New York Journal of Mathematics

New York J. Math. 1 (1995) 184–195.

A Non-quasiconvex Subgroup of a Hyperbolic Group with an Exotic Limit Set

Ilya Kapovich

Abstract. We construct an example of a torsion free freely indecomposable finitely presented non-quasiconvex subgroup H of a word hyperbolic group G such that the limit set of H is not the limit set of a quasiconvex subgroup of G. In particular, this gives a counterexample to the conjecture of G. Swarup that a finitely presented one-ended subgroup of a word hyperbolic group is quasiconvex if and only if it has finite index in its virtual normalizer.

Contents

1. Introduction 184

2. Some Definitions and Notations 185

3. The Proofs 187 References 195

1. Introduction

A subgroup H of a word hyperbolic group G is quasiconvex (or rational) in G if for any finite generating set A of G there is ϵ > 0 such that every geodesic in the Cayley graph Γ(G, A) of G with both endpoints in H is contained in ϵ-neighborhood of H. The notion of a quasiconvex subgroup corresponds, roughly speaking, to that of geometric finiteness in the theory of classical hyperbolic groups (see [Swa], [KS], [Pi]). Quasiconvex subgroups of word hyperbolic groups are finitely presentable and word hyperbolic and their finite intersections are again quasiconvex. Nonquasiconvex finitely generated subgroups of word hyperbolic groups are quite rare and there are very few examples of them. We know only three basic examples of this sort. The first is based on a remarkable construction of E. Rips [R], which allows one, given an arbitrary finitely presented group Q, to construct a word hyperbolic group G and a two-generator subgroup H of G such that H is normal in G and the quotient is isomorphic to Q. The second example is based on the existence of a closed hyperbolic 3-manifold fibering over a circle, provided by results of W. Thurston and T. Jorgensen. The third example is obtained using the result

\begin{matrix}c \\ ◯ \end{matrix}1995 State University of New York ISSN 1076-9803/95

Received June 15, 1995.

Mathematics Subject Classification. Primary 20F32; Secondary 20E06.

Key words and phrases. hyperbolic group, quasiconvex subgroup, limit set.

This research is supported by an Alfred P. Sloan Doctoral Dissertation Fellowship

of M. Bestvina and M. Feign [BF] who proved that if F is a non-abelian free group of finite rank and ϕ is an automorphism of F without periodic conjugacy classes then the HNN-extension of F along ϕ is word hyperbolic.

If G is a word hyperbolic group then we denote the boundary of G (see [Gr],

[GH] and [CDP]) by ∂G. For a subgroup H of G the limit set ∂_{G}(H) of H is the set of all limits in ∂G of sequences of elements of H.

In this note we construct an example of a non-quasiconvex finitely presented one-ended subgroup H of a word hyperbolic group G such that the limit set of H is exotic. By exotic we mean that the limit set of H is not the limit set of a quasiconvex subgroup of G. This result is of some interest since in the previously known examples non-quasiconvex subgroups were normal in the ambient hyperbolic groups and thus (see [KS]) had the same limit sets. Our subgroup H also provides a counter-example to the conjecture of G. A. Swarup [Swa] which stated that a finitely presented freely indecomposable subgroup of a torsion-free word hyperbolic group is quasiconvex if and only if it has finite index in its virtual normalizer (this statement was known to be true for 3-dimensional Kleinian groups). The subgroup H, constructed here, coincides with its virtual normalizer. Here, by the virtual normalizer of a subgroup H of a group G we mean the subgroup

V N_{G}(H) = {g ∈ G | |H : H ∩ gHg^{−1}| < ∞, |gHg^{−1} : H ∩ gHg^{−1}| < ∞}.

2. Some Definitions and Notations

A geodesic in a metric space (X, d) is an isometric embedding α : [0, l] → X where l ≥ 0 and [0, l] is a segment of the real line. We say that a metric space (X, d) is geodesic if any two points of X can be joined by a geodesic path in X. A path β : [0, l] → X is called λ-quasigeodesic if it is parametrized by its arclength and for any t_{1}, t_{2} ∈ [0, l]

|t_{1} − t_{2}| ≤ λ · d(β(t_{1}), β(t_{2})) + λ.

If x, y and z are points in a metric space (X, d) we set

(x, y)_{z} = \frac{1}{2}(d(z, x) + d(z, y) − d(x, y)).

The quantity (x, y)_{z} is called the Gromov inner product of x and y with respect to z.

Let Δ be a triangle in a metric space (X, d) with geodesic sides α, β and γ and vertices x, y, z. (See Figure 1.)

We say that the points p, q, r on α, β and γ are the vertices of the inscribed triangle for Δ if d(x, p) = d(x, r) = (y, z)_{x}, d(y, p) = d(y, q) = (x, z)_{y} and d(z, r) = d(z, q) = (x, y)_{z}. In this situation Δ is called δ-thin if for each t ∈ [0, d(x, p)]

d(p′, r′) ≤ δ

where p′, r′ are points on α, γ with d(x, p′) = d(x, r′) = t and if the symmetric condition holds for y and z.

A geodesic metric space (X, d) is called δ-hyperbolic if there is δ ≥ 0 such that all geodesic triangles are δ-thin.

Figure 1

If G is a finitely generated group and G is a finite generating set for G, we denote the Cayley graph of G with respect to G by Γ(G, G) and denote by d_{G} the word metric on Γ(G, G). Also, for any g ∈ G we define the word length of g as l_{G}(g) = d_{G}(1, g). It is easy to see that (Γ(G, G), d_{G}) is a geodesic metric space. If w is a word in the generators G, we denote by \overline{w} the element of G which w represents.

A finitely generated group G is called word hyperbolic if for each finite generating set G for G there is δ ≥ 0 such that the Cayley graph Γ(G, G) with the word metric d_{G} is δ-hyperbolic.

A subgroup H of a word hyperbolic group G is called quasiconvex in G if for some (and therefore for any) finite generating set G of G there is ϵ > 0 such that every geodesic in the Cayley graph Γ(G, G) of G with both endpoints in H lies in the ϵ-neighborhood of H.

If G is a word hyperbolic group with a finite generating set G, we say that a sequence of points {g_{n} ∈ G | n ∈ ℕ} defines a point at infinity if

lim_{n→∞} inf_{i,j≥n}(g_{i}, g_{j})_{1} = ∞

where the Gromov inner product is taken in d_{G}-metric. Two sequences (a_{n})_{n∈ℕ} and (b_{n})_{n∈ℕ} defining points at infinity are called equivalent if

lim_{n→∞} inf_{i,j≥n}(a_{i}, b_{j})_{1} = ∞.

The boundary ∂G of G is defined to be the set of equivalence classes of sequences defining points at infinity. If a ∈ ∂G is the equivalence class of a sequence (a_{n})_{n∈ℕ}, we say that (a_{n})_{n∈ℕ} converges to a and write lim_{n→∞} a_{n} = a. The boundary ∂G can be endowed with a natural topology which makes it a compact (and metrizable) space. It turns out that the definition of ∂G and the topology on it are independent of the choice of the word metric for G. Moreover, G acts on ∂G by homeomorphisms and the action is given by g · lim_{n→∞} a_{n} = lim_{n→∞} ga_{n} where g ∈ G and (a_{n})_{n∈ℕ} defines a point at infinity. If S is a subset of G (e.g., a subgroup of G), we define the limit set ∂_{G}(S) to be the set of limits in ∂G of sequences of elements of S.

3. The Proofs

Proposition A. Let F be the fundamental group of a closed hyperbolic surface S. Let ϕ be an automorphism of F induced by a pseudo-anosov homeomorphism of S. Take G to be the mapping-torus group of ϕ, that is

G = ⟨F, t | tft^{−1} = ϕ(f), f ∈ F ⟩.

Let x ∈ F be an element which is not a proper power in F (and so, obviously, x is not a proper power in G). Let G_{1} be a copy of G. The group G_{1} contains a copy F_{1} of F and a copy x_{1} of x. Set

M = G ∗_{x=x_{1}} G_{1}  (1)

and H = sgp(F, F_{1}).

Then 1. M is torsion-free and word hyperbolic.

2. H is finitely presented, freely indecomposable and non-quasiconvex in M.

Proof. The group G is torsion free and word hyperbolic since it is the fundamental group of a closed 3-manifold of constant negative curvature (see [Th]). Thus, M is word hyperbolic by the combination theorem for negatively curved groups (see [BF], [KM]). Notice that H = sgp(F, F_{1}) ≅ F ∗_{x=x_{1}} F_{1} and so H is torsion-free,

finitely presentable and freely indecomposable. Moreover, H is word hyperbolic by the same combination theorem.

Suppose H is quasiconvex in M. It is shown in [BGSS] that F is rational with respect to some automatic structure on H since H = F ∗_{C} F_{1} where C = ⟨x⟩ = ⟨x_{1}⟩ is cyclic. Therefore, F is quasiconvex in H (see, for example, [Swa]). Thus, since H is quasiconvex in M and F is quasiconvex in H, the subgroup F is quasiconvex in M. However, F is infinite and has infinite index in its normalizer in M, which (see [KS]) implies that F is not quasiconvex in M. This contradiction completes the proof of Proposition A. □

Theorem B. Let G, G_{1}, M, and H be as in Proposition A. Let K be the limit set ∂_{M}(H) of H in the boundary ∂M of M. Then

H = Stab_{M}(K) = {f ∈ M | fK = K}.

Before proceeding with the proof, we choose a finite generating set G for G and its copy G_{1}. Then G defines the word length l_{G} and the word metric d_{G} for G. Analogously, G ∪ G_{1} is a finite generating set for M = G ∗_{C} G_{1} which defines the word length l_{M} and the word metric d_{M} on M. Fix a δ > 0 such that all d_{M}- geodesic triangles are δ-thin. We also denote by C the subgroup of M generated by x = x_{1}. The element x = x_{1} will sometimes be denoted by c. Thus M = G ∗_{C} G_{1} and H = F ∗_{C} F_{1}. We need to accumulate some preliminary information before proceeding with the proof of Theorem B.

Figure 2

Lemma 1. Let A be a word hyperbolic group with a fixed finite generating set S. Let H and K be quasiconvex subgroups of A such that for every a ∈ A

aHa^{−1} ∩ K = {1}

Then there is a constant r_{0} > 0 such that for every h ∈ H and k ∈ K

l_{A}(hk) ≥ r_{0} · l_{A}(k)

Proof. Fix a finite generating set H for H and a finite generating set K for K. Since K and H are quasiconvex in A, there is ϵ > 0 such that every geodesic [1, h], h ∈ H in the Cayley graph of A lies in the ϵ-neighborhood of H and every geodesic [1, k], k ∈ K lies in the ϵ-neighborhood of K. Also, there is λ > 0 such that every d_{H}-geodesic (d_{K}-geodesic) word defines a λ-quasigeodesic in the Cayley graph of A.

Let h ∈ H and k ∈ K. Fix d_{G}-geodesic representatives u, v, w of h, k and hk respectively. Also, fix a d_{H}-geodesic representative U of h and a d_{K}-geodesic representative V of k. Consider the geodesic triangle Δ in the Cayley graph of A with sides u, v, w (see Figure 2).

Consider the inscribed triangle xyz in the triangle Δ (see Figure 2). It has the following properties:

1. d_{A}(1, x) = d_{A}(1, z), d_{A}(h, x) = d_{A}(h, y), d_{A}(hk, y) = d_{A}(hk, z)

2. the segment [1, u] of U is δ-uniformly close to the segment [1, z] of w and similar conditions hold for the other two corners of Δ; in particular, d_{A}(x, y), d_{A}(x, z), d_{A}(y, z) ≤ δ.

We claim that d_{A}(x, h) = d_{A}(y, h) is small. More precisely, let N be the number of words in the generating set of A of length at most δ + 2ϵ + 2. Suppose d_{A}(h, u) > 4(ϵ + 1)(N + 1). Then there is a sequence of vertices u_{1}, …, u_{N+1} on the segment of u between x and h such that d_{A}(u_{k}, u_{s}) = 4(ϵ + 1)|k − s|. For each k = 1, …, N + 1 there is a vertex U_{k} on U such that d_{A}(U_{k}, u_{k}) ≤ ϵ + 1. Note that when k ≠ s, d_{A}(U_{k}, U_{s}) ≥ 4(ϵ + 1)|k − s| − 2(ϵ + 1) ≥ 2(ϵ + 1) > 0 and therefore all the vertices U_{1}, …, U_{N+1} represent different elements of A. Also, for every k = 1, …, N + 1 there is a unique vertex v_{k} on the segment of V between h and y such that d_{A}(h, u_{k}) = d_{A}(h, v_{k}). Note that since the triangle Δ is δ-thin, we have d_{A}(u_{k}, v_{k}) ≤ δ, k = 1, …, N + 1. Finally, for every k = 1, …, N + 1 there is a vertex V_{k} of V such that d_{A}(v_{k}, V_{k}) ≤ ϵ + 1. Thus d_{A}(U_{k}, V_{k}) ≤ δ + 2(ϵ + 1), k = 1, …, N + 1. For each k we choose a d_{A}-geodesic path α_{k} from U_{k} to V_{k} in the Cayley graph of A. By the choice of N there are i < j such that α_{i} = α_{j} = α. Put a = \overline{α}. Then a^{−1}h′a = k′ where h′ ∈ H is the element represented by the segment of U from U_{j} to U_{i} and k′ ∈ K is the element represented by the segment of V from V_{i} to V_{j}. Since h′ ≠ 1, this contradicts our assumption that any conjugate of H intersects K trivially. Thus we have established that

d_{A}(h, u) ≤ 4(ϵ + 1)(N + 1) = r

Therefore l(w) ≥ d_{A}(z, kh) = d_{A}(v, kh) ≥ l(v) − r and l_{A}(hk) = l(w) ≥ min(1/2, 1/r) · l(v) = min(1/2, 1/r) · l_{A}(k) which concludes the proof of Lemma 1. □

Corollary 2. Let A be a word hyperbolic group with a fixed finite generating set A and let a, b be elements of infinite order in A such that no nontrivial power of a is conjugate in A to a power of b. Then there is a constant r_{1} > 0 such that for every m, n ∈ ℤ

l_{A}(a^{m}b^{n}) ≥ |n| · r_{1}

Proof. This directly follows from Lemma 1 and the fact that cyclic subgroups of word hyperbolic groups are quasiconvex [ABC]. □

Lemma 3. Assume that conditions of Theorem B are satisfied. Let p ∈ C or p = p_{1} … p_{s} be a strictly alternating product of elements of G − C and G_{1} − C, where p_{s} ∈ G_{1} − C. Let y ∈ G be such that no power of y is conjugate in G to a power of x. Then there is a constant D > 0 such that for every n ∈ ℤ

l_{M}(py^{n}) ≥ D · |n|

Proof. Note that if p ∈ C, then, since every conjugate of C in G intersects ⟨y⟩ trivially, Corollary 2 implies that

l_{G}(py^{n}) ≥ r_{1} · |n|

for some constant r_{1} > 0 independent of p, n. Theorem D of [BGSS] implies that G and G_{1} are quasiconvex in M. Therefore there is a constant r > 0 such that for every g ∈ G

l_{M}(g) ≥ r · l_{G}(g)

and therefore

l_{M}(py^{n}) ≥ r · r_{1} · |n|

From now on we assume that p ∉ C that is p = p_{1} … p_{s} is a strictly alternating product of elements of G − C and G_{1} − C with p_{s} ∈ G_{1} − C. To prove Lemma 3 in this case, recall that by the theorem of G. Baumslag, S. Gersten, M. Shapiro, and H. Short [BGSS], cyclic amalgamations of hyperbolic groups are automatic. In the proof of this theorem they construct an actual automatic language for a cyclic amalgam of two hyperbolic groups, which, therefore, consists of quasigeodesic words (see [ECHLPT], Theorem 3.3.4). We will explain how their procedure works in the case of the group M = G ∗_{C} G_{1} (we use the fact that d_{G} |_{C}= d_{G_{1}} |_{C}). Fix a lexicographic order on the generating set G of G and a copy of this order on the generating set G_{1} of G_{1}. We will say that a d_{G}-geodesic word u is minimal in the coset class \overline{u}C if l_{G}(u) ≤ l_{G}(\overline{u} · c) for every c ∈ C and whenever l(u) = l(u′) for some d_{G}-geodesic word u′ with \overline{u′} ∈ \overline{u}C then u is lexicographically smaller than u′. It is clear that any coset class gC, g ∈ G has a unique minimal representative u. Similarily, one defines minimal representatives for coset classes g_{1}C, g_{1} ∈ G_{1}.

Theorem D of [BGSS] provides an explicit construction of an automatic language L in the alphabet M = G ∪ G_{1} for M such that every e ∈ M has a unique representative in L. Note that, in general, Theorem D of [BGSS] gives a construction of such an automatic language for M in a bigger alphabet than G ∪G_{1}. More precisely, they need to find first a generating set G′ containing G for G and a generating set G_{1}′ containing G_{1} for G_{1} such that for some constant ϵ_{1} > 0

|l_{G′}(c) − l_{G_{1}′(c)}| ≤ ϵ_{1}, for every c ∈ C.

Then they construct the automatic language for M in the alphabet G′ ∪ G_{1}′. However, by the choice of M we already have

l_{G}(c) = l_{G_{1}}(c) for every c ∈ C

and so the [BGSS] procedure gives us an automatic language L with uniqueness in the alphabet G ∪ G_{1}. (Although an automatic group has an automatic language over every finite generating set of this group, in this particular case we need not just the fact that M is automatic and possesses an automatic language over M but, rather, the fact that M has an automatic language over M with some very particular properties given by the [BGSS] construction). We will now describe how, given an element e ∈ M, one can find its representative in L.

Suppose e ∈ M. If e ∈ C, then e = x^{k} and we take a d_{G}-geodesic representative of x^{k} to be the representative of e in the automatic language L for M. Suppose e ∉ C. First, write e as a strictly alternating product of elements from

e = e_{1} … e_{j}

of elements from G−C and G_{1} −C. Then express e_{1} as e_{1} = \overline{w_{1}}c^{n_{1}} where w_{1} is the minimal representative in the coset class e_{1}C. Then express c^{n_{1}}e_{2} as c^{n_{1}}e_{2} = \overline{w_{2}}c^{n_{2}} where w_{2} is the minimal representative in the coset class c^{n_{1}}e_{2}C. And so on for i = 1, 2, …, j − 1. Finally, we express c^{n_{j−2}}e_{j−1} as \overline{w_{j−1}}c^{n_{j−1}} where w_{j−1} is the minimal representative in the coset class c^{n_{j−2}}e_{j−1}C.

We put w_{j} to be the lexicographically minimal among all d_{G}-geodesic (d_{G_{1}}- geodesic) representatives of c^{n_{j−1}}e_{j}. As a result we obtain the word w = w_{1} … w_{j} such that \overline{w} = e. This word w is the required representative of e in L.

Note that in the case e ∉ C we have \overline{w_{1}}C = e_{1}C, C\overline{w_{j}} = Ce_{j} and C\overline{w_{i}}C = Ce_{i}C for 1 < i < j. Note also that there is λ > 0 such that all words in L define λquasigeodesics in the Cayley graph of M. This, in particular, means that for every w ∈ L

l(w) ≤ λ · l_{M}(\overline{w}) + λ.

Suppose now that p and y are as in Lemma 3 and n ∈ ℤ, n ≠ 0. We will find the representative w of py^{n} in the automatic language L on M using the procedure described above. Note that ⟨y⟩ ∩ ⟨x⟩ = {1} and so y^{n} ∈ G − C since C = ⟨x⟩. Therefore p_{1} … p_{s}y^{n} is a strictly alternating product of elements of G − C and G_{1} − C. It is clear from the construction that w has the following form:

w = q_{1} … q_{s}v

where

1. q_{i} is a d_{G}-geodesic word when p_{i} ∈ G − C and q_{i} is a d_{G_{1}}-geodesic word

when p_{i} ∈ G_{1} − C;

2. \overline{q_{1}}C = p_{1}C and C\overline{q_{i}}C = Cp_{i}C when i > 1;

3. v is a d_{G}-geodesic word and \overline{v} = cy^{n} for some c ∈ C

4. \overline{w} = p_{1} … p_{s}y^{n} = py^{n} Corollary 2 implies that l(v) = l_{G}(cy^{n}) ≥ r_{1} · |n| for some r_{1} > 0 depending only

on x, y and independent of n.

Therefore l(w) = l(q_{1} … q_{s}v) ≥ l(v) ≥ r_{1} · |n|. Since the language L consists

of λ-quasigeodesics with respect to d_{M}, we conclude that l_{M}(py^{n}) ≥ \frac{l(w)}{λ} − λ ≥

\frac{r_{1}}{λ} · |n| − λ which implies the statement of Lemma 3. □

Lemma 4. Suppose conditions of Theorem B are satisfied. Let u_{1} … u_{m} ∉ H be a strictly alternating product of elements from G − C and G_{1} − C such that u_{m} ∈ G_{1} − C. Let p_{1} … p_{s} be a strictly alternating product of elements of F − C and F_{1} −C. Let q_{0} belong to G_{1} if p_{1} ∈ F −C and q_{0} belong to G when p_{1} ∈ F_{1} −C (we allow q_{0} ∈ C).

Then either q_{0}p_{1} … p_{s}u_{1} … u_{m} ends (when rewritten in the normal form with respect to (1)) in the element of G_{1} − C or q_{0}p_{1} … p_{s}u_{1} … Au_{m} ∈ C.

Proof. Indeed, u_{m} ∈ G_{1} − C and so q_{0}p_{1} … p_{s}u_{1} … u_{m} ends (when rewritten in the normal form with respect to (1)) in the element of G_{1} − C unless u_{m}^{−1} … u_{1}^{−1} is a terminal segment of q_{0}p_{1} … p_{s} that is either m ≥ s and

p_{s−m+1} … p_{s}u_{1} … u_{m} ∈ C  (2)

or

q_{0}p_{1} … p_{s}u_{1} … u_{m} ∈ C  (3)

It is clear that (2) is impossible since p_{s−m+1} … p_{s} ∈ H = gp(F, F_{1}), C ≤ H and u_{1} … u_{m} ∉ H. If (3) holds, we have q_{0}p_{1} … p_{s}u_{1} … u_{m} ∈ C as required. Thus Lemma 4 is established. □

Lemma 5. Let z = u_{1} … u_{m} ∉ H be a strictly alternating product of elements of G − C and G_{1} − C such that u_{m} ∈ G_{1} − C. Let y ∈ G be such that no nontrivial power of y is conjugate in G to a power of x = c. Then there is a constant K_{0} > 0 with the following property.

Let n ∈ ℤ and h ∈ H. Let v be a d_{M}-geodesic from 1 to h and let u be a d_{M}- geodesic from 1 to zy^{n}. Take the vertex v(N) of v at the distance N from 1 and the vertex u(N) of u at the distance N from 1 (see Figure 3). Then

d_{M}(u(N), v(N)) ≥ K_{0} · N.

Proof. Recall that H = sgp(F, F_{1}) = F ∗_{C} F_{1} ≤ M = G ∗_{C} G_{1}. Let λ > 0 be such that any word from the automatic language L on M defines a λ-quasigeodesic in the Cayley graph of M.

Let û_{i} be a d_{G}-geodesic (d_{G_{1}}-geodesic) representative of u_{i}, i ≤ m. Let Y be a d_{G}-geodesic representative of y. Since the element z is fixed and the cyclic subgroup ⟨y⟩ is quasiconvex in M, there is a constant λ_{1} > 0 such that for every k ∈ ℤ the word û_{1} … û_{m−1}û_{m}Y^{k} is a λ_{1}-quasigeodesic with respect to d_{M}. In particular, U = û_{1} … û_{m−1}û_{m}Y^{n} is a λ_{1}-quasigeodesic representative of zy^{n} with respect to d_{M}. Put λ_{2} = max(λ, λ_{1}). Let ϵ > 0 be such that any two λ_{2}-quasigeodesics with common endpoints in the Cayley graph of M are ϵ-Hausdorff-close. We will also assume that ϵ is such that for every k ∈ ℤ and every point x_{0} on a d_{M}-geodesic from 1 to zy^{k} there is k′ ∈ ℤ, k′ ∈ [0, k], such that d_{M}(x_{0}, zy^{k′}) ≤ ϵ.

If h ∈ C and h^{−1} = x^{m}, put p_{1} = h^{−1} and let w_{1} be a d_{G}-geodesic representative of p_{1}.

If h ∉ C let h^{−1} = p_{1}p_{2} … p_{s} be the strictly alternating product of elements of F − C and F_{1} − C. Note that p_{1}p_{2} … p_{s} is also a strictly alternating product of elements of G − C and G_{1} − C. We then can find the representative w of h^{−1} in the automatic language L on M which was described in Lemma 3.

Clearly, w = w_{1}w_{2} … w_{s}, where

1. \overline{w_{1}}C = p_{1}C and w_{1} is minimal in the coset \overline{w_{1}}C

2. for j < s C\overline{w_{j}}C = Cp_{j}C and w_{j} is minimal in the coset \overline{w_{j}}C

3. C\overline{w_{s}} = Cp_{s} that is \overline{w_{s}} = cp_{s} for some c ∈ C

4. each w_{i} is a d_{G} or d_{G_{1}}-geodesic word.

Note that since p_{j} ∈ F ∪ F_{1} and C ≤ F, C ≤ F_{1}, the conditions above imply that \overline{w_{j}} ∈ F ∪ F_{1} for j = 1, …, s.

Let v be a d_{M} geodesic from 1 to h and let u be a d_{M} geodesic from 1 to zy^{n}. Assume N is a positive number such that N ≤ l(v) and N ≤ l(u).

Let v(N) be the point on the geodesic v at the distance N from 1. Let u(N) be the point on the geodesic u at the distance N from 1 (see Figure 3).

Recall that z = u_{1} … u_{m} and y are fixed, u_{m} ∈ G_{1} − C. Recall further that Y is a d_{G}-geodesic representative of y. By the choice of ϵ there is a vertex V (N) of w and a vertex U(N) = zy^{k} of U such that d_{M}(u(N), U(N)) ≤ ϵ, d_{M}(v(N), V (N)) ≤ ϵ (see Figure 3). The segment S_{1} of w from V (N) to 1 is a terminal segment of w = w_{1} … w_{s}, and it has the form

S_{1} = q_{i}w_{i+1} … w_{s}

Figure 3

where i ≤ s and q_{i} is a nonempty terminal segment of w_{i}. The segment S_{2} of U from 1 to U(N) is an initial segment of U = û_{1} … û_{m−1}û_{m}Y^{n} of the form

S_{2} = û_{1} … û_{m−1}û_{m}Y^{k}

for some integer k ∈ [0, n]. Notice that d_{M}(1, u_{1} … u_{m−1}û_{m}y^{k}) ≥ N − ϵ and therefore

|k|l_{M}(y) ≥ l_{A}(y^{k}) ≥ N − l(û_{1} … û_{m−1}û_{m}) − ϵ

and

|k| ≥ (1/l_{A}(y))(N − l(û_{1} … û_{m−1}û_{m}) − ϵ).

Thus, for some constant K_{1} > 0 independent of h, n, we have

|k| ≥ K_{1} · N.

By Lemma 4, either \overline{q}_{i}\overline{w}_{i+1} … \overline{w}_{s}u_{1} … u_{m} ∈ C or \overline{q}_{i}\overline{w}_{i+1} … \overline{w}_{s}u_{1} … u_{m} ends in the element of G_{1} − C, when rewritten in normal form with respect to (1). Therefore, by Lemma 3, there is a constant D > 0 independent of h, n such that

l_{M}(\overline{q}_{i}\overline{w}_{i+1} … \overline{w}_{s}u_{1} … u_{m}y^{k}) = l_{M}(\overline{S_{1}S_{2}}) ≥ D|k|

and hence l_{M}(\overline{q}_{i}\overline{w}_{i+1} … \overline{w}_{s}u_{1} … u_{m}y^{k}) ≥ K_{1} · D · N. It remains to recall that |d_{M}(u(N), v(N)) − l_{M}(\overline{q}_{i}\overline{w}_{i+1} … \overline{w}_{s}u_{1} … u_{m}y^{k})| ≤ 2ϵ to conclude that there is a constant K_{2} > 0 independent of h, n such that d_{M}(u(N), v(N)) ≥ K_{2} · N. This completes the proof of Lemma 5. □

Proof of Theorem B. Suppose z ∈ Stab_{M}(K). We will show that z ∈ H by induction on the syllable length of z with respect to presentation (1). When the syllable length of z is 0, that is z ∈ C, the statement is obvious. Suppose now that z ∈ Stab_{M}(K) − H, the syllable length of z is m > 0 and the statement has been proved for elements of Stab_{M}(K) of smaller syllable length. Write z as a strictly alternating product z = u_{1} … u_{m} of elements from G − C and G_{1} − C. If u_{m} ∈ F ∪ F_{1}, then u_{m} ∈ H ∩ Stab_{M}(K), and so u_{1} … u_{m−1} ∈ Stab_{M}(K). Therefore, u_{1} … u_{m−1} ∈ H by the inductive hypothesis, u_{m} ∈ H, and so z ∈ H. Thus, u_{m} ∈ (G − F) ∪ (G_{1} − F_{1}). Assume for definiteness that u_{m} ∈ G_{1} − F_{1}, that is, u_{m} = f_{1}t_{1}^{j} for some j ≠ 0, f_{1} ∈ F_{1}.

Choose y ∈ F so that no power of y is conjugate in G to a power of x. Fix a d_{G}-geodesic representative Y of y.

Let y^{+} = lim_{n→∞}y^{n} ∈ ∂M. By definition of K we have y^{+} ∈ K and therefore

zy^{+} ∈ K. This means that for any N > 0 there is an element h ∈ H and a positive power y^{n} of y such that (h, zy^{n})_{1} > N, the Gromov inner product taken in the d_{M}- metric. This means that l_{M}(h) ≥ N, l_{M}(zy^{n}) ≥ N and d_{M}(h(N), (zy^{n})(N)) ≤ δ where h(N) and (zy^{n})(N) are elements of M represented by initial segments of length N of d_{M}-geodesic representatives of h and zy^{n}.

Then l_{M}(h(N), (zy^{n})(N)) ≥ K_{0} · N where K_{0} is the constant independent of h, n which is provided by Lemma 5. Thus,

δ ≥ l_{M}(h(N), (zy^{n})(N)) ≥ K_{0} · N

and therefore N ≤ (1/K_{0}) · δ. This contradicts the fact that N can be chosen arbitrarily big.

Therefore, z ∉ Stab_{M}(K), which completes the proof of Theorem B. □

Corollary 6. Let M, G, G_{1}, C and H be as in Theorem B. Then (a) the limit set of H is not the limit set of a quasiconvex subgroup of M;

(b) the virtual normalizer V N_{M}(H) of H in M is equal to H.

Proof.

(a) Suppose there is a quasiconvex subgroup Q_{1} of M such that ∂_{M}(H) = ∂_{M}(Q_{1}) = K. Clearly, Q_{1} is infinite since K is nonempty. Set

Q = Stab_{M}(K) = {y ∈ M|yK = K}.

Since Q_{1} is infinite and quasiconvex in M and Q = Stab_{M}(∂_{M}(Q_{1})), it follows from Lemma 3.9 of [KS] that Q contains Q_{1} as a subgroup of finite index and therefore Q is also quasiconvex in M. On the other hand, Theorem B implies that H = Stab_{M}(K), and so H = Q. This contradicts the fact that H is not quasiconvex in M by Proposition A.

(b) It is not hard to see that A ≤ V N_{B}(A) ≤ Stab_{B}(∂_{B}(A)) when A is an infinite subgroup of a word hyperbolic group B. Indeed, if g ∈ V N_{B}(A), then A_{0} = A ∩ gAg^{−1} has finite index n in A. Let A = A_{0} ∪ A_{0}c_{1} ∪ ⋯ ∪ A_{0}c_{n−1}, and let D = max{l_{A}(c_{i}) | i = 1, …, n − 1}. Suppose p ∈ ∂_{B}(A). Then there is a sequence a_{m} ∈ A such that p = lim_{m→∞}a_{m}. For each m there is b ∈ B with

A Non-quasiconvex Subgroup 195

l_{B}(b) ≤ D + l_{B}(g) such that ga_{m}b = a′_{m} ∈ A_{0}. Therefore gp ∈ ∂_{B}(A_{0}) = ∂_{B}(A). Since p ∈ ∂_{B}(A) was chosen arbitrarily, we have g∂_{B}(A) ⊆ ∂_{B}(A). Since by the same argument g^{−1}∂_{B}(A) ⊆ ∂_{B}(A), we conclude that g∂_{B}(A) = ∂_{B}(A). Thus, A ≤ V N_{B}(A) ≤ Stab_{B}(∂_{B}(A)).

For the subgroup H of M we have H ≤ V N_{M}(H) ≤ Stab_{M}(∂_{M}(H)). On the other hand, Stab_{M}(∂_{M}(H)) = H by Theorem B. Therefore, H = V N_{M}(H). □

References

[ABC] J. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short, Notes on hyperbolic groups, Group Theory from a Geometric Viewpoint, Proc. ICTP. Trieste, World Scientific, Singapore, 1991, pp. 3–63.

[BF] M. Bestvina and M. Feign, The combination theorem for negatively curved groups, J. of Diff. Geom. 35 (1992), 85–101.

[BGSS] G. Baumslag, S. Gersten, M. Shapiro, and H. Short, Automatic groups and amalgams, J. of Pure and Appl. Algebra 76 (1991), 229–316.

[CDP] M. Coornaert, T. Delzant, and A. Papadopoulos, Notes sur les Groupes Hyperboliques de Gromov, Lect. Notes in Math., no. 1441, Springer-Verlag, Berlin, 1990.

[ECHLPT] D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson, and W. P. Thurston, Word Processing in Groups, Jones and Bartlett, Boston, 1992.

[GH] E. Ghys and P. de la Harpe (editors), Sur les Groupes Hyperboliques d’aprés Mikhael Gromov, Progress in Mathematics, no. 83, Birkhäuser, Boston, 1990.

[Gr] M. Gromov, Hyperbolic groups, Essays in Group Theory (S. M. Gersten, ed.), MSRI Publ., no. 8, Springer-Verlag, Berlin, 1987, pp. 75–263.

[KM] O. Kharlampovich and A. Miasnikov, Hyperbolic groups and amalgams, Trans. Amer. Math. Soc. (to appear).

[KS] I. Kapovich and H. Short, Greenberg’s theorem for quasiconvex subgroups of word hyperbolic groups, preprint.

[Pi] Ch. Pittet, Surface groups and quasiconvexity, Geometric Group Theory, Vol. 1 (Sussex 1991), London Math. Soc. Lecture Notes, no. 181, Cambridge University Press, Cambridge, 1993, pp. 169–175.

[R] E. Rips, Subgroups of small cancellation groups, Bull. London. Math. Soc. 14 (1982), 45–47.

[Swa] G. A. Swarup, Geometric finiteness and rationality, J. of Pure and Appl. Algebra 86 (1993), 327–333.

[Th] W. Thurston, Geometry and Topology of 3-manifolds, Lecture Notes, Princeton University Press, Princeton, 1977.

City College, 138th Street and Convent Avenue, New York, NY 10031 ilya@groups.sci.ccny.cuny.edu

Typeset by AMS-TEX