 0 (as ∆ is P ′ − transverse). Suppose that ∆ is P ′ − transverse, and let λ, σ be as in the definition of being strongly asymp-totically transverse-free. Let K n = σc n and notice that ∆ n must be µ − a.e. P− almost-transverse with constants K n , D n , where µ − lim D n /r n = 0. Inparticular, ∆ n is κ n − thin, where κ n = λ ( D n + c n ) so that µ − lim κ n /r n = 0.This implies that ∆ is a tripod, and hence we showed that P ′ is transverse-free. We proved that both conditions of Theorem 1.4 are satisfied for Y and ROJECTIONS AND RELATIVE HYPERBOLICITY 11 P ′ , therefore Y is tree-graded with respect to P ′ . As Y was any asymptoticcone of X , the proof is complete. ⇒ : For each P ∈ P , define π P in such a way that for each x ∈ X we have d ( π P ( x ) , x ) ≤ d ( x, P ) + 1. Property ( AP ′
1) is obvious. Property ( AP ′ − (1).Let us prove ( AP
3) (we will use the lemma once again). Let B be auniform bound on the diameters of N H ( P ) ∩ N H ( Q ) for P = Q ∈ P (seeLemma 1.10), where H = max { tM, L } for t as in Lemma 1.11. Fix P, Q ∈ P , P = Q . Suppose that there exist x, y ∈ Q such that d ( π P ( x ) , π P ( y )) ≥ L + B + 1. Consider a geodesic [ x, y ]. It is contained in N tM ( Q ). Considerpoints x ′ , y ′ on [ x, y ] such that d ( x ′ , π P ( x )) ≤ L , d ( y ′ , π P ( y )) ≤ L . Then d ( x ′ , y ′ ) ≥ d ( π P ( x ) , π P ( y )) − L ≥ B + 1. This is in contradiction with diam ( N H ( P ) ∩ N H ( Q )) ≤ B .These considerations readily imply ( AP P is asymptotically transverse-free. Supposethat there is no λ such that P satisfies the definition of being asymptoticallytransverse-free with σ = tM for M as in Lemma 1.9 and t as in Lemma 1.11.Then we have a diverging sequence ( r ′ n ) and geodesic triangles ∆ n which are P− almost-transverse with constants K, D n and optimal thinness constant r n = r ′ n D n . Let α n , β n , γ n be the sides of ∆ n . We can assume that thereexists p n ∈ α n with d ( p n , β n ∪ γ n ) = r n . Consider Y = C ( X, ( p n ) , ( r n )), andlet α, β, γ be the geodesics (or geodesic rays, or geodesic lines) in Y inducedby ( α n ) , ( β n ) , ( γ n ). Also, let P ′ be the collection of pieces for Y as in thedefinition of asymptotic tree-gradedness. We claim that for each P ∈ P ′ , | α ∩ P | ≤ β, γ ). This easily leads to a contradiction. Infact, suppose that α, β, γ all have finite length. Then they form a transversegeodesic triangle that is not a tripod, a contradiction. If at least one ofthem is infinite, we can reduce to the previous case observing that transversegeodesic rays in Y at finite Hausdorff distance eventually coincide, so thatwe can cut off parts of α, β, γ to get once again a transverse geodesic trianglethat is not a tripod.So, suppose that the claim does not hold. Then we can find sequencesof points ( x n ) , ( y n ) on ( α n ) and a sequence ( P n ) of elements of P so that µ − lim d ( x n , P n ) /r n , µ − lim d ( y n , P n ) /r n = 0 but µ − lim d ( x n , y n ) /r n > α n between x n and y n intersects N M ( P n ), so that it contains a subgeodesic in N tM ( P n ). It is easily seen thatthe the length l n of the maximal such subgeodesic has the property that µ − lim l n /r n >
0, in contradiction with diam ( N tM ( P n ) ∩ α ) ≤ D n . (cid:3) Distance formula
Let G be a relatively hyperbolic group and let P be the collection of allleft cosets of peripheral subgroups. For P ∈ P , let π P be a closest pointprojection map onto P . Denote by ˆ G the coned-off graph of G . Let (cid:8)(cid:8) x (cid:9)(cid:9) L ROJECTIONS AND RELATIVE HYPERBOLICITY 12 denote x if x > L , and 0 otherwise. We write A ≈ λ,µ B if A/λ − µ ≤ B ≤ λA + µ . Theorem 3.1 (Distance formula for relatively hyperbolic groups) . Thereexists L so that for each L ≥ L there exist λ, µ so that the following holds.If x, y ∈ G then (2) d ( x, y ) ≈ λ,µ X P ∈P (cid:8)(cid:8) d ( π P ( x ) , π P ( y )) (cid:9)(cid:9) L + d ˆ G ( x, y ) . Proof.
Let us start with a preliminary fact. There exists σ so that whenever γ i , for i = 1 ,
2, is a geodesic with endpoints in N D ( P i ) for some P i ∈ P with P = P we have diam ( γ ∩ γ ) ≤ σ = σ ( D ). (This is similar to [ ? , Lemma8.10], which could also be used for our purposes.) This follows from quasi-convexity of the peripheral subgroups (Lemma 1.11) combined with theexistence of a bound depending only on δ on the diameter of N δ ( P ) ∩ N δ ( P )(Lemma 1.10). So, we have the following estimate for D , M as in Lemma1.15 − (1) for K = 1 and C = 0 and σ = σ ( D ):(3) d ( x, y ) ≥ X P ∈P d ( π P ( x ) ,π P ( y )) ≥ σ +2 M (cid:0) d ( π P ( x ) , π P ( y )) − σ − M (cid:1) . Write A . λ,µ B or B & λ,µ A if A ≤ λB + µ . In view of (3) and the factthat the inclusion G → ˆ G is Lipschitz we have the inequality & λ,µ in (2).Hence we just need to show that any lift ˜ α of a geodesic α in ˆ G satisfies l ( ˜ α ) . λ,µ R , where R denotes the right hand side of (2), with x, y theendpoints of ˜ α . Let α , . . . , α n be all maximal subgeodesics of ˜ α of lengthat least some large L ′ contained in some left cosets P , . . . , P n . We have l ( ˜ α ) ≈ λ,µ X l ( α i ) + d ˆ G ( x, y ) . The endpoints of α i have uniformly bounded distance from π P i ( x ) , π P i ( y )respectively by Lemma 1.15 − (2). (cid:3) Sample application of the distance formula.
We now provide anapplication of the distance formula. We first need a preliminary lemma. Wekeep the notation set above.
Proposition 3.2.
Let φ : G → G be a ( K, C ) − quasi-isometric embeddingbetween relatively hyperbolic groups so that the image of any left coset of pe-ripheral subgroup of G is mapped in the C − neighborhood of a left coset of aperipheral subgroup of G . Then φ is a ( K ′ , K ′ ) − quasi-isometric embeddingat the level of the coned-off graphs, where K ′ = K ′ ( K, C ) .Proof. In view of the characterization of projections given in Lemma 1.15 − (1)and the fact that left cosets of peripheral subgroups are coarsely preserved,we see that for each x ∈ G and left coset P of a peripheral subgroup of G we have that π φ ( P ) ( φ ( x )) is at uniformly bounded distance from φ ( π P ( x )),where φ ( P ) is a left coset of a peripheral subgroup of G containing φ ( P )in its C − neighborhood. ROJECTIONS AND RELATIVE HYPERBOLICITY 13
Fix x, y and let ˆ γ be a geodesic in ˆ G connecting them. Let ˆ γ , . . . , ˆ γ n bethe maximal sub-geodesics of ˆ γ that do not contain an edge contained in anyleft coset of peripheral subgroup P so that d ( π P ( x ) , π P ( y )) is larger thansome suitable constant M . The lift of ˆ γ i is a quasi-geodesic, and in particularthe image γ ′ i of the lift via φ is also a quasi-geodesic. The observation wemade at the beginning of the proof and the distance formula imply that γ ′ i is a quasi-geodesic in ˆ G as well. We see then that the image of ˆ γ through φ is made of a collection of quasi-geodesics of ˆ G (with uniformly boundedconstants) and if M was chosen large enough those quasi-geodesics connectpoints on a geodesic ˆ α in ˆ G from φ ( x ) to φ ( y ) by Lemma 1.17. It is nothard to check that φ (ˆ γ ) crosses these points in the same order as ˆ α does,which implies that φ (ˆ γ ) is a quasi-geodesic (again, with uniformly boundedconstants). In fact, it suffices to show that γ ′ i does not connect points onopposite sides in ˆ α of some φ ( P ), where d ( π P ( x ) , π P ( y )) > M . If it did,we would have that the projections of the endpoints of γ ′ i on φ ( P ) are farapart, which implies that the same holds for the endpoints of ˆ γ i , but this isnot the case in view of Lemma 1.15 − (2). (cid:3) Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, United Kingdom
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