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Quasi-hyperbolic planes in relatively hyperbolic groups

We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific embeddings we find remain quasi-isometric embeddings when composed with the inclusion map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to"almost every"peripheral (Dehn) filling. We apply our theorem to study the same question for fundamental groups of 3-manifolds. The key idea is to study quantitative geometric properties of the boundaries of relatively hyperbolic groups, such as linear connectedness. In particular, we prove a new existence result for quasi-arcs that avoid obstacles.


Introduction
A well known question of Gromov asks whether every (Gromov) hyperbolic group which is not virtually free contains a surface group. While this question is still open, its geometric analogue has a complete solution. Bonk and Kleiner [BK05], answering a question of Papasoglu, showed the following.
Theorem 1.1 (Bonk-Kleiner [BK05]). A hyperbolic group G contains a quasi-isometrically embedded copy of H 2 if and only if it is not virtually free.
In this paper, we study when a relatively hyperbolic group admits a quasi-isometrically embedded copy of H 2 . Our main result is the following.
Theorem 1.2. Let (G, P) be a finitely generated relatively hyperbolic group, where all peripheral subgroups are virtually nilpotent. Then there is a quasi-isometrically embedded copy of H 2 in G if and only if G does not split as a non-trivial graph of groups where the edge groups are finite and the vertex groups virtually nilpotent.
This result generalizes Theorem 1.1 since a finitely generated group is virtually free if and only if it admits a graph of groups decomposition with all vertex and edge groups finite.
For a hyperbolic group G, quasi-isometrically embedded copies of H 2 in G correspond to quasi-symmetrically embedded copies of the circle S 1 = ∂ ∞ H 2 in the boundary of the group. Bonk and Kleiner build such a circle when a hyperbolic group has connected boundary by observing that the boundary is doubling (there exists N so that any ball can be covered by N balls of half the radius) and linearly connected (there exists L so that any points x and y can be joined by a continuum of diameter at most Ld(x, y)). For such spaces, a theorem of Tukia applies to find quasi-symmetrically embedded arcs, or quasi-arcs [Tuk96].
We note that this proof relies on the local connectedness of boundaries of one ended hyperbolic groups, a deep result following from work of Bestvina and Mess, and Bowditch and Swarup [BM91,Proposition 3.3], [Bow98,Theorem 9.3], [Bow99b, Corollary 0.3], [Swa96].
Our strategy is similar to that of Bonk and Kleiner, but to implement this we have to prove several basic results regarding the geometry of the boundary of a relatively hyperbolic group, which we believe are of independent interest.
The model for the boundary that we use is due to Bowditch, who builds a model space X(G, P) by gluing horoballs into a Cayley graph for G, and setting ∂ ∞ (G, P) = ∂ ∞ X(G, P) [Bow99c] (see also [GM08]).
We fix a choice of X(G, P) and, for suitable conditions on the peripheral subgroups, we show that the boundary ∂ ∞ (G, P) has good geometric properties. For example, using work of Dahmani and Yaman, such boundaries will be doubling if and only if the peripheral subgroups are virtually nilpotent. We establish linear connectedness when the peripheral subgroups are one ended. (See Section 4 for precise statements.) At this point, by Tukia's theorem, we can find quasi-isometrically embedded copies of H 2 in X(G, P), but these can stray far away from G into horoballs in X(G, P). To prove our theorem we must ensure that this does not occur, and we do so by building a quasi-arc in the boundary that suitably avoids the parabolic points. This requires both additional geometric properties of the boundary (see Section 5) and the following generalisation of Tukia's theorem which builds quasi-arcs that avoid certain kinds of obstacles. (For definitions, see Section 6.) Theorem 6.2. Let (Z, ρ) be a compact, N -doubling and L-linearly connected metric space. Suppose V is a collection of compact subsets of Z with scale function D : V → (0, ∞).
Suppose V is L-separated, and each V ∈ V is both L-porous and Lavoidable on scales below D(V ). Then for a constant λ = λ(N, L) there exists a λ-quasi-arc γ in Z which satisfies diam(γ) ≥ 1 2 diam(Z), and ρ(γ, V ) ≥ 1 λ D(V ) for each V ∈ V. Our methods are able to find embeddings that avoid more subgroups than just virtually nilpotent peripheral groups. Theorem 1.3. Suppose both (G, P 1 ) and (G, P 1 ∪ P 2 ) are finitely generated relatively hyperbolic groups, where all peripheral subgroups in P 1 are virtually nilpotent and non-elementary, and all peripheral subgroups in P 2 are hyperbolic. Suppose G is one-ended and does not split over a subgroup of a conjugate of some P ∈ P 1 .
Finally, suppose that ∂ ∞ H ⊂ ∂ ∞ (G, P 1 ) does not locally disconnect the boundary, for any H ∈ P 2 (see Definition 4.2). Then there is a transversal quasi-isometric embedding of H 2 in G.
If both P 1 and P 2 are empty the group is hyperbolic and the result is a corollary of Theorem 1.1. If P 1 is empty, but P 2 is not, then the group is hyperbolic, but the quasi-isometric embeddings we find avoid the hyperbolic subgroups conjugate to those in P 2 .
Example 1.4. Let M be a compact hyperbolic 3-manifold with a single, totally geodesic surface as boundary ∂M . The fundamental group G = π 1 (M ) is hyperbolic, and also is hyperbolic relative to F = π 1 (∂M ) (see, for example, [Bel07,Proposition 13.1]).
The hypotheses of Theorem 1.3 are satisfied for P 1 = ∅ and P 2 = {F }, since ∂ ∞ G = ∂ ∞ (G, ∅) is a Sierpinski carpet, with the boundary of conjugates of F corresponding to the peripheral circles of the carpet. Thus, we find a transversal quasi-isometric embedding of H 2 into G.
Roughly speaking, a quasi-isometric embedding fails to be transversal if the image contains points which are far apart from each other, but both close to the same left coset of a peripheral subgroup (see Definition 2.7). The notion of transversality is interesting for us in view of the following results, which show that the embeddings constructed above persist both in the coned-off (or "electrified") graph of G, and in certain peripheral fillings of G. Proposition 2.10. LetΓ be the coned-off graph of a relatively hyperbolic group (G, P) and let c : Γ →Γ be the natural map. Suppose Z is a geodesic metric space. If a quasi-isometric embedding f : Z → G is transversal then c • f : Z →Γ is a quasi-isometric embedding.
Proposition 2.11. Let G be hyperbolic relative to P 1 , . . . P n (with a fixed system of generators) and let N i P i . Let Z be a geodesic metric space, and suppose that f : Z → G is a transversal quasi-isometric embedding. There exists R 0 , depending only on suitable data, so that if Recall that G/ {N i } is usually referred to as a peripheral (or Dehn) filling of G, and these are relatively hyperbolic (with peripheral groups {P i /N i }) under the hypotheses described above [Osi07,GM08] (see Section 2.3).
When combined, Theorem 1.3 and Proposition 2.11 provide interesting examples of relatively hyperbolic groups containing quasi-isometrically embedded copies of H 2 that do not have virtually nilpotent peripheral subgroups. A key point here is that Theorem 1.3 provides embeddings transversal to hyperbolic subgroups, and so one can find many interesting peripheral fillings. See Example 2.13 for details.
Using our results, we give a simple proof showing when the fundamental group of a closed, oriented 3-manifold contains a quasi-isometrically embedded copy of H 2 . Determining which 3-manifolds (virtually) contain immersed or embedded π 1 -injective surfaces [KM09, CLR97, Lac10, BS04] is a very difficult problem. The following theorem essentially follows from known results, in particular work of Masters and Zhang [MZ08,MZ11]. However, our proof is a simple consequence of Theorem 1.2 and the geometrisation theorem. Theorem 8.2. Let M be a closed 3-manifold. Then π 1 (M ) contains a quasi-isometrically embedded copy of H 2 if and only if M does not split as the connected sum of manifolds each with geometry S 3 , R 3 , S 2 × R or Nil.
Notice that the geometries mentioned above are exactly those that give virtually nilpotent fundamental groups.
As a final observation, we note that Leininger and Schleimer recently proved a result similar to Theorem 1.2 for Teichmüller spaces [LS11], using very different techniques.
1.1. Outline. In Section 2 we define relatively hyperbolic groups and their boundaries, and discuss transversality and its consequences. In Section 3 we give preliminary results linking the geometry of the boundary of a relatively hyperbolic group to that of its model space. Further results on the boundary itself are found in Sections 4 and 5, in particular, how sets can be connected (and avoided) in a controlled manner.
The existence of quasi-arcs that avoid obstacles is proved in Section 6. The proofs of Theorems 1.2 and 1.3 are given in Section 7. Finally, connections with 3-manifold groups are explored in Section 8.
1.2. Notation. The notation x C y (occasionally abbreviated to x y) signifies x ≥ y − C. Similarly, x C y signifies x ≤ y + C. If x C y and x C y we write x ≈ C y.
Throughout, C, C 1 , C 2 , etc., will refer to appropriately chosen constants. The notation C 3 = C 3 (C 1 , C 2 ) indicates that C 3 depends on the choices of C 1 and C 2 .
For a metric space (Z, d), the open ball with centre z ∈ Z and radius r > 0 is denoted by B r (z) or B(z, r). The closed ball with the same centre and radius is denoted by B r (z) = B(z, r). We write d(z, V ) for the infimal distance between a subset V ⊂ Z and a point z ∈ Z. The open neighbourhood of V ⊂ Z of radius r > 0 is the set The closed neighbourhood N r (V ) = N (V, r) is defined analogously.

Relatively hyperbolic groups and transversality
In this section we define relatively hyperbolic groups and their (Bowditch) boundaries. We introduce the notion of a transversal embedding, and show that such embeddings persist into the coned-off graph of a relatively hyperbolic group, or into suitable peripheral fillings of the same.
2.1. Basic definitions. There are many definitions of relatively hyperbolic groups. We give one here in terms of actions on a cusped space. First we define our model of a horoball. x, y ∈ Γ, identified with (x, 1), (y, 1) ∈ B(Γ), we have Proof. Any geodesic in B(Γ) will project to the image of a geodesic in Γ, so it suffices to check the bound in the hyperbolic plane, for points (0, 1) and (t, 1), with t ≥ 1. But then d B ((0, 1), (t, 1)) = arccosh(1+ t 2 2 ), and is bounded (by 1) for t ≥ 1.
Definition 2.3. Suppose G is a finitely generated group, and P = {P 1 , P 2 , . . . , P n } a collection of finitely generated subgroups of G. Let S be a finite generating set for G, so that S ∩ P i generates P i for each i = 1, . . . , n.
Let Γ(G) = Γ(G, S) be the Cayley graph of G with respect to S. Let X = X(G, P) = X(G, P, S) be the space resulting from gluing to Γ(G, S) a copy of B(Γ(P i , S ∩ P i )) to each coset gP i of P i , for each i = 1, . . . , n.
We say that (G, P) is relatively hyperbolic if X is Gromov hyperbolic, and call the members of P peripheral subgroups. This is equivalent to the other usual definitions of (strong) relative hyperbolicity; see [Bow99c], [GM08,Theorem 3.25]  Let O be the collection of all disjoint (open) horoballs in X, that is, the collection of all connected components of X \ Γ(G, S). Note that G acts properly and isometrically on X, cocompactly on X \ O∈O O , and the stabilizers of the sets O ∈ O are precisely the conjugates of the peripheral subgroups. Subgroup of conjugates of peripheral subgroups are called parabolic subgroups.
The boundary of (G, P) is the set ∂ ∞ (G, P) = ∂ ∞ X(G, P) = ∂ ∞ X. Choose a basepoint w ∈ X, and denote the Gromov product in X by (·|·) = (·|·) w ; this can be defined as (a|b) = inf{d(w, γ)}, where the infimum is taken over all (bi-infinite) geodesic lines γ from a to b; such a geodesic is denoted by (a, b).
Fix a visual metric ρ on ∂ ∞ X with parameters C 0 , > 0, that is for all a, b ∈ ∂ ∞ X, we have e − (a|b) /C 0 ≤ ρ(a, b) ≤ C 0 e − (a|b) . These choices of ρ, and C 0 are fixed for the rest of the paper.
We denote the (path) metric on X by d, while the word metric on G will be denoted by d G . Notice that d ≤ d G . For Horoballs can also be viewed as sub-level sets of Busemann functions. We will use this fact frequently in the remainder of this paper.
Definition 2.4. Given a point c ∈ ∂ ∞ X, and basepoint w ∈ X, let γ : [0, ∞) → X be a geodesic from w to c. The Busemann function corresponding to c and w is the function β c (·, w) : X → R defined by Note that this is well defined as it is the limit of a monotone function.
Proof. Note that −β a O (·, w) is a horofunction according to Bowditch's definition, and so the claim follows from the discussion before [Bow99c,Lemma 5.4].
Finally, we extend the distance estimate of Lemma 2.2 to X.
Lemma 2.6. Let (G, P) be relatively hyperbolic, and X = X(G, P) as above. There exists A = A(δ X ) < ∞ so that for any left coset gP of some P ∈ P, with metric d P , for any distinct x, y ∈ gP , we have d(x, y) ≈ A 2 log(d P (x, y)).
Proof. Follows from Lemma 2.2 and the hyperbolicity of X.

2.2.
Transversality and coned-off graphs. Our goal here is to define transversality of quasi-isometric embeddings, and show that a transversal quasi-isometric embedding of H 2 in Γ will persist if we coneoff the Cayley graph. We continue with the notation of Definition 2.3.
Definition 2.7. Let (G, P) be a relatively hyperbolic group. Let Z be a geodesic metric space. Given a function η : LetΓ be the coned-off graph of G and let c : Γ →Γ be the natural map. Essentially,Γ is the graph obtained by adding an edge of length one between any two members of the same coset of the same peripheral subgroup, see [Far98]. Recall that a λ-quasi-geodesic is a (λ, λ)-quasiisometric embedding of an interval.
Proof. By Lemma 2.8, whenever γ is a geodesic in Z, c • f (γ) is a quasi-geodesic with uniformly bounded constants inΓ.
Suppose N i P i are normal subgroups. The peripheral filling of G with respect to {N i } is defined as (See [Osi07] and [GM08].) Let p : G → G({N i }) be the quotient map.
Proposition 2.11. Let (G, P) be a relatively hyperbolic group, and G({N i }) a peripheral filling of G, as defined above. Let Z be a geodesic metric space, and suppose that f : Proof. We will use results from [DGO11] (see also [DG08,Cou11]). For any sufficiently large R 0 we can combine Propositions 7.6 and 5.20 in [DGO11] to show that G({N i }) acts on a certain δ -hyperbolic space Y (namely Y = X /Rot for X and Rot = (C, R) as in Proposition 7.6), with the following properties: (1) δ only depends on the hyperbolicity constant of X = X(G, P), (2) there is a 1-Lipschitz map ψ : G({N i }) → Y (this follows from the construction of Y and the fact that the inclusion of G in X is 1-Lipschitz), (3) there is a map φ : (N L (G) ⊆ X) → Y such that, for each g ∈ G, φ| B(g,L) is an isometry, where L = L(R 0 , X) → ∞ as R 0 → ∞.
Suppose that L as in (3) satisfies L ≥ C 1 + 8δ + 1. Then for each x ∈ α we have that φ| B(x,8δ +1) is an isometry, and so [BH99, . Let x 1 , x 2 be the endpoints of γ. Using (4) above, we see that On the other hand, recall that p is 1-Lipschitz, so As γ was arbitrary, we are done.
As discussed in the introduction, we can use Proposition 2.11 to find interesting examples of relatively hyperbolic groups with quasiisometrically embedded copies of H 2 , but whose peripheral groups are not virtually nilpotent. We note the following lemma.
Lemma 2.12. Let F 4 be the free group with four generators, and let R be fixed. Then there are normal subgroups {K α } α∈R of F 4 so that the quotient groups F 4 /K α are amenable but not virtually nilpotent, so that if α = β then F 4 /K α and F 4 /K β are not quasi-isometric, and so Proof. It is shown in [Gri84] that there is an uncountable family of 4-generated groups {F 4 /K α } α∈R of intermediate growth with distinct growth rates. In particular, these groups are amenable, not virtually nilpotent and pairwise non-quasi-isometric. To conclude the proof, let K be a finite index normal subgroup of F 4 so that K ∩ B R (e) = {e}, and let K α = K α ∩ K F 4 . As F 4 /K α is finite index in F 4 /K α , it inherits all the properties above.
Example 2.13. Let M be a closed hyperbolic 3-manifold so that G = π 1 (M ) is hyperbolic relative to a subgroup P ≤ G that is isomorphic to F 4 . For example, let M be a compact hyperbolic manifold whose boundary ∂M is a totally geodesic surface of genus 3, and let M be the double of M along ∂M . Observe that π 1 (M ) is hyperbolic relative to π 1 (∂M ), and π 1 (∂M ) is hyperbolic relative to a copy of π 1 (S ) = F 4 , where S ⊂ ∂M is a punctured genus 2 subsurface. Thus π 1 (M ) is hyperbolic relative to π 1 (S ).
Since G is hyperbolic with 2-sphere boundary, and P is quasi-convex in G with Cantor set boundary, the hypotheses of Theorem 1.3 are satisfied for P 1 = ∅ and P 2 = {P }. Therefore, we find a transversal quasi-isometric embedding of H 2 into G.
Let R 0 be chosen by Proposition 2.11. As P is quasi-convex in G, we choose R so that for x ∈ P , if d P (e, x) ≥ R then d G (e, x) ≥ R 0 . Now let {K α } be the subgroups constructed in Lemma 2.12. By Proposition 2.11, for each α ∈ R the peripheral filling G α = G/ K α is relatively hyperbolic and contains a quasi-isometrically embedded copy of H 2 .
As P/K α is non-virtually cyclic and amenable, it does not have a non-trivial relatively hyperbolic structure. Therefore, G α is not hyperbolic relative to virtually nilpotent subgroups, for in any peripheral structure P, some peripheral group H ∈ P must be quasi-isometric to P/K α by [BDM09, Theorem 4.8].
Finally, if α = β then G α is not quasi-isometric to G β by [BDM09, Theorem 4.8] as P/K α and P/K β are not quasi-isometric.

Separation of parabolic points and horoballs
In this section we study how the boundaries of peripheral subgroups are separated in ∂ ∞ X. We also establish a preliminary result on quasiisometrically embedding copies of H 2 .
3.1. Separation estimates. We begin with the following lemma.
Lemma 3.1. Let (G, P 1 ) and (G, P 1 ∪ P 2 ) be relatively hyperbolic groups, where all peripheral subgroups in P 2 are hyperbolic groups (P 2 is allowed to be empty), and set X = X(G, P 1 ). Let H denote the collection of the horoballs of X and the left cosets of the subgroups in P 2 ; more precisely, the images of those left cosets under the natural inclusion G → X.
Then the collection of subsets H has the following properties. ( (3) There exists R such that, given any H ∈ H and any geodesic ray γ connecting w to a ∈ ∂ ∞ H, the subray of γ whose starting point has distance d H from w is entirely contained in N R (H).
Proof. Claim (1) follows from the corresponding fact in the Cayley graph [DS05, Theorem 4.1(α 1 )]. Let us show (2). Uniform quasi-convexity of the horoballs is clear. If H is a left coset of a peripheral subgroup in P 2 , then it is quasiconvex in the Cayley graph Γ of G [DS05, Lemma 4.15]. What is more, by [DS05, Theorem 4.1(α 1 )], geodesics in Γ connecting points of H are transversal with respect to P 1 . Therefore, by Lemma 2.8 they are quasi-geodesics in X(G). We conclude that H is quasi-convex in X since pairs of points of H can be joined by quasi-geodesics (with uniformly bounded constants) which stay uniformly close to H.
We now show (3). As a ∈ ∂ ∞ H and H is quasi-convex, there exists C = C(X(G, P)) so that d(x, H) ≤ C for each x ∈ γ with d(x, w) large enough. Fix such an x.
Let y ∈ γ be such that d(y, w) = d H . Let p ∈ H be chosen so that d(w, p) ≤ d H + 1. By the defining property of y and the thinness of the geodesic triangle with vertices w, p, x, we have d(y, H) ≤ d(y, p) ≤ C (δ X ). The desired property now follows from (2).
From this lemma, we can deduce separation properties for the boundaries of sets in H.
Lemma 3.2. We make the assumptions of Lemma 3.1. Then there Conversely, we show that separation properties of certain points in the boundary ∂ ∞ X have implications for the intersection of sets in X.
Lemma 3.3. We make the assumptions of Lemma 3.1.
Let γ be a geodesic line connecting w to a ∈ ∂ ∞ X. Suppose that for some H ∈ H and r ≥ 1 we have d(a, ∂ ∞ H) ≥ e − d H /r. Then γ intersects N L (H) in a set of diameter bounded by K + 2L, for K = K(r, X) and any L ≥ 0.
In order to prove the lemma we will use [GdlH90, Lemma 2.12], which states that any finite configuration of points and geodesics in a δ X -hyperbolic space X is approximated by a tree, up to an additive error that depends only on δ X and the number of points and geodesics involved. So, we only need to show a result in the case of trees, and (up to controlled error) we will have obtained the same result for all hyperbolic spaces. In the notation of Figure 1, notice that β c (x, w) = d(x, p)−d(w, p) (the point p can be chosen to be the point on a geodesic from x to w such that d(w, p) = (c|x); this makes sense in all hyperbolic spaces, not just trees.) Proof of Lemma 3.3. We will treat the horoball case and the left coset case separately, beginning with the latter.
We can assume that H is not bounded, for otherwise the lemma is trivially true. Due to quasi-convexity, each point on H is at uniformly bounded distance from a geodesic line connecting points a 1 , a 2 in ∂ ∞ H, and thus also at uniformly bounded distance, say C 1 = C 1 (X), from either a ray connecting w to a 1 or a ray connecting w to a 2 .
Let c ∈ ∂ ∞ H and let γ be a ray connecting w to c. We have that (a|c) ≤ d H + r + C 2 , for r = log(r)/ and C 2 = C 2 (δ X , , C 0 ). There exists C 3 = C 3 (C 1 , X) so that any point x on γ such that d(x, w) ≥ (a|c) + L + C 3 has the property that d(x, γ ) ≥ L + C 1 . This applies to all rays connecting w to some c ∈ ∂ ∞ H, and so d( We are left to deal with the horoball case. Let c and γ be as above. Once again, (a|c) ≤ d H +r +C 2 , for r = log(r)/ . In an approximating tree for γ and γ (see Figure 1 Therefore, using Lemma 2.2 and [GdlH90, Lemma 2.12], one finds Arguing as before, one sees that, for 3.2. Embedded planes. We finish this section by noting that in order to find a quasi-isometrically embedded copy of H 2 in a relatively hyperbolic group, we need only to embed a quadrant of H 2 into our model space X, provided that the embedding does not go too far into the horoballs. (Compare with [BK05].) As we see later, this means that we do not need to embed a quasi-circle into the boundary of our group, merely a quasi-arc.
Let G, P 1 , P 2 , H and X = X(G, P 1 ) be as in Lemma 3.1.

Then,
(1) there exists a quasi-isometric embedding g : from a geodesic γ x from w to f (a x ). By Lemma 3.3 all points on γ x are C 2 = C 2 (r, X) close to a point in Γ(G), and therefore points in f (Q) are C 1 + C 2 close to points in Γ(G). Notice that Q contains balls {B n } of H 2 of arbitrarily large radius, each of which admits a (λ, λ)-quasi-isometric embedding f n : B n → X so that each point in f n (B n ) is a distance C 1 + C 2 close to a point in Γ(G). In particular, translating those embeddings appropriately using the action of G on X we can and do assume that the center of B n is mapped at uniformly bounded distance from w. As X is proper, we can use Arzelà-Ascoli to obtain the required (λ , λ )-quasiisometric embedding g as the limit of a subsequence of {f n } (more (2) By Lemma 3.3, f is transversal, so g and any map at finite distance from g will also be transversal. By (1) we can defineĝ : so it is enough to show that for some λ , Let γ be the geodesic in H 2 connecting x to y. Let γ be a geodesic in X fromĝ(x) toĝ(y), which is at Hausdorff distance at most C 4 = C 4 (λ , C 3 , δ X ) fromĝ(γ). In order to proceed we need the following.
Remark 3.6. There exists some increasing function θ such that Each maximal subpath β ⊆ γ contained in some H ∈ H has length at most C 5 = C 5 (C 4 ). Choose M ≥ θ(C 5 ) such that M ≥ sup{θ(x)/x : x ∈ (0, 1]}. We can substitute β by a subpath in Γ(G) of length at most M l(β), where l(β) is the length of β. Let α be the path in Γ(G) obtained from γ in this way.

Boundaries of relatively hyperbolic groups
In this section we study the geometry of the boundary of a relatively hyperbolic group, endowed with a visual metric ρ as in Section 2.
Before we begin, we summarize some known results about the topology of such boundaries.
Theorem 4.1 (Bowditch). Suppose (G, P) is a one-ended relatively hyperbolic group which does not split over a subgroup of a conjugate of some P ∈ P, and every group in P is finitely presented, one or two ended, and contains no infinite torsion subgroup. Then ∂ ∞ (G, P) is connected, locally connected and has no global cut points. Recall that a point p in a connected, locally connected topological space Z is not a local cut point if for every connected neighbourhood U of p, the set U \ {p} is also connected. If, in addition, Z is compact, then Z is locally path connected, so p is not a local cut point if and only if every neighbourhood U of p contains an open V with p ∈ V ⊂ U and V \ {p} path connected.
More generally, we have the following definition, used in the statement of Theorem 1.3.
Definition 4.2. A closed set V in a connected, locally connected topological space Z does not locally disconnect Z if for any p ∈ V and connected neighbourhood U of p, the set U \ V is also connected.
For relatively hyperbolic groups, we note the following. Proof. The lemma follows, similarly to the proof of [Dah05, Proposition 3.3], from the fact that the parabolic subgroup P corresponding to p acts properly discontinuously and cocompactly on ∂ ∞ (G, P)\{p}, which is connected and locally connected. Let us make this precise.
Choose an open set K 0 with compact closure in ∂ ∞ (G, P)\{p}, so that P K 0 = ∂ ∞ (G, P)\{p}. Then define K 1 = q∈P :d(q,e)≤1 qK 0 . As ∂ ∞ (G, P)\{p} is connected and locally path connected, and K 1 is compact, one can easily find an open, path connected K so that K 1 ⊂ K ⊂ K ⊂ ∂ ∞ (G, P)\{p}. Now suppose that P has one end. Let U be a neighbourhood of p. As P acts properly discontinuously on ∂ ∞ (G, P)\{p}, there exists R so that if d P (e, g) > R, then gK ⊂ U . Let Q be the unbounded connected component of P \ B(e, R). Then QK is path connected as for g, h ∈ P , if d P (g, h) ≤ 1, gK ∩hK = ∅. Finally, observe that V = QK ∪{p} ⊂ U is a neighbourhood of p so that V \ {p} = QK is connected.
Conversely, suppose that p is not a local cut-point. Let D be so that if qK ∩ rK = ∅ then d P (q, r) ≤ D. Suppose we are given R > 0. We can find a connected neighbourhood U of p in ∂ ∞ (G, P) so that U \ {p} is path connected and gK ∩ U = ∅ for all g ∈ B(e, R + D) ⊂ P . Let R ≥ R+D be chosen so that gK ∩U = ∅ for all g ∈ P \B(e, R ). Given g, h ∈ P \B(e, R ) we can find a path in U \{p} connecting gK to hK. So, there exists a sequence g = g 0 , g 1 , . . . , g n = h in P \ B(e, R + D) so that g i K ∩ g i+1 K = ∅ for all i = 0, . . . , n − 1. Thus as d P (g i , g i+1 ) ≤ D, we have that g and h can be connected in P outside B(e, R). As R was arbitrary, P is one ended. Every hyperbolic group has doubling boundary, but this is not the case for relatively hyperbolic groups. Recall that all relatively hyperbolic groups we consider are finitely generated, with P a finite collection of finitely generated peripheral groups.
Proof. By [DY05, Theorem 0.1], every peripheral subgroup is virtually nilpotent if and only if X = X(G, P) has bounded growth at all scales: for every 0 < r < R there exists some N so that every radius R ball in X can be covered by N balls of radius r.
If X has bounded growth at some scale then ∂ ∞ X is doubling [BS00, Theorem 9.2].
On the other hand, if ∂ ∞ X is doubling, then ∂ ∞ X quasi-symmetrically embeds into some S n−1 (see [Ass83], or [Hei01, Theorem 12.1]). Therefore, X quasi-isometrically embeds into some H n [BS00, Theorems 7.4, 8.2], which has bounded growth at all scales. We conclude that X has bounded growth at all scales (for small scales, the bounded growth of X follows from the finiteness of P, and the finite generation of G and all peripheral groups).

4.2.
Partial self-similarity. The boundary of a hyperbolic group G with a visual metric ρ is self-similar: there exists a constant L so that for any ball B(z, r) ⊂ ∂ ∞ G, with r ≤ diam(∂ ∞ G), there is a L-bi-Lipschitz map f from the rescaled ball (B(z, r), 1 Relatively hyperbolic groups do not have the same self-similarity. However, a kind of partial self-similarity holds that will be essential in this and the following section.
The following lemma follows [BK11, Proposition 3.3] closely, adapted to the situation where G does not act cocompactly on X.
Lemma 4.6. Suppose X is a δ X -hyperbolic, proper, geodesic metric space with base point w ∈ X. Let ρ be a visual metric on the boundary ∂ ∞ X with parameters C 0 , . Suppose a group G acts isometrically on X, with Gw the orbit of w.
For all z ∈ ∂ ∞ X and r ≤ diam(∂ ∞ X), let y ∈ [w, z) ⊂ X be the point, if it exists, with d(w, y) = − 1 log(2rC 0 )−δ X −1. Suppose we take y ∈ [w, y] so that d(Gw, y ) ≤ D, and set r = re d(y,y ) . Then there exists a L 0 -bi-Lipschitz map f from the rescaled ball (B(z, r ), 1 r ρ) to an open set U ⊂ ∂ ∞ G, so that B(f (z), 1 L 0 ) ⊂ U . (If such a y does not exist, the same conclusion holds upon setting r = r.) Proof. Up to changing the constant L 0 , we may assume that the point y exists, for if it does not, then r is bounded away from zero, so we may set f to be the identity map.

Linear Connectedness.
Under the hypotheses of Theorem 4.1, we saw that ∂ ∞ (G, P) was connected and locally connected. In this section we show that ∂ ∞ (G, P) will satisfy a quantitatively controlled version of this property.
Definition 4.9. We say a complete metric space (X, d) is L-linearly connected for some L ≥ 1 if for all x, y ∈ X there exists a compact, connected set J x, y of diameter less than or equal to Ld(x, y). We can assume that J is an arc (up to slightly increasing L).
For more discussion of this property, including the proof that we can assume J is an arc, see [Mac08, Page 3975].
Proposition 4.10. If (G, P) is relatively hyperbolic, ∂ ∞ (G, P) is connected and locally connected with no global cut points, then ∂ ∞ (G, P) is linearly connected.
If P is empty then G is hyperbolic, and this case is already known by work of Bonk   Proof. Let X = Γ(G) by a Cayley graph of G with visual metric ρ, and let L 0 be chosen by Lemma 4.6 for D = 1. The boundary of G is locally connected [BM91,Bow98,Bow99b,Swa96], so for every z ∈ ∂ ∞ G, we can find an open connected set V z satisfying z ∈ V z ⊂ B(z, 1 2L 0 ). The collection of all V z forms an open cover for the compact space ∂ ∞ G, and so this cover has a Lebesgue number α > 0.
Otherwise, by Lemma 4.6 (with y = y ), there exists an L 0 -bi-Lipschitz map f : (B(z, r) . Therefore f −1 (J) ⊂ B(z, r) joins z to z , and has diameter at most 2r = 4L 0 α ρ(z, z ). So ∂ ∞ G is 4L 0 /α-linearly connected. The key step in the proof of Proposition 4.10 is the construction of chains of points in the boundary. We defer the proof of this lemma.
Proof of Proposition 4.10. Given a, b ∈ ∂ ∞ (G, P), apply Lemma 4.12 to get a chain of points J 1 = {c 0 , . . . , c n }. For j ≥ 1, we define J j+1 iteratively by applying Lemma 4.12 to each pair of consecutive points in J j , and concatenating these chains of points together. Notice that diam(J j+1 ) ≤ diam(J j ) + 2K 1 2 j ρ(a, b). This implies that the diameter of J = J j is linearly bounded in ρ(a, b), and J is clearly compact and connected as desired.
We require two further lemmas before commencing the proof of Lemma 4.12. The first is an elementary lemma on the geometry of infinite groups.
Lemma 4.13. Let P be an infinite group (with a fixed, finite system of generators). Then for each p, q ∈ P there exists a geodesic ray α starting from p and such that d(q, α) ≥ d(p, q)/3.

QUASI-HYPERBOLIC PLANES IN RELATIVELY HYPERBOLIC GROUPS 21
The next lemma describes the geometry of geodesic rays passing through a horoball.
Lemma 4.14. Fix O ∈ O and a, b ∈ ∂ ∞ (G, P)\{a O }. Let γ a , γ b be geodesics from a O to a, b and let q a , q b be the last points in γ a ∩O, γ b ∩O, which we assume to be both non-empty. Also, let γ be a geodesic from w to a O and let q be the first point in γ ∩ O (so that d O ≈ d(w, q)). Then there exists E = E(X) < ∞ so that the following holds. ( ( (4) Let z be a point on O with d(z, q) ≥ E, let γ 0 be a geodesic ray starting at z and such that γ 0 ∩ O = {z} and let γ 1 be a geodesic ray from z to a O . Then the concatenation of γ 1 and γ 0 lies within distance E from a bi-infinite geodesic.
Proof. As in Lemma 3.3 we only need consider the case of trees, illustrated by Figure 2.
(2) The figure illustrates one of the cases to consider, the other being when q a , q b are between x and y.
In the other case we obtain a strict inequality, but also q a = q b .
(3) In this case γ a diverges from γ before q.
(4) In the corresponding situation in a tree, the said concatenation actually is a bi-infinite geodesic.
For each peripheral subgroup P ∈ P we denote by d P the path metric on any left coset of P .
Proof of Lemma 4.12. We need to find chains of points joining distinct points a, b ∈ ∂ ∞ (G, P), as described in the statement of the lemma.
Let p a,b ∈ [w, a) denote the centre of the quasi-tripod w, a, b, i.e. the point on [w, a) ⊂ X = X(G, P) such that d(w, p a,b ) = (a|b).
Let R = R(X) be a large constant to be determined by Case 1 below. All constants may depend tacitly on C 0 , , δ X . Case 1: We first assume that there exists O ∈ O such that p a,b ∈ O and d(p a,b , gP ) > R, where gP is the left coset of the peripheral subgroup P corresponding to O.
Suppose that b = a O . Let γ a be a geodesic from w to a and let q a be the last point in γ a ∩ O. Let γ be a geodesic from w to a O and let q be the first point in γ ∩ O. Notice that q, q a lie on gP , and by Lemma 4.14(1) By Lemma 4.13, there exists a geodesic ray α in gP starting at q a such that d P (q, α) ≥ d P (q a , q)/3. Therefore, by Lemma 2.6 and (4.15), (4.16) d(q, α) ≈ A 2 log(d P (α, q)) ≥ 2 log(d P (q a , q)/3) ≈ A+3 d(q a , q) ≥ 2R.
Let q a = q 0 , . . . , q n , . . . be the points of α ∩ gP . For each i consider the point c i ∈ ∂ ∞ (G, P) which is the endpoint of q i q −1 0 γ a . By Lemma 4.14(4) and (4.16), for R ≥ R 0 (E, A), a geodesic ray γ c i from w to c i will pass a point within E from q i , and hence by Lemma 4.14(3) we can assume that ( Using Lemma 4.14(1) and (4.16), there exists C 2 = C 2 (C 1 , A) so that And consequently there exists C 3 = C 3 (C 2 , E) so that for each i By Lemma 4.14(2), (4.15) and (4.16), we have for C 4 = C 4 (E, C 1 , A) Let γ a , q a be as above and let γ b , q b be defined analogously. As R ≥ 2E, by Lemma 4.14(3) we have (a|a O ), (b|a O ) ≥ d O . Using Lemma 4.14(1) and the approximate equality case of Lemma 4.14(2), (4.19) Let α be a geodesic in gP connecting q a to q b . Similarly to the previous case, let q a = q 0 , . . . , q n = q b be the points of α ∩ gP and consider the points c i ∈ ∂ ∞ (G, P) which are the endpoints of q i q −1 0 γ a . Notice that by Lemma 2.2 and (4.19), so for S = S(E, A) large enough, In particular, if S = S(E, A) is large enough we have (c i |a O ) ≥ d O for each i, by Lemma 4.14(3). From Lemma 4.14(2) we have, for C 5 = C 5 (E, C 1 ) giving the diameter bound ρ(a, c i ) ≤ C 6 ρ(a, b), for C 6 = C 6 (C 5 , E, A).
for R large enough, depending on C 7 and E.
Finally, if ρ(a, b) ≥ ρ(a, a O )/S or ρ(a, b) ≥ ρ(b, a O )/S, we can find chains connecting a to a O and a O to b and concatenate them. Case 2: We assume that d(p a,b , Γ(G)) < R.
Let L 0 > 1 and α > 0 be chosen as in the proof of Corollary 4.11 on setting D = R.
Let r 1 = 2 L 0 α ρ(a, b) and let y 1 ∈ [w, a) be chosen so that d(w, y 1 ) = − 1 log(2r 1 C 0 ) − δ X − 1. (We can assume, as in the proof of Corollary 4.11, that such a y 1 exists.) Let t 0 be a large constant to be determined later. Case 2a: If there exists y ∈ [w, y 1 ] so that d(y 1 , y ) ≤ 3t and d(y , Γ(G)) ≤ D, then we can apply Lemma 4.6 as in the proof of Corollary 4.11 to show that a and b can be joined by a set of diameter at most 4L 0 α e 3t ρ(a, b). Case 2b: If no such y exists, we are in the situation of Figure 3. Let y 2 ∈ [w, y 1 ) be chosen so that d(y 1 , y 2 ) = t and let O ∈ O be the horoball containing [y 2 , y 1 ], which corresponds to the coset gP . Let d O = d(w, O), and let p 1 ∈ gP be chosen so that d(p 1 , p a,b ) < R. (In the figure, p 1 = p a,b .) Let ρ 1 be a visual metric on ∂ ∞ (G, P) based at p 1 . We can assume that (∂ ∞ (G, P), ρ) and (∂ ∞ (G, P), ρ 1 ) are isometric, with the isometry induced by the action of p 1 . In the metric ρ 1 , we have that a, b and a O are points separated by at least δ 0 = δ 0 (R).
In our situation, δ 1 may be chosen to be independent of the choice of (finitely many) c = a O satisfying d(O, p 1 ) = 0, so δ 1 = δ 1 (δ 0 , ∂ ∞ (G, P)). Therefore, a and b can be joined by a compact arc J in (∂ ∞ (G, P), ρ 1 ) that does not enter B ρ 1 (a O , δ 1 ). So geodesic rays from p 1 to points in J are at least 2δ X from the geodesic ray [p a,b , a O ) outside the ball B(p 1 , t), for t = − 1 log(δ 1 ) + C 8 , for C 8 = C 8 (C 0 , , δ X ).
Translating this back into a statement about (∂ ∞ (G, P), ρ), we see that geodesics from w to points in J must branch from [w, p a,b ] after y 2 , that is, the set J lies in the ball B(a, C 9 e t d(a, b)), for C 9 = C 9 (C 0 ).
From these connected sets of controlled diameter, it is easy to extract chains of points satisfying the conditions of the lemma.

Avoidable sets in the boundary
In order to build a hyperbolic plane that avoids horoballs, we need to build an arc in the boundary that avoids parabolic points. In Theorem 1.3, we also wish to avoid the specified hyperbolic subgroups. We have topological conditions such as the no local cut points condition which help, but in this section we find more quantitative control.
Given p ∈ X, and 0 < r < R, the annulus A(p, r, R) is defined to be B(p, R) \ B(p, r). More generally, we have the following.
Definition 5.1. Given a set V in a metric space Z, and constants 0 < r < R < ∞, we define the annular neighbourhood If an arc passes through (or close to) a parabolic point in the boundary, we want to reroute it around that point. The following definition will be used frequently in the following two sections.
Definition 5.2 ([Mac08]). For any x and y in an embedded arc A, let A[x, y] be the closed, possibly trivial, subarc of A that lies between them.
An arc B ι-follows an arc A if there exists a (not necessarily continuous) map p : B → A, sending endpoints to endpoints, such that for all x, y ∈ B, B[x, y] is in the ι-neighbourhood of A[p(x), p(y)]; in particular, p displaces points at most ι.
We now define our notion of avoidable set, which is a quantitatively controlled version of the no local cut point and not locally disconnecting conditions. Definition 5.3. Suppose (X, d) is a complete, connected metric space. A set V ⊂ X is L-avoidable on scales below δ for L ≥ 1, δ ∈ (0, ∞] if for any r ∈ (0, δ/2L), whenever there is an arc I ⊂ X and points x, y ∈ I ∩ A(V, r, 2r) so that I[x, y] ⊂ N (V, 2r), there exists an arc J ⊂ A(V, r/L, 2rL) with endpoints x, y so that J (4rL)-follows I[x, y].
The goal of this section is the following proposition.
Proposition 5.4. Let (G, P 1 ) and (G, P 1 ∪ P 2 ) be relatively hyperbolic groups, where all groups in P 2 are proper infinite hyperbolic subgroups of G (P 2 may be empty), and all groups in P 1 are finitely presented and one ended. Let X = X(G, P 1 ), and let H be the collection of all horoballs of X and left cosets of the subgroups of P 2 . (As usual we regard G as a subspace of X.) Suppose that ∂ ∞ X is connected and locally connected, with no global cut points. Suppose that ∂ ∞ P does not locally disconnect ∂ ∞ X for each P ∈ P 2 . Then there exists L ≥ 1 so that for every H ∈ H, ∂ ∞ H ⊂ ∂ ∞ X is L-avoidable on scales below e − d(w,H) .
This proposition is proved in the following two subsections. 5.1. Avoiding parabolic points. We prove Proposition 5.4 in the case H is a horoball. This is the content of the following proposition.
Proposition 5.5. Suppose (G, P) is relatively hyperbolic, ∂ ∞ (G, P) is connected and locally connected with no global cut points, and all peripheral subgroups are one ended and finitely presented. Then there exists L ≥ 1 so that for any horoball The reason for restricting to this scale is that this is where the geometry of the boundary is determined by the geometry of the peripheral subgroup. Recall from Proposition 4.3 that such parabolic points are not local cut points.
The first step is the following simple lemma about finitely presented, one ended groups.
Lemma 5.6. Suppose P is a finitely generated, one ended group, given by a (finite) presentation where all relators have length at most M , and let Γ(P ) be its Cayley graph. Then any two points x, y ∈ Γ(P ) such that 2M ≤ r x ≤ r y , where r x = d(e, x) and r y = d(e, y), can be connected by an arc in A(e, r x /3, 3r y ) ⊂ Γ(P ).
Proof. By Lemma 4.13, we can find an infinite geodesic ray in Γ(P ) from x which does not pass through B(e, r x /3). Let x be the last point on this ray satisfying d(e, x ) = 2r y . Do the same for y, and let y denote the corresponding point. Note that x and y lie on the boundary of the unique unbounded component of {z ∈ Γ(P ) : d(e, z) ≥ 2r y }, which we denote by Z. We prove the lemma by finding a path from x to y contained in A(e, 2r y − M, 2r y + M ).
Let β 1 be a path joining x and y in Γ(P ) ∩ Z. We can assume that β 1 is an arc, and β 1 ∩ B(e, 2r y ) = {x , y }. Let p be the first point of [y , e] that meets [e, x ] in Γ(P ). Then the concatenation of β 1 , [y , p] and [p, x ] forms a simple, closed loop β 2 in Γ(P ).
As β 2 represents the identity in P , there exists a diagram D for β 2 : a connected, simply connected, planar 2-complex D together with a map of D into the Cayley complex Γ 2 (P ) sending cells to cells and ∂D to β 2 .
Let D ⊂ D be the union of closed faces B ⊂ D which have a point u ∈ ∂B with d(u, e) = 2r y . Let D be the connected component of x in D . Let γ ⊂ ∂D be the simple closed curve in R 2 bounding D .
If either β 1 or [y , p]∪[p, x ] live in A(p, 2r y −M, 2r y +M ), we are done. Otherwise, as we travel around γ from x , in one direction we must take a value > 2r y , and in the other a value < 2r y , thus there is a point v ∈ γ \ {x } with d(e, v) = 2r y . If v is in the interior of D, the adjacent faces are in D , giving a contradiction. So v ∈ β 2 , and v must be y . Thus there is a path from x to y in D ⊂ A(e, 2r y − M, 2r y + M ).
We can now prove the proposition.
Proof of Proposition 5.5. By Proposition 4.3, parabolic points in the boundary are not local cut points.
We claim that there exists an L ≥ 1 so that for any parabolic point a O , any r ≤ e − d(w,O) /2L, and any x, y ∈ A(a O , r, 2r), there exists an arc J ⊂ A(a O , r/L, 2rL) joining x to y.
This claim suffices to prove the proposition, because the 4rL-following property is automatic since diam (B(a O , 2rL)) ≤ 4rL.
Suppose the claim is false. Then there exist annuli A(a On , r n , 2r n ) ⊂ ∂ ∞ G centered on parabolic points associated to horoballs O n ⊂ X, which contain points a n , b n that cannot be joined in A(a On , r n /n, 2r n n).
Using Lemma 4.6, we may assume that d(w, O n ) ≤ C 1 (X), and further that every O n = O, for some horoball O. Moreover, we may assume that r n → 0.
Let gP g −1 be the corresponding parabolic subgroup, with p O ∈ gP a closest point to w. Fix some n ≥ N , where N is to be determined below, and let a = a n and b = b n , r = r n . Let q a , q b be the last points of (a O , a), (a O , b) contained in O. We will write x C y if the quantities with error C 2 = C 2 (C 1 , E), so by Lemma 2.6 we have, for C 3 = C 3 (C 2 , A), d gP (q a , p O ) C 3 r −1/ n , and the same for q b . Let r P be the smaller of d gP (q a , p O ), d gP (q b , p O ), and notice that their ratio is controlled by C 2 3 independent of n. By Lemma 5.6 there is a chain of points q a = q 0 , q 1 , . . . , q n = q b in gP joining q a to q b in gP so that, in the path metric on gP , q i ∈ A(p O , r P /3, 3r P C 2 3 ). We now follow part of the proof of Lemma 4.12, Case 1. Notice that by Lemma 4.14(4) each geodesic (a O , c i ) passes close to q i , where c i = q i q −1 0 a. Note that (q i |q i+1 ) C 4 (a|a O )+((a|a O )−d O ) ≈ C 4 2(a|a O ), with C 4 = C 4 (E, C 1 ). If we choose N so that r n ≤ r * , for r * = r * (C 3 ), we have (q i |a O ) ≈ C 5 (a|a O ) for C 5 = C 5 (δ X ) and all i, so ρ(c i , a O ) C 6 r n , for C 6 = C 6 (C 5 ). Moreover, for some C 7 = C 7 (C 4 , C 5 ). By Proposition 4.10, ∂ ∞ G is L-linearly connected. Since r n goes to zero as n → ∞, for n ≥ N = N (C 3 , C 6 ), we have ρ(c i , c i+1 ) ≤ ρ(c i , a O )/(2LC 6 ). Therefore we can join together the chain of points c 0 = a, c 1 , . . . , c n = b inside A(a O , r n /2C 6 , (C 6 + 1)r n ). This gives us a contradiction.

5.2.
Avoiding hyperbolic subgroups. In this section we complete the proof of Proposition 5.4 for H = gP , where P ∈ P 2 . By assumption, ∂ ∞ H ⊂ ∂ ∞ X does not locally disconnect ∂ ∞ X. First, a preliminary lemma.
Proof. If not, we can find a sequence of cosets H n = g n P n , points a n ∈ ∂ ∞ H n and scales r n so that N (∂ ∞ H n , r n /n) ⊃ B(a n , r n ). As in the proof of Proposition 5.5, we may assume that d(w, H n ) is bounded, using the action of G and Lemma 4.6, and therefore, after taking subsequences, that H n = H is constant. Likewise, using the action of g n P n g −1 n and Lemma 4.6, we can assume that r n is bounded away from zero.
Using compactness, we may also assume that a n converges to some a ∈ ∂ ∞ H. So, we have that a is in the interior of ∂ ∞ H ⊂ ∂ ∞ X, since ∂ ∞ H is closed in ∂ ∞ X. This is a contradiction because ∂ ∞ H is not all of ∂ ∞ G (proper peripheral subgroups of a relatively hyperbolic group are of infinite index), so if a is a point of ∂ ∞ H, one can use the action of H to find points in ∂ ∞ G \ ∂ ∞ H that are arbitrarily close to a.
We continue with the proof of Proposition 5.4. Suppose that the conclusion to the proposition is false. That is, there exists a sequence of configurations {(H n = g n P n , x n , y n , I n , r n )} n∈N where I n ⊂ N (∂ ∞ H n , 2r n ) ⊂ ∂ ∞ X is an arc with endpoints x n , y n ∈ A(∂ ∞ H n , r n , 2r n ), and r n ≤ e − d(w,Hn) /2n, but there is no arc J ⊂ A(∂ ∞ H n , r n /n, 2r n n) with endpoints x n and y n that 4r n n-follows I n [x n , y n ].
Note that, as in Lemma 5.7, we may assume that d(w, H n ) is bounded, and in fact that H n = H is constant, and that ρ(x n , y n ) is bounded away from zero. This is because we can use the action of G to shift H close to the origin, and the action of g n P n g −1 n to "zoom in" on x n and y n , both using Lemma 4.6.
We fix n, and build our desired arc J from x n to y n in stages. Let L 1 ≥ 2 be as in Lemma 5.7. We can find a chain of points x n = z 0 , z 1 , z 2 , . . . , z m = y n in A(∂ ∞ H n , r n /L 1 , 2r n ) so that for every i, ρ(z i , z i+1 ) ≤ 4r n , and so that {z i } 2r n -follows I n [x n , y n ]. (The definition of ι-follows is extended from arcs to chains in the obvious way.) This is done as follows: extract from I n [x n , y n ] a chain of points {z i } that 0-follows I n [x n , y n ] and so that ρ(z i , z i+1 ) ≤ r n . Then whenever ρ(z i , ∂ ∞ H) ≤ r n /L 1 , use Lemma 5.7 to find a point z i at most r n /L 1 +r n away from z i , and outside N (∂ ∞ H, r n /L 1 ). Otherwise let z i = z i . Adjacent points in this chain are at most r n + 2(r n /L 1 + r n ) ≤ 4r n separated, and the new chain 2r n -follows the previous chain.
We claim there exists L 2 = L 2 (L 1 , X), independent of n and H, so that any two points u, v ∈ A(∂ ∞ H, r n /L 1 , 2r n ) satisfying ρ(u, v) ≤ 4r n can be joined by an arc K ⊂ A(∂ ∞ H, r n /L 2 , 2L 2 r n ) of diameter at most 2L 2 r n .
In fact, the compactness of ∂ ∞ X and the fact that ∂ ∞ H ⊂ ∂ ∞ X does not locally disconnect ∂ ∞ X give the existence of a constant L 2 depending on n satisfying the requirement. An argument based on self-similarity of ∂ ∞ X around points of ∂ ∞ H, i.e. an application of Lemma 4.6 as in Corollary 4.11, lets us rescale r n to size at least C(X, H, L 0 ) > 0, and hence choose L 2 uniformly in n.
Using this, we find arcs J i joining each z i and z i+1 together in our chain. From this, we extract an arc J by cutting out loops: travel along J 0 until you meet J j for some j ≥ 1, and at that point cut out the rest of J 0 and all J k for 1 ≤ k < j. Concatenate the remainders of J 0 and J j together, and continue along J j .
The resulting arc J will 2L 2 r n -follow the chain {z i }, and so it will 4L 2 r n -follow I n [x n , y n ].
This provides a contradiction for large enough n.

Quasi-arcs that avoid obstacles
In this section we build quasi-arcs in a metric space that avoid specified obstacles.
Definition 6.1. Let (Z, ρ) be a compact metric space. Let V be a collection of compact subsets of Z provided with some map D : V → (0, ∞), which we call a scale function.
The (modified) relative distance function ∆ : As we saw in Section 5, we often only have control on topology on a sufficently small scale. The purpose of the scale function is to determine the size of the neighbourhood of each V ∈ V on which we have this control. An example of a scale function is D(V ) = diam(V ), if every V ∈ V has |V | > 1. In this case, ∆ is the usual relative distance.
The goal of this section is the following result.
Theorem 6.2. Let (Z, ρ) be a compact, N -doubling and L-linearly connected metric space. Suppose V is a collection of compact subsets of Z with scale function D : V → (0, ∞).
Suppose V is L-separated, and each V ∈ V is both L-porous and Lavoidable on scales below D(V ). Then for a constant λ = λ(N, L) there exists a λ-quasi-arc γ in Z which satisfies diam(γ) ≥ 1 2 diam(Z), and ρ(γ, V ) ≥ 1 λ D(V ) for each V ∈ V. Before continuing, we recall some results from [Mac08]. Recall that for any x and y in an embedded arc A, we denote by A[x, y] the closed, possibly trivial, subarc of A that lies between them. We say that an arc A in a doubling and complete metric space is an ι-local λ-quasi-arc if diam(A[x, y]) ≤ λd(x, y) for all x, y ∈ A such that d(x, y) ≤ ι. (See Definition 5.2 for the notion of ι-following.) Proposition 6.3 ([Mac08, Proposition 2.1]). Given a complete metric space (Z, ρ) that is L-linearly connected and N -doubling, there exist constants s = s(L, N ) > 0 and S = S(L, N ) > 0 with the following property: for each ι > 0 and each arc A ⊂ X, there exists an arc J that ι-follows A, has the same endpoints as A, and satisfies (6.4) ∀x, y ∈ J, ρ(x, y) < sι =⇒ diam(J[x, y]) < Sι.
Proof of Theorem 6.2. Let r = r(L, N ) > 0 be fixed sufficiently small as determined later in the proof. Let D 0 = sup{D(V ) : V ∈ V}; if V = ∅, set D 0 = diam(Z). Observe that as every V ∈ V is L-porous, we have D 0 ≤ L diam(Z). (We assume L ≥ 10. ) We filter V according to size. For i ∈ N, let V n = {V ∈ V : r n < D(V )/D 0 ≤ r n−1 } and C n (K) = {N (V, D 0 r n /K) : V ∈ V i }. As V is L-separated, for any K > 2L, each C n (K) consists of disjoint neighbourhoods.
We start with any arc J 0 in Z joining points separated by diam(Z), and build arcs J n in Z by induction on n.
Assume we have been given an arc J n−1 . We will modify J n−1 independently inside the (disjoint) sets in C n (4L). Let r n = D 0 r n 16L 2 , and observe that for any V ∈ V n , A(V, r n /L, 2r n /L) ⊂ A(V, r n /L 2 , 3r n L) ⊂ N (V, D 0 r n /4L) ∈ C n (4L).
First, we modify the arc J n−1 to ensure that its endpoints lie outside N (V, 2r n /L), for any V ∈ V n . If an endpoint lies within 2r n /L of V , use porosity to find a point in A(V, 2r n /L, 2r n ) within a distance of 2r n + 2r n /L of this endpoint, and connect these points inside N (V, 3r n L) using linear connectedness. Now perform suitable cancellation to obtain a new arc J n which 3r n L-follows J n−1 .
Second, given V ∈ V n , each time J n meets N (V, r n /L 2 ), the arc J n travels through A = A(V, r n /L, 2r n /L) both before and after meeting N (V, r n /L 2 ). For each such meeting, we use that V is L-avoidable with "r" equal to r n /L to find a detour path in A(V, r n /L 2 , 4r n ). After doing so, we concatenate the paths found into an arc J n , as at the end of the proof of Proposition 5.4. Note that J n will 4r n -follow J n . Now apply Proposition 6.3 to J n with ι = r n /2L 2 . Call the resulting arc J n : it ι-follows J n , so it (D 0 r n /4L)-follows J n−1 . Observe that J n avoids the neighbourhood of V of size r n /2L 2 = D 0 r n /32L 4 .
We now find the limit of the arcs J n . For every n, J n (D 0 r n /4L)follows J n−1 . Let s = s/8L 3 , where s is given by Proposition 6.3, then observe that J n satisfies (6.4) with ι = D 0 r n /4L, and where s is replaced by s .
We can assume that r ≤ min s 4+2S , 1 10 , where S is given by Proposition 6.3, since s and S depend only on L and N . Now apply Lemma 6.5 to the arcs J n to find a quasi-arc γ. Provided r ≤ 1 2 , the endpoints of γ are at most from those of J 0 , so diam(γ) ≥ 1 2 diam(Z). Finally, for each n, γ lies in a neighbourhood of J n of size at most where this last inequality holds for r ≤ 1/32L 3 . We conclude by observing that for any V ∈ V n ⊂ V, we have ρ(γ, V ) ≥ D 0 r n 64L 4 ≥ r 64L 4 D(V ).

Building hyperbolic planes
In this section we prove Theorems 1.2 and 1.3. The boundary ∂ ∞ (G, P 1 ) is N -doubling, for some N , by Proposition 4.5.
Theorem 4.1 implies that ∂ ∞ (G, P 1 ) is connected and locally connected, with no global cut points. By Proposition 4.10 ∂ ∞ (G, P 1 ) is L 2 -linearly connected for some L 2 ≥ 1, while by Proposition 5.4 there exists L 3 ≥ 1 so that for every H ∈ H, ∂ ∞ H is L 3 -avoidable on scales below e − d(w,H) . Let L 1 be the porosity constant set by Lemma 5.7; note that points in a connected space are automatically porous. Lemma 3.2 implies that V is L 4 -separated, for some L 4 .
We set L to be the maximum of L 1 , L 2 , L 3 and L 4 . We apply Theorem 6.2 to build a quasi-arc γ in ∂ ∞ (G, P 1 ).
Finally, the separation condition on γ and Proposition 3.5 give us a transversal, quasi-isometric embedding of H 2 in X(G, P).
It only remains to prove Theorem 1.2.
Proof of Theorem 1.2. The only if direction of the proof is elementary, for if G admits a graph of groups decomposition with finite edge groups and virtually nilpotent vertex groups, then any quasi-isometrically embedded copy of H 2 must map, up to finite distance, into (a left coset of) a vertex group, which is impossible.
We now reduce the if direction of Theorem 1.2 to a special case of Theorem 1.3. We can assume that P does not contain virtually cyclic subgroups as those can be removed from the list of peripheral subgroups preserving relative hyperbolicity. As the peripheral subgroups are finitely presented, G is finitely presented as well (see for example [Osi06]), therefore it admits a graph of groups decomposition where all edge groups are finite and all vertex groups have at most one end [Dun85, Theorem 5.1]. As all edge groups are finite, the vertex groups are undistorted and therefore they are themselves relatively hyperbolic with the obvious peripheral structure [DS05, Theorem 1.8]. By the assumptions of the theorem, one of the vertex groups, say G , is properly relatively hyperbolic, that is the peripheral subgroups are proper (virtually nilpotent) subgroups.
Bowditch shows [Bow01, Theorem 1.4] that G admits a maximal peripheral splitting, and so we can find a vertex group G of this splitting so that G does not admit a splitting over parabolic subgroups. Such a vertex group is one ended, hyperbolic relative to one-ended virtually nilpotent subgroups P (we can remove all two-ended ones, if any). Therefore, all hypotheses of Theorem 1.3 are fulfilled for G 3 with P 1 = P and P 2 = ∅. Unfortunately G may be distorted in G , but the statement of Theorem 1.3 gives the existence of a transversal quasi-isometric embedding of H 2 . As the coned-off graph of a vertex group isometrically embeds (given a suitable choice of generators) in the coned-off graph of the whole group, an easy argument based on Proposition 2.10 shows that a transversal quasi-isometric embedding of H 2 in a vertex group gives a quasi-isometric embedding in the whole group.

Application to 3-manifolds
In this final section, we consider which 3-manifold groups contain a quasi-isometrically embedded copy of H 2 .
Lemma 8.1. Let M be a closed graph manifold. Then π 1 (M ) contains a quasi-isometrically embedded copy of H 2 .
Proof. All fundamental groups of closed graph manifolds are quasiisometric [BN08, Theorem 2.1], so we can choose M . Consider a splitting of the closed genus 2 surface S into an annulus A and a twicepunctured torus S , as in Figure 4 below. Let M be obtained from two copies M 1 , M 2 of S × S 1 , in such a way that the gluings interchange the S 1 factor and the surface factor. Now, embed S in the following way. Let γ be path connecting the boundary components of S depicted in Figure 4, and embed A in M 1 as γ × S 1 . We can assume, up to changing the gluings, that there exists p such that A ∩ M 2 ⊆ S × {p}. Embed S in M 2 as S × {p}. Now, we claim that S is a retract of M . If this is true then first of all π 1 (S) injects in π 1 (M ), and this gives us a map f : H 2 → M .