Relatively hyperbolic extensions of groups and Cannon Thurston maps

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1 Proc. Indian Acad. Sci. (Math. Sci.) Vol. 20, No., February 200, pp Indian Academy of Sciences Relatively hyperbolic extensions of groups and Cannon Thurston maps ABHIJIT PAL Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata , India abhijit MS received 7 September 2009; revised 28 October 2009 Abstract. Let (K, K ) (, N (K )) (Q, Q ) be a short exact sequence of pairs of finitely generated groups with K a proper non-trivial subgroup of K and K strongly hyperbolic relative to K. Assuming that, for all g, there exists k g K such that gk g = k g K kg, we will prove that there exists a quasi-isometric section s: Q. Further, we will prove that if is strongly hyperbolic relative to the normalizer subgroup N (K ) and weakly hyperbolic relative to K, then there exists a Cannon Thurston map for the inclusion i: Ɣ K Ɣ. Keywords. Cannon Thurston maps; relatively hyperbolic groups.. Introduction Let us consider the short exact sequence of finitely generated groups K i p Q such that K is non-elementary word hyperbolic. In [2], Mosher proved that if is hyperbolic, then Q is hyperbolic. To prove that Q is hyperbolic, Mosher (in [2]) constructed a quasi-isometric section from Q to, that is, a map s: Q satisfying k d Q(q, q ) ɛ d (s(q), s(q )) kd Q (q, q ) + ɛ, for all q,q Q, where d and d Q are word metrics and k,ɛ 0 are constants. In [6], the existence of a Cannon Thurston map for the embedding i: Ɣ K Ɣ was proved, where Ɣ K and Ɣ are respectively the Cayley graphs of K and. In this paper, we will generalize these results to the case where the kernel is strongly hyperbolic relative to a cusp subgroup. One of our main theorems states: Theorem 2.0. Suppose we have a short exact sequence of finitely generated groups K p Q, with K strongly hyperbolic relative to a non-trivial proper subgroup K and suppose that, for all g, there exists k g K such that gk g = k g K kg. Then there exists a (k, ɛ)-quasi-isometric section s: Q for some constants k,ɛ 0. 57

2 58 Abhijit Pal As a corollary, under the hypotheses as in the above theorem, we can take the image of the quasi-isometric section to lie in N (K ). Let S be a once-punctured torus. Then its fundamental group π (S) = F(a, b) is strongly hyperbolic relative to the peripheral subgroup H = aba b. Let M be a 3-manifold fibering over the unit circle with fiber S such that the fundamental group π (M) is strongly hyperbolic relative to the subgroup H Z. Then we have a short exact sequence of pairs of finitely generated groups: (π (S), H ) (π (M), H Z) (Z, Z). Let K = π (S), = π (M); and let Ɣ K and Ɣ be the Cayley graphs of K and respectively. Bowditch [3] and Mj [9], proved the existence of a Cannon Thurston map for the embedding i: Ɣ K Ɣ. Motivated by this example, we will prove the following theorem: Theorem 3.. Suppose we have a short exact sequence of pairs of finitely generated groups (K, K ) i (, N (K )) p (Q, Q ) with K a proper non-trivial subgroup of K and K strongly hyperbolic relative to K. Assume further that, for all g, there exists k g K such that gk g = k g K kg.if is strongly hyperbolic relative to N (K ) and weakly hyperbolic relative to the subgroup K, then there exists a Cannon Thurston map for the embedding i: Ɣ K Ɣ, where Ɣ K and Ɣ are Cayley graphs of K and respectively. 2. Relative hyperbolicity and quasi-isometric section For basic notions and properties about hyperbolic metric spaces, refer to []. Let K and ɛ 0. A (K, ɛ)-quasigeodesic in a metric space X is a (K, ɛ)-quasi-isometric embedding γ : J X, where J is an interval (bounded or unbounded) of the real line. A (K, K)- quasigeodesic in X will be called as K-quasigeodesic. We recall romov s definition of a strongly relatively hyperbolic group. DEFINITION 2. [5] Let be a finitely generated group acting freely and properly discontinously by isometries on a proper and δ-hyperbolic metric space X, such that the quotient space X/ is quasiisometric to [0, ). Let H denote the stabilizer subgroup of the endpoint on X of a lift of this ray to X. Then is said to be strongly hyperbolic relative to H. The subgroup H is said to be a parabolic or cusp subgroup. The end points on X of lifts of [0, ) will be called as parabolic end points. For a group strongly hyperbolic relative to a subgroup H, the stabilizer subgroup of a parabolic end point is aha for some a. DEFINITION 2.2 [4] Let be a finitely generated group, and let H be a finitely generated subgroup of. Let Ɣ be the Cayley graph of. Let ˆƔ be a new graph obtained from Ɣ as follows:

3 Relatively hyperbolic extensions of groups 59 For each left coset gh of H in, we add a new vertex v(gh) to Ɣ, and add an edge e(gh) of length /2 from each element gh of gh to the vertex v(gh). We call this new graph the coned-off Cayley graph of with respect to H, and denote it by ˆƔ = ˆƔ (H ). We say that is weakly hyperbolic relative to the subgroup H if the coned-off Cayley raph ˆƔ is hyperbolic. eodesics in the coned-off space ˆƔ will be called as electric geodesics. For a path γ Ɣ, there is an induced path ˆγ in Ɣ obtained by replacing the portion of γ inside a left coset by an edge path of length passing through the cone point corresponding to that left coset. If ˆγ is an electric geodesic (resp. P -quasigeodesic), γ is called a relative geodesic (resp. relative P -quasigeodesic). If ˆγ passes through some cone point v(gh), we say that ˆγ penetrates the coset gh. DEFINITION 2.3 [4] ˆγ is said to be an electric (K, ɛ)-quasigeodesic in (the electric space) Ɣ without backtracking if ˆγ is an electric (K, ɛ)-quasigeodesic in Ɣ and ˆγ does not return to any left coset after leaving it. DEFINITION 2.4 [4] Bounded coset penetration: The pair (, H ) is said to satisfy the bounded coset penetration property if, for every P, there is a constant D = D(P) such that, whenever α and β are two electric P -quasigeodesics without backtracking starting and ending at same points, the following conditions hold:. If α penetrates a coset gh but β does not penetrate gh, then α travels a Ɣ -distance of at most D in gh. 2. If both α and β penetrate a coset gh, then the vertices in Ɣ, at which α and β first enter gh, lie at a Ɣ -distance of at most D from each other; similarly for the last exit vertices. We state below Farb s definition of a strongly relatively hyperbolic group. DEFINITION 2.5 [4] is said to be strongly hyperbolic relative to H if is weakly hyperbolic relative to H and the pair (, H ) satisfies the bounded coset penetration property. Theorem 2.6 [2, 3]. is strongly hyperbolic relative to the cusp subgroup H in the sense of Farb if and only if is strongly hyperbolic relative to H in the sense of romov. Let be a group strongly hyperbolic relative to a subgroup. Let X = Ɣ and H g be the closed set in Ɣ corresponding to the left coset g of in. Let H = {H g : g }. Let X h be the space obtained from X by gluing H g [0, ) to H g for all H g H, where H g [0, ) is equipped with the path metric d h induced from the following two pieces of data: (a) d h,t ((x, t), (y, t)) = 2 t d H (x, y), where d h,t is the path metric on H t = H {t}. (b) d h ((x, t), (x, s)) = t s for all x H and for all t,s [0, ).

4 60 Abhijit Pal Then it is shown by Bowditch (in [2]) that X h is hyperbolic. H g H were referred to as horosphere-like sets by Mj and Reeves in [] and H g [0, ) was referred to as hyperbolic cones or horoball-like sets in [0]. DEFINITION 2.7 [2] Relative hyperbolic boundary: For the relatively hyperbolic group (, ), the boundary X h of X h will be called as relative hyperbolic boundary of (, ) and will be denoted by Ɣ(, ). Bowditch in [2] showed that if acts properly discontinuously by isometries on a proper hyperbolic space Z and the action of on Z is geometrically finite (i.e. every point of Z is either a conical limit point or a bounded parabolic point) and minimal (i.e. if the limit set = Z) then Z is homeomorphic to Ɣ(, H ). DEFINITION 2.8 Let K Q be a short exact sequence of finitely generated groups with K strongly hyperbolic relative to K. We say that preserves cusps, if for all g, there exists a g K such that gk g = a g K ag. DEFINITION 2.9 [2] Quasi-isometric section: Let K Q be a short exact sequence of finitely generated groups. A map s: Q is said to be a (R, ɛ) quasi-isometric section if R d Q(q, q ) ɛ d (s(q), s(q )) Rd Q (q, q ) + ɛ, for all q,q Q, where d and d Q are word metrics and R, ɛ 0 are constants. Let K be a strongly hyperbolic group relative to a cusp subgroup K. For each parabolic point α Ɣ(K,K ), there is a unique subgroup of the form ak a. Now the Hausdorff distance between two sets ak and ak a is uniformly bounded by the length of the word a. Hence α corresponds to a left coset ak of K in K. Let K i p Q be a short exact sequence of finitely generated groups with K non-elementary and strongly hyperbolic relative to a subgroup K. We use the following notations for our further purpose: Let be the set of all parabolic end points for the relatively hyperbolic group K with cusp subgroup as K. Let 2 ={(α,α 2 ): α and α 2 are distinct elements in }. For a K, let i a : K K denote the inner automorphism and L a : K K the left translation. For g, let I g : K K be the outer automorphism, that is, I g (k) = gkg and L g : be the left translation. preserves cusps, so for each g, there exists a g K such that ag g N (K ). If b K, then it can be easily proved that d K (a g K,gbg a g K ) d K (K,bK ) + 2l K (ag g). Since I g(bk ) = g(bk )g = gbg a g K ag and the Hausdorff distance between gbg a g K and gbg a g K ag is bounded, I g will induce a map I g :

5 Relatively hyperbolic extensions of groups 6 and I g is a bijection. Therefore, I g will induce a bijective map I g 2: 2 2. For the sake of convenience of notation we will use I g for I g and I g 2. Similarly, for a K, i a and L a will induce bijective maps (with same notation) from to and 2 to 2. In the following theorem we generalize Mosher s construction of quasi-isometric section to the relatively hyperbolic case. Theorem 2.0. Suppose we have a short exact sequence of finitely generated groups K i p Q, such that K is strongly hyperbolic relative to a non-trivial proper subgroup K and preserves cusps. Then there exists an (R, ɛ) quasi-isometric section s: Q for some R,ɛ 0. Proof. Let α = (α,α 2 ) 2, then the stabilizer subgroup of each α i is a i K ai for some a i K, i =, 2. Let λ, μ be two relative geodesics in Ɣ K starting from a point of a K and ending at some point of a 2 K and further assume that ˆλ, ˆμ passes through the cone points v(a i K ), i =, 2. Let x,y be the exit points of λ, μ from the left coset a K respectively. Then due to bounded coset penetration property 2, d K (x, y) D, where D is the constant as in Definition 2.4. Let B α be the set of all exit points from a K of relative geodesics λ starting at some point of a K and ending at some point of a 2 K such that ˆλ passes through v(a i K ) s, i =, 2. Then B α is a bounded set with diameter less than or equal to D. Let C ={α 2 : e K B α }, where e K is the identity element in K. We fix an element η = (η,η 2 ) 2. Let ={g : η I g (C)}. A subset of will be proved to be the image of a quasi-isometric section. Step. We first prove that, for any g, a K I ga (C) = 2. Let α = (α,α 2 ) 2. Each α i corresponds to a left coset a i K, i =, 2. Let λ be a relative geodesic in Ɣ K starting at some point of a K and ending at some point of a 2 K such that ˆλ passes through cone points v(a i K ), i =, 2. Let x α be the exit point of λ from a K. Then x α B α. Now there exists k K such that L k (x α ) = e K. Since L k is an isometry, L k (λ) will be a relative geodesic joining point of ka K and ka 2 K with L k (λ) containing the cone points v(ka K ), v(ka 2 K ) and e K the exit point of L k (λ) from ka K. Now there exists β i such that β i corresponds to the left coset ka i K, i =, 2. Therefore β = (β,β 2 ) 2, e K B β and L k (α) = β C. Since L k and i k are same on the relative hyperbolic boundary, we have i k (α) C. Thus a K (i a (C)) = 2. Consequently, for any g, we have a K I ga (C) = a K I g i a (C) = I g ( a K (i a (C))) = I g ( 2 ) = 2. Step 2. Now we prove that p( ) = Q. Let q Q. Then there exists g such that p(g) = q. Now a K I ga (C) = 2 for any g. Therefore, for η 2, there exists a K such that η I ga (C). Hence ga and p(ga) = p(g) = q. Step 3. Now we prove that there exist constants R,ɛ 0 such that, for all g, g, the following holds: R d Q(p(g), p(g )) ɛ d (g, g ) Rd Q (p(g), p(g )) + ɛ.

6 62 Abhijit Pal We can choose a finite symmetric generating set S of such that p(s)is also a generating set for Q. Obviously, d Q (p(g), p(g )) d (g, g ) for all g, g.toproved (g, g ) Rd Q (p(g), p(g )) + ɛ for all g, g, it suffices to prove that there exists R such that d (g, g ) R whenever d Q (p(g), p(g )) for some g, g. Let d Q (p(g), p(g )) for some g, g. Then g g = ka for some k K and a is either the identity element of or a generator of. Since g, g, I g (C) I g (C). This implies I ka (C) C = I g g (C) C.NowI ka = i k (I a ). Therefore i k (I a (C)) C. For each α 2, we choose an element a α B α. Define a map F : 2 Ɣ K by F(α) = a α. Since L k is an isometry, for k K, ka α B kα and hence d K (a kα,ka α ) = d K (F(kα), kf(α)) D, () where kα denotes the image of α under the map L k : 2 2. Let B D (e K ) be the closed D-neighborhood of e K.NowF(C)is contained in the union of the B α s containing identity e K. Therefore F(C) is contained in B D (e K ). Since preserves cusps, there exists s K such that F(I a (C)) is contained in the union of the B α s containing s and hence F(I a (C)) B D (s), where B D (s) is a closed D-neighborhood of s. From (), the Hausdorff distance between two sets F(kI a (C)) and kf(i a (C)) is bounded by D. ForasetA Ɣ K, let N D (A) denote the closed D-neighborhood of A. Thus F(kI a (C)) N D (kf (I a (C))) = kn D (F (I a (C))) kb 2D (s). Now K acts properly discontinuously on Ɣ K. Therefore B D (e K ) kb 2D (s) for finitely many k s in K. This implies F(C) F(kI a (C)) for finitely many k s in K. And hence C L k (I a (C)) = C ki a (C) for finitely many k s in K. L k = i k on the relative hyperbolic boundary. Hence C (I ka (C)) for finitely many k s in K. Thus g g = ka for finitely many k s. Since the number of generators of is finite, there exists a constant R such that d (g, g ) R. Now we define s: Q as follows: Let q Q and let there exist g, g such that p(g) = p(g ) = q. Then by the above inequality d (g, g ) R. We choose one element g p (q) for each q Q and define s(q) = g. Then s defines a single valued map satisfying: R d Q(q, q ) ɛ d (s(q), s(q )) Rd Q (q, q ) + ɛ for some constants R, ɛ 0 and for all q,q Q. COROLLARY 2. Suppose we have a short exact sequence of pairs of finitely generated groups (K, K ) (, N (K )) p (Q, Q ) with K strongly relatively hyperbolic with respect to the cusp subgroup K.Ifpreserves cusps, then Q = Q and there is a quasi-isometric section s: Q N (K ) satisfying R d Q(q, q ) ɛ d N (K )(s(q), s(q )) Rd Q (q, q ) + ɛ

7 Relatively hyperbolic extensions of groups 63 where q,q Q and R,ɛ 0 are constants. Further, if is weakly hyperbolic relative to K, then Q is hyperbolic. Proof. Let q Q. Then there exists g such that p(g) = q. Since preserves cusps, gk g = ak a for some a K. Therefore, a g N (K ) and q = p(a g) Q. Thus Q = Q. Let 2 K ={(α,α 2 ) 2 : α corresponds to subgroup K } and C ={α 2 K : e K B α }, where B α is defined as in the above theorem. We fix an element η 2 K and set ={g N (K ): η I g (C)}. We can choose a finite symmetric generating set S of such that p(s) is a generating set of Q and S contains the generators of N (K ). Using the same argument as in the above theorem, by replacing with N (K ), we get a quasiisometric section s: Q N (K ) satisfying: R d Q(q, q ) ɛ d N (K )(s(q), s(q )) Rd Q (q, q ) + ɛ. for some constants R, ɛ 0 and for all q,q Q. Since d Q (q, q ) d (s(q), s(q )) d N (K )(s(q), s(q )), we can take the quasiisometric section s: Q N (K ) such that R d Q(q, q ) ɛ d (s(q), s(q )) Rd Q (q, q ) + ɛ. Now, let Ɣ pel denote the space obtained from Ɣ by coning left cosets of K in. Since is weakly hyperbolic with respect to K, Ɣ pel is hyperbolic. We will prove that Q is hyperbolic. The quasi-isometric section s: Q N (K )( ) will induce a map ŝ: Q Ɣ pel. Now, for all q,q Q, d pel(ŝ(q),ŝ(q )) d (s(q), s(q )) Rd Q (q, q ) + ɛ, where d pel is the metric on Ɣ pel. Obviously, d Q(q, q ) d pel(ŝ(q),ŝ(q )). Hence ŝ is a quasiisometric section from Q to Ɣ pel. Therefore, s(q) is quasiconvex in Ɣpel. Since Ɣpel is hyperbolic, Q is hyperbolic. 3. Existence of Cannon Thurston maps i Consider the inclusion between pairs of relatively hyperbolic groups (H, H ) (, ). i will induce a proper embedding i: Ɣ H Ɣ. Let X = Ɣ and Y = Ɣ H. Recall that X h is the space obtained from X by gluing the hyperbolic cones. Inclusion of a horospherelike set in its hyperbolic cone is uniformly proper. Therefore, the inclusion of X in X h is uniformly proper, i.e., for all M > 0 and x,y X, there exists N > 0 such that d (x, y) N whenever d Xh (x, y) M, where d is the word metric corresponding to. Since preserves cusps, i will induce a proper embedding i h : Y h X h. DEFINITION 3. A Cannon Thurston map for i: (Ɣ H, H H ) (Ɣ, H ) is said to exist if there exists a continuous extension i h : Y h Y h X h X h of i h : Y h X h. To prove the existence of a Cannon Thurston map for the inclusion i: (K, K ) (, N (K )), we need the notion of partial electrocution.

8 64 Abhijit Pal DEFINITION 3.2 [] (Partial electrocution) Let (X, H,, L) be an ordered quadruple such that the following holds:. X is a proper geodesic metric space, H is a collection of subsets H α of X, and X is hyperbolic relative to H, i.e., the space X h, obtained from X by gluing H α [0, ) to each H α with the metric d h satisfying properties and 2 above, is hyperbolic. 2. There exists δ>0 such that L is a collection of δ-hyperbolic metric spaces L α and is a collection of (uniformly) coarse Lipschitz maps g α : H α L α. Note that the indexing set for H α,l α,g α is common. The partially electrocuted space or partially coned off space corresponding to (X, H,, L) is obtained from X by gluing in the (metric) mapping cylinders for the maps g α : H α L α. Lemma 3.3 []. iven K,ɛ 0, there exists C>0 such that the following holds: Let γ pel and γ denote respectively a (K, ɛ) partially electrocuted quasigeodesic in (X, d pel ) and a hyperbolic (K, ɛ)-quasigeodesic in (X h,d h ) joining a,b. Then γ X lies in a (hyperbolic) C-neighborhood of (any representative of) γ pel. Further, outside of a C-neighborhood of the horoballs that γ meets, γ and γ pel track each other. Let be a group strongly hyperbolic relative to the subgroup. Let X = Ɣ be the Cayley graph of and X h be the complete hyperbolic space obtained from X. We describe a special type of quasigeodesic in X h which will be essential for our purpose: DEFINITION 3.4 [8] We start with an electric quasi-geodesic ˆλ in the electric space ˆX without backtracking. For any horosphere-like set H penetrated by ˆλ, let x H and y H be the respective entry and exit points to H. We join x H and y H by a hyperbolic geodesic segment in H [0, ). This results a path, say λ,inx h. The path λ will be called an electro-ambient path. Lemma 3.5 [8]. The Hausdorff distance between any subsegment λ st of the electroambient path λ and the geodesic in X h joining the end points of λ st is bounded by some constant L, where L depends only upon the hyperbolicity constant of X h. For the rest of the paper, we will work with the following pair of short exact sequence of finitely generated groups: (K, K ) (, N (K )) p (Q, Q ). Since all the above groups are finitely generated, we can choose a finite symmetric generating set S of such that S contains generators of K,K,N (K ) and p(s) is also a finite generating set of Q. We will assume the hypotheses of Corollary 2.. As a consequence, Q = Q and there exists an (R, ɛ) quasi-isometric section s: Q N (K ) such that R d Q(q, q ) ɛ d (s(q), s(q )) Rd Q (q, q ) + ɛ for all q,q Q. Further, we assume that is strongly hyperbolic relative to the subgroup N (K ). Also, using a left translation L k by an element k K, we can assume that s(q) contains the identity element e K of

9 Relatively hyperbolic extensions of groups 65 K and s(q) N (K ). We have assumed that is weakly hyperbolic relative to the subgroup K and hence the coned-off space Ɣ pel obtained by coning left cosets gk of K to a point v(gk ) is hyperbolic. As Ɣ Q is quasi-isometrically embedded in Ɣ pel, Q is hyperbolic. We have also assumed that is strongly hyperbolic relative to N (K ). Thus Ɣ pel becomes a partially electrocuted space obtained from Ɣ by partially electrocuting the closed sets (horosphere-like sets) H gn (K ) in Ɣ corresponding to the left cosets gn (K ) to the hyperbolic space g(s(q)), where g(s(q)) denotes the image of s(q) under the left translation L g for g. Let λ b = ˆλ\H K denote the portions of ˆλ that do not penetrate horosphere-like sets in H K. The following lemma gives a sufficient condition for the existence of a Cannon Thurston map for the inclusion i: (Ɣ K, H K ) (Ɣ, H ). Lemma 3.6.[0]. A Cannon Thurston map for i: (Ɣ K, H K ) (Ɣ, H ) exists if there exists a non-negative function M(N)with M(N) as N such that the following holds: iven y 0 Ɣ K, and an electric quasigeodesic segment ˆλ in ˆƔ K, if λ b = ˆλ\H K lies outside an N-ball around y 0 Ɣ K then, for any partially electrocuted quasigeodesic β pel in Ɣ pel joining end points of ˆλ, β b = β pel \H lies outside an M(N)-ball around i(y 0 ) in Ɣ. 3. Construction of quasiconvex sets and retraction map Recall that for g, L g : denotes the left translation by g and I g : K K denotes the automorphism I g (k) = gkg. Let φ g = I g then φ g (a) = g ag. Since L g is an isometry, L g preserves distance between left cosets of in. Hence L g induces an isometry. The embedding i: Ɣ K Ɣ will induce an embedding î: ˆƔ K Ɣ pel ˆL g : Ɣ pel Ɣpel. The embedding i: Ɣ K Ɣ will induce an embedding î: ˆƔ K Ɣ pel. Let ˆλ be an electric geodesic segment in ˆƔ K with end points a and b in Ɣ K. Let ˆλ g be an electric geodesic in ˆƔ K joining φ g (a) and φ g (b). Define Bˆλ = g s(q) ˆL g.î(ˆλ g ). On ˆƔ K, define a map πˆλ g : ˆƔ K ˆλ g taking k ˆƔ K to one of the points on ˆλ g closest to k in the metric d ˆK. Lemma 3.7 [7]. For πˆλ g defined above, d ˆK (πˆλ g (k), πˆλ g (k )) Cd ˆK (k, k ) + C for all k, k ˆƔ K, where C depends only on the hyperbolic constant of ˆƔ K. DEFINITION 3.8 (Retraction map) Define ˆλ : Ɣpel Bˆλ as follows: Let x Ɣ pel. Then there exists a unique g s(q) such that L g (î(k)) = x for some unique k K, define ˆλ (x) = L g (î(ˆπˆλ g (k))). ˆλ will be called a retraction map..

10 66 Abhijit Pal Theorem 3.9 [6, 7]. There exists C 0 > 0 such that dĝ( ˆλ (g), ˆλ (g )) C 0 dĝ(g, g ) + C 0 for all g, g Ɣ pel. In particular, if Ɣpel is hyperbolic then Bˆλ is uniformly (independent of ˆλ) quasiconvex. 3.2 Proof of Theorem 3. Since i: Ɣ K Ɣ is an embedding, we identify k K with its image i(k). Let ˆμ g = ˆL g (ˆλ g ), where g s(q). μ b g =ˆμ g\h. B λ b = g s(q) μb g. Y = Ɣ K and X = Ɣ. Lemma 3.0. There exists A>0such that for all x μ b g B λ b Bˆλ, if λb lies outside B N (p) for a fixed reference point p Ɣ K then x lies outside an f(n) A+ ball about p in Ɣ, where f(n) as N. Proof. Let x μ b g for some g s(q). Let γ be a geodesic path in Ɣ Q joining the identity element e Q of Ɣ Q and p(x) Ɣ Q. Order the vertices on γ so that we have a finite sequence e Q = q 0,q,...,q n = p(x) = p(g) such that d Q (q i,q i+ ) = and d Q (e Q, p(x)) = n. Since s is a quasi-isometric section, this gives a sequence s(q i ) = g i such that d (g i,g i+ ) R + ɛ = R (say). Observe that g n = g and g 0 = e. Let B R (e ) be a closed ball around e of radius R. Then B R (e ) is finite. Now for each g, the outer automorphism φ g is a quasi-isometry. Thus there exists K and ɛ 0 such that for all g B R (e ), φ g is a (K, ɛ) quasi-isometry and K,ɛ are independent of elements of. Let s i = g i+ g i, then s i B R (e ), where i = 0,...,n. Hence φ si is a (K, ɛ) quasi-isometry. Therefore, φ si will induce a ( ˆK, ˆɛ) quasi-isometry ˆφ si from ˆƔ K to ˆƔ K, where ˆK, ˆɛ depends only on K and ɛ. Now x μ b g n and L g preserves distance between left cosets for all g. Hence there exists x λ b g n such that x = L gn (x ). Let [p, q] gn λ b g n be the connected portion of λ b g n on which x lies. Since ˆφ sn is a quasi-isometry, ˆφ sn ([p, q] gn ) will be an electric quasigeodesic lying outside horospherelike sets and hence it is a quasigeodesic in Y h lying at a uniformly bounded distance C from λ h g n in Y h (and hence in X h ), where λ h g n is the electroambient representative of ˆλ gn and Y h,x h are respectively the complete hyperbolic metric spaces corresponding to Y, X. Thus there exists x 2 λ h g n such that d Xh (φ sn (x ), x 2 ) C. But x 2 may lie inside horoball-like set penetrated by ˆλ gn. Due to bounded coset (horosphere) penetration properties, there exists y λ b g n such that d Xh (x 2,y) D. Thus d Xh (φ sn (x ), y) C + D. Since X = Ɣ is properly embedded in X h, there exists M>0 depending only upon C,D such that d (φ sn (x ), y) M. Hence d (L gn (φ sn (x )), L gn (y)) = d (φ sn (x ), y) M and L gn (y) μ b g n.

11 Relatively hyperbolic extensions of groups 67 Let z = L gn (y). Then d (x, z) d (x, L gn (φ sn (x ))) + d (L gn (φ sn (x )), L gn (y)) d (x, xs n ) + M R + M = A (say). Thus, we have shown that, for x μ b g n there exists z μ b g n such that d (x, z) A. Proceeding in this way, for each y μ b g i there exists y μ b g i such that d (y, y ) A. Hence there exists x λ b such that d (x, x ) An. Since Ɣ K is properly embedded in Ɣ, there exists f(n)such that λ b lies outside an f(n)-ball about p in Ɣ and f(n) as N. Therefore, d (x,p) f(n). Thus d (x, p) f(n) d (x, x ) f(n) An. Also d (x, p) n. Therefore, d (x, p) f(n) f(n) A+, that is, x lies outside the A+ -ball about p in Ɣ. Theorem 3.. Consider a short exact sequence of pairs of finitely generated groups (K, K ) (, N (K )) p (Q, Q ) with K strongly hyperbolic relative to the non-trivial proper subgroup K.If preserves cusps, strongly hyperbolic relative to N (K ) and weakly hyperbolic relative to the subgroup K, then there exists a Cannon Thurston map for the embedding i: Ɣ K Ɣ, where Ɣ K and Ɣ are Cayley graphs of K and respectively. Proof. It suffices to prove the condition of Lemma 3.6. So, for a fixed reference point p Ɣ K, we assume that ˆλ is an electric geodesic segment in ˆƔ K such that λ b ( Ɣ K ) lies outside an N-ball B N (p) around p. Let β pel be a quasigeodesic in the partially electrocuted space Ɣ pel be a nearest point projection from Ɣ pel then β pel is a quasigeodesic in Ɣpel joining the end points of ˆλ. Let Prˆλ onto the quasiconvex set Bˆλ. Let β pel = Prˆλ (β pel), lying on Bˆλ.Soβ pel lies in a P -neighborhood of β pel in Ɣ pel. But β pel might backtrack. β pel can be modified to form a quasigeodesic γ pel in Ɣ pel of the same type (i.e., lying in a P -neighborhood of β pel ) without backtracking with end points remaining the same. By Lemma 3.3, β pel and γ pel satisfy bounded coset (horosphere) penetration properties with the closed sets (horosphere-like sets) in Ɣ corresponding to the left cosets of N (K ) in. Thus if γ pel penetrates a horosphere-like set C gn (K ) corresponding to the left coset gn (K ) of N (K ) in and β pel does not, then the length of the geodesic traversed by γ h, where γ h is the electroambient path representative of γ pel, inside C gn (K ) [0, ) is uniformly bounded. Let C ={C gn (K ): g }. Thus there exists C 0 such that if x βpel b = β pel\c, then there exists y γpel b = γ pel\c such that d (x, y) C. Since y γpel b B λ b, by Lemma 3.0, d (y, p) f(n) A+. Therefore, d (x, p) f(n) A+ C (= M(N), say) and M(N) as N. By Lemma 3.6, a Cannon Thurston map for i: Ɣ K Ɣ exists.

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