Contemporary Mathematics

On definitions of relatively hyperbolic groups

Inna Bumagin

Abstract. The purpose of this note is to provide a short alternate proof of the fact that [9, Question 1] has an affirmative answer. Our proof combined with the result of Szczepanski [9] shows that a group which is relatively hyperbolic in the sense of the definition of Gromov is relatively hyperbolic in the sense of the definition of Farb.

1. Definitions

The notion of a relatively hyperbolic group was introduced by Gromov [7] as a generalization of the concept of a (word) hyperbolic group.

1.1. The Gromov definition. Let X be a hyperbolic (in the Gromov sense) complete locally compact geodesic metric space. Let x ∈ X, suppose z is a point at infinity, and γ is a geodesic ray from x to z. By a horosphere through x with center z we mean the limit as t → ∞ of the sphere of radius t in X with center γ(t). A horosphere is the level surface of the horofunction h(x) corresponding to the ray γ. By the radius of a horosphere through x we mean the value h(x). A horoball is the interior of a horosphere.

Suppose a group G admits a properly discontinuous isometric action on X so that the quotient space Y = X/G is quasi-isometric to the union of k copies of [0, ∞) joined at zero. Assume that the action of G on X is free. Lift the rays in Y to the rays γ_{i}: [0, ∞) → X for i = 1, 2, …, k. Let H_{i} be the isotropy subgroup of γ_{i}(∞); assume that H_{i} preserves h_{i}. Assume that in X there exists a G-invariant system GB of disjoint horoballs, and the action of G on X \ GB is cocompact.

Then Gromov calls G hyperbolic relative to the subgroups H_{1}, …, H_{k}.

1.2. Definitions proposed by Farb.

Definition 1.1. [6](Relatively hyperbolic group in a weak sense) Let G be a finitely generated group, and let H be a finitely generated subgroup of G. Fix a set A of generators of G. In the Cayley graph Γ(G, A) add a vertex v(gH) for each left coset gH of H, and connect v(gH) with each x ∈ gH by an edge of length \frac{1}{2}. The obtained graph Γ̂ is called a coned-off graph of G with respect to H.

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1991 Mathematics Subject Classification. 20F67,20F65,05C25.

Key words and phrases. Relatively hyperbolic group, Gromov hyperbolic space, Cayley graph, bounded coset penetration.

The group G is called weakly hyperbolic relative to H if Γ̂ is a hyperbolic metric space.

Remark. The terminology in Definition 1.1 was part of what was suggested by Bowditch in [1]. In [6] and [9] a group G that satisfies Definition 1.1 is termed simply “hyperbolic relative to H”.

Theorem 1.2. [9, Theorem 1] Let G be a finitely generated group, and let H_{1}, …, H_{r} be a finite set of finitely generated subgroups of G. If G is hyperbolic relative to H_{1}, …, H_{r} in the sense of Gromov’s definition, then G is weakly hyperbolic relative to H_{1}, …, H_{r}.

Furthermore, in [9, Example 3] Szczepanski shows that the class of weakly relatively hyperbolic groups is strictly larger than the class of groups relatively hyperbolic in the sense of Gromov’s definition.

In [6] Farb shows solvability of the word problem for weakly relatively hyperbolic groups that have a property which he calls the Bounded Coset Penetration property. The Bounded Coset Penetration property appears to be crucial for solvability of the conjugacy problem for relatively hyperbolic groups [2].

Definition 1.3. [6](Bounded Coset Penetration property) Let a group G be weakly hyperbolic relative to a finitely generated subgroup H. A path u in Γ is a relative P -quasigeodesic, if its projection û to Γ̂ is a P -quasigeodesic. The path u is a path without backtracking if u never returns to a subset which u penetrates. The pair (G, H) is said to satisfy the Bounded Coset Penetration (BCP) property if ∀P ≥ 1, there is a constant c = c(P) so that for every pair u, v of relative P -quasi-geodesics without backtracking, with same endpoints, the following conditions hold:

(1) If u penetrates a coset gH and v does not penetrate gH, then u travels a Γ-distance of at most c in gH.

(2) If both u and v penetrate a coset gH, then the vertices in Γ at which u and v first enter (last exit) gH lie a Γ-distance of at most c from each other.

Thus one considers the class of groups which can be defined as follows.

Definition 1.4. (Relatively hyperbolic group by Farb) Let G be a finitely generated group, and let H be a finitely generated subgroup of G. We say that G is hyperbolic relative to H in the sense of Farb, if G is weakly hyperbolic relative to H and the pair (G, H) has the BCP property.

In [9] Szczepanski asks the following question.

Question 1.5. [9, Question (1)] Let a group G be hyperbolic relative to a finitely generated subgroup H in the sense of Gromov’s definition. Does the pair (G, H) have the BCP property?

It is already known that Question 1.5 has an affirmative answer. In fact, results of Bowditch [1] and Dahmani [4],[5] imply that the class of groups that are relatively hyperbolic in the sense of Gromov’s definition and the class of groups that are relatively hyperbolic in the sense of Farb’s definition coincide. We provide a direct proof of the affirmative answer to Question 1.5. Combined with the proof of Theorem 1.2 by Szczepanski, this gives a short alternate proof of the following theorem.

Theorem 1.6. Let G be a finitely generated group, and let H be a finitely generated subgroup of G. If the group G is hyperbolic relative to H in the sense of the Gromov definition, then G is hyperbolic relative to H in the sense of the definition of Farb.

2. Quasiconvex subsets in a hyperbolic space

Our approach is to generalize the arguments that Farb used to prove [6, Theorem 4.11] (cf. [8, Theorem 5.10]).

Notation 2.1. Let (X, d) be a δ-hyperbolic metric space (δ ≥ 0) with a collection Σ of closed disjoint ϵ-quasiconvex subsets. We assume that the distance between any two subsets S_{1}, S_{2} ∈ Σ is bounded below by a constant R > 24δ + 4ϵ. We obtain a space X_{S} by deleting the interiors of all of the sets in Σ. The boundary of X_{S} consists of disjoint connected components, each component is the boundary of a set S ∈ Σ. We give X_{S} the path metric d_{S}. Next, we obtain the quotient X̂ of X_{S} by identifying points which lie in the same boundary component of X_{S}. Hence X̂ is equipped with a path pseudometric d̂ induced from the path metric d_{S}.

Lemma 2.2. [9, Proposition 1]. The space (X̂, d̂) is hyperbolic in the Gromov sense.

Definition 2.3. (Projections onto quasiconvex sets) Fix a subset S ∈ Σ. Let x and y be two points in X, and let P r(x) and P r(y) denote the projections of these points onto S. Let α be a path in X with the initial point i(α) = x and the terminal point t(α) = y. By the projection P r(α, S) of α onto S we mean an X-geodesic connecting P r(x) and P r(y), and by the X-length l_{X}(P r(α, S)) of the projection P r(α, S) we mean the X-length of that geodesic:

l_{X}(P r(α, S)) = d(P r(x), P r(y)).

Given S_{1} ∈ Σ, by the length of the projection of S_{1} onto S we mean the maximum length of the projection of a path in S_{1} onto S:

l_{X}(P r(S_{1}, S)) = max{l_{X}(P r(α, S)) | α ⊂ S_{1}}.

Notation 2.4. Let γ be a path in X, and let l_{X}(γ) be the X-length of γ. We denote by γ̂ the projection of γ into X̂, and by lX̂(γ̂) the X̂-length of γ̂. Obviously, lX̂(γ̂) ≤ l_{X}(γ). Also, given a path γ̂ in X̂, by γ we mean a path in X whose projection into X̂ is γ̂. Given γ̂, the path γ is not unique in general, but the following definition does not depend on particular choice of γ.

Definition 2.5. (Intersections with quasiconvex sets) We will say that γ̂ (or γ) intersects a subset S ∈ Σ, if γ intersects the boundary of S and travels a non-zero X-distance in the interior of S.

By [7, Lemma 7.3D] and [3, Chapter 10, Proposition 2.1], we have the inequality

(1)  d(P r(x), P r(y)) ≤ max(C, C + d(x, y) − d(x, P r(x)) − d(y, P r(y))),

where C = 2ϵ + 12δ. Moreover, in [9] Szczepanski shows the following. If the geodesic segment P r(α, S) does not intersect the 2δ-neighborhood of α in X, then

(2)  l_{X}(P r(α, S)) ≤ C.

Assume that each S ∈ Σ is a convex set. In the proof of Theorem 1.6 (Section 3 below) we use the fact that horoballs are convex sets. Let ∂S denote the boundary of a set S ∈ Σ. Observe that ∂S may be not convex (for instance, horospheres are not convex). For a set U ⊂ X, denote by Nb_{X}(U, λ) the λ-neighborhood of U in X.

Lemma 2.6. Let S ∈ Σ be a set, and let β be an X-geodesic that does not intersect Nb_{X}(S, 2δ). If x and y are the endpoints of β, then

d_{S}(P r(x), P r(y)) ≤ C + 16δ.

Proof. Let x_{s} = P r(x, S) and y_{s} = P r(y, S), so that [x, x_{s}] ∩ S = x_{s} and [y, y_{s}] ∩ S = y_{s}. By the inequality (2), d(x_{s}, y_{s}) ≤ C. Moreover, X-geodesic [x_{s}, y_{s}] stays 4δ-close to the union of the geodesics [x_{s}, x] ∪ β ∪ [y, y_{s}]. Hence, there exists a path in X \ S which joins x_{s} and y_{s}, with X-length bounded by C + 16δ, as claimed. □

Lemma 2.7. Let γ̂ be a P -quasigeodesic (P ≥ 1) in X̂. Assume that γ̂ does not intersect any subset S ∈ Σ. Given a subset S_{0} ∈ Σ, the projection P r(γ, S_{0}) of γ onto S_{0} has d_{S}-length at most D = D(P).

Proof. Let x and y denote the endpoints of γ̂, and let β be an X-geodesic joining x and y. Observe that γ = γ̂, so that l_{X}(γ) = lX̂(γ̂). Since d̂(x, y) ≤ d(x, y), γ is a P -quasigeodesic in X. If β does not intersect Nb_{X}(S, 2δ), then the claim follows from Lemma 2.6. In what follows, we assume that β intersects Nb_{X}(S, 2δ). Let z (or w) be the point where β first enters (or last exits) Nb_{X}(S, 2δ), and let z_{s} = P r(z, S_{0}) and w_{s} = P r(w, S_{0}). It remains to show that d_{S}(z_{s}, w_{s}) is bounded. We argue as follows.

The Hausdorff X-distance between β and γ is bounded by a constant N(P), hence there are points a, b ∈ γ so that d(a, z) ≤ N(P) and d(b, w) ≤ N(P). Denote by γ_{0} the subsegment of γ between a and b. Having projected γ to X̂, we have the following inequalities:

d̂(a, b) ≤ d(a, P r(a, S)) + d(b, P r(b, S)) ≤ 2(N(P) + 2δ), so that
lX̂(γ̂_{0}) ≤ 2P (N(P) + 2δ).

As l_{X}(γ_{0}) = lX̂(γ̂_{0}) and d_{S}(z_{s}, w_{s}) ≤ d(z_{s}, a) + l_{X}(γ_{0}) + d(b, w_{s}), we have that

(3)  d_{S}(z_{s}, w_{s}) ≤ 2(N(P) + 2δ) + 2P (N(P) + 2δ).

Since Lemma 2.6 applies to the segments [x, z] and [w, y], we conclude that the d_{S}-length of the projection P r(γ, S_{0}) of γ onto S_{0} is bounded by

(4)  D(P) = 2(C + 16δ) + 2(P + 1)(N(P) + 2δ)).
□

For each P -quasi-geodesic γ̂ in X̂, we are able to bound the d_{S}-length of the projection of γ̂ onto a quasiconvex set in terms of the length lX̂(γ̂).

Lemma 2.8. Fix a subset S_{0} ∈ Σ. Let P ≥ 1 and γ̂ be a P -quasigeodesic in X̂, which does not intersect S_{0}. Then the projection P r(γ, S_{0}) of γ onto S_{0} has d_{S}-length at most

(5)  l_{S}(P r(γ), S_{0}) ≤ E(γ̂, P) = (\frac{2lX̂(γ̂)}{R} + 3)(C + 16δ + D(P)).

Proof. Let S ∈ Σ be a subset different from S_{0}. Since S and S_{0} stay distance at least R apart, by Lemma 2.6, the projection of S onto S_{0} has d_{S}-length bounded by C + 16δ. Moreover, γ̂ intersects at most n = \frac{lX̂(γ̂)}{R} + 1 quasiconvex subsets from Σ, and has at most n + 2 subsegments which do not intersect any S ∈ Σ. By Lemma 2.7, the claim follows. □

Lemma 2.9. [9, Lemma 1] For any K ≥ 2δ+ϵ, there is a constant L = L(R, K) so that in the X̂-metric, whenever α̂ is an X̂-geodesic, α̂ stays (K + L/2)-close to an X-geodesic β with the same endpoints as α̂.

As a consequence of Lemma 2.8 and Lemma 2.9, we have a uniform bound on the projection length of an X̂-geodesic, which does not depend on the length of this geodesic.

Lemma 2.10. Let K ≥ 2δ+ϵ, and let α̂ be a X̂-geodesic so that α does not intersect the K-neighborhood Nb_{X}(S_{0}, K) of S_{0} in X. Then the d_{S}-length of P r(α, S_{0}) is bounded by a constant E_{1} = E_{1}(R, K, δ) which does not depend on the length of α̂.

Proof. Whenever a ∈ X is a point, by a_{s} we mean the projection of a onto S_{0}. Let x and y be the endpoints of α̂, and let β be an X-geodesic joining x and y. Observe that if β does not intersect Nb_{X}(S_{0}, 2δ), then by Lemma 2.6, d_{S}(x_{s}, y_{s}) ≤ C + 16δ. In what follows, we assume that β intersects Nb_{X}(S_{0}, 2δ). Let z (or w) be the point where β first enters (or last exits) Nb_{X}(S_{0}, 2δ). By Lemma 2.9, α̂ stays (K + L/2)-close to β in the X̂-metric. Therefore, there are points a, b ∈ α̂ so that d̂(a, z) ≤ K + L/2 and d̂(b, w) ≤ K + L/2. Hence,

(6)  lX̂(α̂_{0}) ≤ 2(K + L/2 + 2δ),

where α̂_{0} is the segment of α̂ joining a and b. By Lemma 2.8, we have that

(7)  d_{S}(a_{s}, b_{s}) ≤ E(α̂_{0}) ≤ E(2(K + L/2 + 2δ), 1).

Let α̂_{1} (or α̂_{2}) be the segment of α̂ joining x and a (or b and w). It remains to show that the projections of α̂_{1} and of α̂_{2} have bounded d_{S}-length. Let β_{i} be an X-geodesic joining the endpoints of α̂_{i}. As long as β_{1} intersects Nb_{X}(S_{0}, 2δ), we are able to find a point a_{1} in α̂_{1} which is (K + L/2 + 2δ)-close to S_{0}, so that the X̂-length of the segment of α̂ joining a_{1} and b satisfies the inequality (6) above. Therefore, without loss of generality, we can assume that neither β_{1} nor β_{2} intersects Nb_{X}(S_{0}, 2δ). Hence Lemma 2.6 applies to these segments, so that the length of the projection of α̂ is bounded as follows:

(8)  l_{S}(P r(α), S_{0}) ≤ E_{1} = E(2K + L + 4δ, 1) + 2(C + 16δ).

This upper bound does not depend on the length of α̂, as claimed. □

Lemma 2.11. (Bounded Subset Penetration) Let P ≥ 1 be a constant, and let ξ̂ and τ̂ be two P -quasigeodesics without backtracking in X̂, with common endpoints. Then there exists a constant B = B(δ, ϵ, P) such that the following holds.

(1) If ξ̂ intersects a subset S ∈ Σ and τ̂ does not intersect S, then the d_{S}-distance s between the points where ξ first enters and last exits S is bounded by B.

(2) If both ξ̂ and τ̂ intersect a subset S ∈ Σ, then the d_{S}-distance s between the points where ξ̂ and τ̂ first enter (or last exit) S is bounded by B.

Proof. Fix K = 2δ + ϵ. In the case (1), observe that s is bounded by the d_{S}-length of the projection of the path which is the concatenation of τ̂ and the two subsegments of ξ̂ that lie outside S. Let α̂ denote either of these P -quasigeodesic segments, and let β̂ be a X̂-geodesic with the same endpoints as α̂. Observe that if a subsegment β̂′ of β̂ does not intersect NbX̂(S, K), then β′ does not intersect Nb_{X}(S, K), hence by Lemma 2.10, we have that l_{S}(P r(β′)) ≤ E_{1}.

Next , assume that β̂ intersects Nb_{X}(S, K), but does not intersect S. Let x be the point where β̂ first enters NbX̂(K, S), and let y be the point where β̂ last exits NbX̂(S, K). Denote by β̂_{xy} the segment of β̂ joining x and y. Then lX̂(β̂_{xy}) ≤ d̂(x, y) ≤ 2K. Since R > 3K, the geodesic β̂_{xy} does not leave NbX̂(S, R), so that β̂_{xy} does not intersect any subset S′ ∈ Σ, and by Lemma 2.7, l_{S}(P r(β_{xy}, S)) ≤ D(1). Observe that β̂ is the concatenation of β̂_{xy} and other two (possibly, degenerate) segments that do not intersect NbX̂(S, K). Therefore, we have that l_{S}(P r(β̂, S)) ≤ D(1) + 2E_{1} in this case.

Finally, assume that β̂ intersects S. Let x_{s} (or y_{s}) be the point where β̂ first enters (or last exits) S. As X̂ is a hyperbolic space, α̂ and β̂ stay a bounded distance apart; let M(P) be a constant that bounds this distance. There are points z, w ∈ α̂ with d̂(z, x_{s}) ≤ M(P) and d̂(w, y_{s}) ≤ M(P). Let α̂_{zw} be the segment of α̂ between z and w, we have that lX̂(α̂_{zw}) ≤ 2P M(P). By Lemma 2.8,

l_{S}(P r(α_{zw}), S) ≤ E(α̂_{zw}) ≤ E(2P M(P)).

Let α̂_{1} (or α̂_{2}) be the segment of α̂ joining the initial point of α̂ and z (or w and the terminal point of α̂). Let β_{i} be an X-geodesic joining the endpoints of α̂_{i}. W.l.o.g. (cf. the proof of Lemma 2.10), we can assume that β_{1}, β_{2} do not intersect S. As we have shown, l_{S}(P r(β_{i}, S)) ≤ D(1) + 2E_{1}, for i = 1, 2.

Therefore, l_{S}(P r(α, S)) ≤ 2(D(1) + 2E_{1}) + E(2P M(P)), so that

(9)  s ≤ 3(2(D(1) + 2E_{1}) + E(2P M(P)))

which finishes the proof in the case (1).

Now, we prove (2). Let s be the X-distance between the points where ξ and τ first enter the subset S, and let ξ_{1} and τ_{1} be the initial segments of ξ and τ with ends at the points where these paths first enter S. Observe that s ≤ l_{S}(P r(ξ_{1}^{−1} ∘ τ_{1}), S). The arguments used in the proof of case (1) show that

s ≤ 2(2(D(1) + 2E_{1}) + E(2P M(P))).

Obviously, the distance between the points where ξ and τ last exit the subset S is bounded by 2(2(D(1) + 2E_{1}) + E(2P M(P))) as well.

Therefore, one can set B = 3(2(D(1) + 2E_{1}) + E(2P M(P))) to finish the proof of the lemma. □

3. Proof of Theorem 1.6

First, consider the case of one subgroup H so that G is hyperbolic relative to H. By [6, Corollary 3.2 ], we can assume that the generating set A of G contains a generating set A_{H} of H as a subset. We define a map f from the Cayley graph Γ(G, A) to X as follows. Lift the ray in the quotient space Y to a ray γ: [0, ∞) → X and choose a horoball S so that the boundary of S is the level surface of the horofunction which corresponds to the ray γ, and the images GS of S under the action of G form a G-invariant system of disjoint horoballs. We assume that the minimum distance between two horoballs is R. Notifying that a horoball is a convex subset of X, we define X′ to be the space obtained from X by deleting the interiors of all the horoballs in Σ = GS (cf. notation 2.1). Clearly, the action of G on X′ is cocompact.

Pick a point x on the boundary of S. Let g.x be the image of x under the action of g ∈ A. Hence, g.x ∈ g.S. Observe that the horoball g.S corresponds to the left coset gH in the following meaning. If g ∈ A \ A_{H}, then g.S ≠ S, and we join x and g.x by a segment which does not intersect any horoball. If g ∈ A_{H}, then g.S = S, and we join x and g.x by a segment which lies in S. Define a map f: Γ(G, A) → X as follows. Let id ∈ Γ(G, A) be the point that corresponds to 1_{G}. The map f sends id ∈ Γ(G, A) to the point x ∈ X and [id, g.id] to the segment in X that joins x and g.x, for each g ∈ A. Extend f to Γ(G, A) equivariantly. Since G acts on X by isometries, and the action of G on X̂ is cocompact, the map f induces a quasi-isometry f̂ between Γ̂ and X̂. Therefore, [9, Proposition 1] (cf. Lemma 2.2) implies that Γ̂ is a hyperbolic metric space i.e., G is weakly hyperbolic relative to H ([9, Theorem 1], cf. Theorem 1.2).

Furthermore, let ρ be a path in Γ(G, A), and let f(ρ) = β ∈ X. If ρ penetrates a coset gH, then β intersects the horosphere g.S; moreover, if the distance between the points where β first enters and last exits the boundary of g.S is bounded by l, then the Γ-distance that ρ travels in gH is bounded in terms of l and the constants of quasi-isometry f̂. Therefore, the pair (G, H) has the BCP property.

Now, assume that G is hyperbolic relative to H_{1}, …, H_{k}, in the sense of the Gromov definition. We assume that the generating set A of G contains a generating set A_{i} ⊂ A for H_{i}, for all i = 1, …, k. Lift the k rays in the quotient space Y to rays γ_{i}: [0, ∞) → X and choose horoballs S_{i} so that the boundary of S_{i} is the level surface of the horofunction which corresponds to the ray γ_{i}, and the images Σ = ⋃_{i=1}^{k} GS_{i} of S_{i} under the action of G form a G-invariant system of disjoint horoballs. Assume that the distance between two horoballs in Σ is bounded below by R. Pick a point x_{i} in S_{i}, for each i = 1, …, k, and connect each x_{j} to x_{1} by an edge e_{j}, for j = 2, …, k. The union of the edges e_{j} is a finite tree which we denote by T. Each generator h ∈ A_{i} (for some i) maps x_{i} ∈ S_{i} to another point y_{i} ∈ S_{i}, and x_{j} ∈ S_{j} (j ≠ i) to a point h.x_{i} ∈ h.S_{i} ≠ S_{i}, so that hT and T are disjoint. Obviously, T and g.T are disjoint for each generator g ∈ A \ ⋃_{i=1}^{k} A_{i}. We proceed as in the case of one subgroup and get a map from Γ̂ to X̂. Contract the tree T and all its images GT to a point, in order to see that f̂ is a quasi-isometry between Γ̂ and X̂. The same argument as above shows that the BCP property holds.

Acknowledgements

This paper was written during my stay at McGill University as a postdoctoral fellow. I am thankful to my supervisors O. Kharlampovich, A. Miasnikov and D. Wise for their warm hospitality. I am also thankful to François Dahmani for kindly submitting me his thesis, and to the referee for useful comments.

References

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Department of Mathematics and Statistics, McGill University, Montréal, Québec H3A 2K6, Canada

E-mail address: bumagin@math.mcgill.ca