A Non-quasiconvex Subgroup of a

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 nitely presented non-quasiconvex subgroup^H of a word hyperbolic group^Gsuch 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 nitely presented one-ended subgroup of a word hyperbolic group is quasiconvex if and only if it has nite index in its virtual normalizer.

Contents

1. Introduction 184

2. Some Denitions and Notations 185

3. The Proofs 187

References 195

1. Introduction

A subgroupH of a word hyperbolic groupGis quasiconvex (or rational) inGif for any nite generating setAof Gthere is >0 such that every geodesic in the Cayley graph ;(GA) ofGwith both endpoints inH is contained in-neighborhood ofH. The notion of a quasiconvex subgroup corresponds, roughly speaking, to that of geometric niteness in the theory of classical hyperbolic groups (see Swa], KS], Pi]). Quasiconvex subgroups of word hyperbolic groups are nitely presentable and word hyperbolic and their nite intersections are again quasiconvex. Non- quasiconvex nitely 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 rst is based on a remarkable construction of E. Rips R], which allows one, given an arbitrary nitely presented group Q, to construct a word hyperbolic groupGand a two-generator subgroupH ofGsuch thatH 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 bering over a circle, provided by results of W. Thurston and T. Jorgensen. The third example is obtained using the result

Received June 15, 1995.

Mathematics Subject Classication. 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

c

1995StateUniversityofNewYork

ISSN1076-9803/95

184

of M. Bestvina and M. Feign BF] who proved that ifF is a non-abelian free group of nite rank and is an automorphism of F without periodic conjugacy classes then the HNN-extension ofF alongis 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 subgroupH of Gthe limit set@^G(H) of H is the set of all limits in@Gof sequences of elements ofH.

In this note we construct an example of a non-quasiconvex nitely 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 ofG. 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 subgroupH also provides a counter-example to the conjecture of G. A. Swarup Swa] which stated that a nitely presented freely indecomposable subgroup of a torsion-free word hyperbolic group is quasiconvex if and only if it has nite 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 subgroupH of a groupGwe mean the subgroup

V NG(H) =^fg²G ^{j j}H :H^\gHg^;1j<^1jgHg^;1:H^\gHg^;1j<^1g:

2. Some Denitions and Notations

A geodesic in a metric space (Xd) is an isometric embedding : 0l] ^! X where l 0 and 0l] is a segment of the real line. We say that a metric space (Xd) is geodesic if any two points ofX can be joined by a geodesic path inX. A path : 0l] ^!X is called -quasigeodesic if it is parametrized by its arclength and for anyt¹t²²0l]

jt^1;t^2j d((t¹)(t²)) +: Ifxy andzare points in a metric space (Xd) we set

(xy)z= 12(d(zx) +d(zy)^;d(xy)):

The quantity (xy)z is called the Gromov inner product of xandywith respect to z.

Let be a triangle in a metric space (Xd) with geodesic sides , and and verticesxyz. (See Figure 1.)

We say that the points pqr on , and are the vertices of the inscribed trianglefor ifd(xp) =d(xr) = (yz)x, d(yp) =d(yq) = (xz)y andd(zr) = d(zq) = (xy)z. In this situation is called-thin if for eacht²0d(xp)]

d(p⁰r⁰)

where p⁰r⁰ are points on , with d(xp⁰) = d(xr⁰) = t and if the symmetric condition holds fory andz.

A geodesic metric space (Xd) is called -hyperbolic if there is0 such that all geodesic triangles are-thin.

x

p

r

z p’

y

γ

β α

q

r’

Figure 1

If G is a nitely generated group and ^G is a nite generating set for G, we denote the Cayley graph ofGwith respect to ^G by ;(G^G) and denote byd^G the word metric on ;(G^G). Also, for any g ² G we dene the word length of g as l^G(g) =d^G(1g). It is easy to see that (;(G^G)d^G) is a geodesic metric space. Ifw is a word in the generators^G, we denote bywthe element ofGwhichwrepresents.

A nitely generated groupGis called word hyperbolic if for each nite generating set^GforGthere is0 such that the Cayley graph ;(G^G) with the word metric d^G is-hyperbolic.

A subgroup H of a word hyperbolic group G is called quasiconvex in G if for some (and therefore for any) nite generating set ^G ofG there is >0 such that every geodesic in the Cayley graph ;(G^G) of Gwith both endpoints inH lies in the-neighborhood ofH.

If G is a word hyperbolic group with a nite generating set ^G, we say that a sequence of points^fgn²G^jn^2Ng denes a point at innityif

nlim^!1_ijinf

n(gigj)¹=¹

where the Gromov inner product is taken in d^G-metric. Two sequences (a_n)n^2N

and (bn)n^2N dening points at innity are called equivalent if

nlim^!1_ijinf

n(aibj)¹=¹:

The boundary@GofGis dened to be the set of equivalence classes of sequences dening points at innity. Ifa²@Gis the equivalence class of a sequence (an)n^2N, we say that (an)n^2N converges toaand write lim_n

!1

an =a. The boundary @Gcan be endowed with a natural topology which makes it a compact (and metrizable) space. It turns out that the denition of@Gand the topology on it are independent of the choice of the word metric forG. Moreover,Gacts on@Gby homeomorphisms and the action is given byg limn^!1an = limn^!1gan where g²Gand (an)n^2N

denes a point at innity. IfS is a subset of G(e.g., a subgroup ofG), we dene the limit set@^G(S) to be the set of limits in@Gof sequences of elements ofS.

3. The Proofs

Proposition A.

LetF be the fundamental group of a closed hyperbolic surfaceS. Letbe an automorphism of F induced by a pseudo-anosov homeomorphism of S. TakeGto be the mapping-torus group of , that is

G=^hFt^jtft^;1=(f) f²Fⁱ:

Letx²F be an element which is not a proper power in F (and so, obviously,x is not a proper power in G). LetG¹ be a copy of G. The group G¹ contains a copy F¹ of F and a copyx¹ ofx. Set

M=G_x

=x¹G¹ (1)

andH =sgp(FF¹).

Then1. M is torsion-free and word hyperbolic.

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

Proof.

^{The groupG}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(FF¹) = F _x

=x¹F¹ and so H is torsion-free, nitely presentable and freely indecomposable. Moreover,H is word hyperbolic by the same combination theorem.

SupposeH is quasiconvex in M. It is shown in BGSS] thatF is rational with respect to some automatic structure onH sinceH =F_CF¹whereC=^hxⁱ=^hx¹ⁱ is cyclic. Therefore,F is quasiconvex in H (see, for example, Swa]). Thus, since H is quasiconvex inM andF is quasiconvex inH, the subgroupF is quasiconvex in M. However,F is innite and has innite index in its normalizer inM, which (see KS]) implies thatF is not quasiconvex in M. This contradiction completes the proof of Proposition A.

Theorem B.

^{LetG,G1,M, andH} be as in Proposition A. LetK be the limit set

@M(H) of H in the boundary @M ofM. Then

H =StabM(K) =^ff ²M^jfK=K^g:

Before proceeding with the proof, we choose a nite generating set^G forGand its copy ^G1. Then ^G denes the word length l^G and the word metric d^G for G. Analogously,^GG1 is a nite generating set forM =GCG¹ which denes the word length l^M and the word metric d^M on M. Fix a > 0 such that all d^M- geodesic triangles are-thin. We also denote by C the subgroup of M generated byx=x¹. The elementx=x¹will sometimes be denoted byc. ThusM =G_CG¹ and H = F _CF¹. We need to accumulate some preliminary information before proceeding with the proof of Theorem B.

h

x

z y u

u

v V

V

i

i j i

j j

vj

Ui

U

V v

w U

u

1

hk

Figure 2

Lemma 1.

LetA be a word hyperbolic group with a xed nite generating set S. LetH andK be quasiconvex subgroups ofA such that for every a²A

aHa^;1\K=^f1^g

Then there is a constantr⁰>0 such that for everyh²H andk²K l^A(hk)r⁰ l^A(k)

Proof.

Fix a nite generating set ^H for H and a nite generating set^K for K. SinceK andH are quasiconvex inA, there is >0 such that every geodesic 1h], h²H in the Cayley graph ofAlies in the-neighborhood ofH and every geodesic 1k], k²K lies in the-neighborhood ofK. Also, there is >0 such that every d^H-geodesic (d^K-geodesic) word denes a -quasigeodesic in the Cayley graph of A.

Let h ² H and k ² K. Fix d^G-geodesic representatives uvw of h, k and hk respectively. Also, x a d^H-geodesic representative U of h and ad^K-geodesic representativeV ofk. Consider the geodesic triangle in the Cayley graph ofA with sidesuvw(see Figure 2).

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

1. d^A(1x) =d^A(1z),d^A(hx) =d^A(hy),d^A(hky) =d^A(hkz)

2. the segment 1u] of U is -uniformly close to the segment 1z] of w and similar conditions hold for the other two corners of in particular, d^A(xy) d^A(xz) d^A(yz).

We claim thatd^A(xh) =d^A(yh) is small. More precisely, let N be the num- ber of words in the generating set of A of length at most + 2+ 2. Suppose d^A(hu)>4(+ 1)(N+ 1). Then there is a sequence of verticesu¹:::uN⁺¹ on the segment of u between x and h such that d^A(ukus) = 4(+ 1)^jk^;s^j. For

each k = 1:::N + 1 there is a vertex Uk on U such that d^A(Ukuk) + 1.

Note that whenk⁶=s,d^A(UkUs)4(+ 1)^jk^;s^j;2(+ 1)2(+ 1)>0 and therefore all the verticesU¹:::UN⁺¹ represent dierent elements of A. Also, for everyk = 1:::N+ 1 there is a unique vertex vk on the segment of V between handy such thatd^A(huk) =d^A(hvk). Note that since the triangle is-thin, we haved^A(ukvk),k= 1:::N+ 1. Finally, for everyk= 1:::N+ 1 there is a vertexVk ofV such that d^A(vkVk)+ 1. Thus d^A(UkVk)+ 2(+ 1), k= 1:::N+1. For eachkwe choose ad^A-geodesic path k fromU_k toV_k in the Cayley graph ofA. By the choice ofN there arei < j such that i= j= . Put a= . Thena^;1h⁰a=k⁰ where h^{0 2}H is the element represented by the segment of U from Uj to Ui and k^{0 2} K is the element represented by the segment of V fromVi toVj. Sinceh^{0 6}= 1, this contradicts our assumption that any conjugate of H intersectsK trivially. Thus we have established that

d^A(hu)4(+ 1)(N+ 1) =r

Therefore l(w) d^A(zkh) = d^A(vkh) l(v)^;r and l^A(hk) = l(w) min(1=21=r) l(v) = min(1=21=r) l^A(k) which concludes the proof of Lemma 1.

Corollary 2.

LetAbe a word hyperbolic group with a xed nite generating set^A and letabbe elements of innite order inA such that no nontrivial power of ais conjugate inAto a power ofb. Then there is a constant r¹>0 such that for every mn^2Z

l^A(a^mbⁿ)^jn^j r¹

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 satised. Let p ² C or p= p¹:::ps be a strictly alternating product of elements of G^;C and G^1;C, whereps²G^1;C. Let y ²Gbe such that no power of y is conjugate in G to a power ofx. Then there is a constantD >0 such that for every n^2Z

l^M(pyⁿ)D ^jn^j

Proof.

Note that if p ² C, then, since every conjugate of C in G intersects ^hyⁱ trivially, Corollary 2 implies that

l^G(pyⁿ)r^{1 j}n^j

for some constantr¹ >0 independent of pn. Theorem D of BGSS] implies that GandG¹ are quasiconvex inM. Therefore there is a constantr >0 such that for everyg²G

l^M(g)r l^G(g) and therefore

l^M(pyⁿ)r r^{1 j}n^j

From now on we assume thatp⁶²Cthat is p=p¹:::ps is a strictly alternating product of elements of G^;C and G^1;C with ps²G^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_CG¹ (we use the fact that d^{G j}_C= d^{G1 j}_C). Fix a lexicographic order on the generating set^G of Gand a copy of this order on the generating set^G1 ofG¹. We will say that ad^G-geodesic word uis minimal in the coset classuC ifl^G(u) l^G(u c) for every c ² C and wheneverl(u) = l(u⁰) for somed^G-geodesic wordu⁰withu⁰²uC thenuis lexicographically smaller thanu⁰. It is clear that any coset class gC, g ²G has a unique minimal representativeu. Similarily, one denes minimal representatives for coset classesg¹C,g¹²G¹.

Theorem D of BGSS] provides an explicit construction of an automatic language Lin the alphabet ^M=^GG1 forM such that everye²M has a unique repre- sentative inL. Note that, in general, Theorem D of BGSS] gives a construction of such an automatic language forMin a bigger alphabet than^GG1. More precisely, they need to nd rst a generating set^G0 containing^G forGand a generating set

G

1

0 containing^G1 forG¹ such that for some constant¹>0

jl^G0(c)^;l^G10(_c^)j1 for everyc²C:

Then they construct the automatic language forM in the alphabet^G0G10. How- ever, by the choice ofM we already have

l^G(c) =l^G1(c) for everyc²C

and so the BGSS] procedure gives us an automatic language L with uniqueness in the alphabet^GG1. (Although an automatic group has an automatic language over every nite 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 thatM has an automatic language over^M with some very particular properties given by the BGSS] construction). We will now describe how, given an elemente²M, one can nd its representative inL.

Supposee²M. Ife²C, thene=x^k and we take ad^G-geodesic representative of x^k to be the representative of e in the automatic languageL for M. Suppose e⁶²C. First, writeeas a strictly alternating product of elements from

e=e¹:::e_j

of elements fromG^;CandG^1;C. Then expresse¹ase¹=w¹cⁿ¹ wherew¹is the minimal representative in the coset classe¹C. Then expresscⁿ¹e²ascⁿ¹e²=w²cⁿ² where w² is the minimal representative in the coset class cⁿ¹e²C. And so on for i= 12:::j^;1. Finally, we expressc^nj;2ej^;1 aswj^;1c^nj;1 where wj^;1 is the minimal representative in the coset classc^nj;2ej^;1C.

We put wj to be the lexicographically minimal among all d^G-geodesic (d^G1- geodesic) representatives ofc^nj;1ej. As a result we obtain the word w=w¹:::wj

such thatw=e. This wordwis the required representative ofein L.

Note that in the casee⁶²Cwe havew¹C=e¹C,Cwj =Cej andCwiC=CeiC for 1 < i < j. Note also that there is >0 such that all words in L dene - quasigeodesics in the Cayley graph ofM. This, in particular, means that for every w²L

l(w) l^M(w) +:

Suppose now thatpand y are as in Lemma 3 and n^{2 Z}n⁶= 0. We will nd the representativewofpyⁿin the automatic languageLonM using the procedure described above. Note that ^hy^i\hxⁱ =^f1^gand so y^{n 2} G^;C since C = ^hxⁱ. Therefore p¹:::psyⁿ is a strictly alternating product of elements of G^;C and G^1;C. It is clear from the construction thatwhas the following form:

w=q¹:::qsv where

1. q_i is a d^G-geodesic word when p_i ² G^;C and q_i is a d^G1-geodesic word whenp_i²G^1;C

2. q¹C=p¹C andCq_iC=Cp_iC wheni >1

3. v is ad^G-geodesic word andv=cyⁿ for some c²C 4. w=p¹:::psyⁿ =pyⁿ

Corollary 2 implies thatl(v) =l^G(cyⁿ)r^{1 j}n^jfor somer¹>0 depending only onxyand independent ofn.

Therefore l(w) = l(q¹:::q_sv) l(v) r^{1 j}n^j. Since the language L consists of-quasigeodesics with respect tod^M, we conclude thatl^M(pyⁿ) l(w)

^;

r¹

^jn^j;which implies the statement of Lemma 3.

Lemma 4.

Suppose conditions of Theorem B are satised. Let u¹:::um ⁶² H be a strictly alternating product of elements from G^;C and G^1;C such that um ²G^1;C. Let p¹:::ps be a strictly alternating product of elements of F ^;C andF^1;C. Letq⁰ belong toG¹ifp¹²F^;Candq⁰belong toGwhenp¹²F^1;C (we allow q⁰²C).

Then either q⁰p¹:::psu¹:::um ends (when rewritten in the normal form with respect to(1)) in the element of G^1;C or q⁰p¹:::psu¹:::Aum²C.

Proof.

^Indeed,um² G^1;C and soq⁰p¹:::psu¹:::umends (when rewritten in the normal form with respect to (1)) in the element ofG^1;C unlessu^;1_m :::u^;11 is a terminal segment ofq⁰p¹:::ps that is eithermsand

ps^;m⁺¹:::psu¹:::um²C (2)

or q⁰p¹:::p_su¹:::u_m²C (3)

It is clear that (2) is impossible since ps^;m⁺¹:::ps ² H = gp(FF¹), C H andu¹:::um⁶²H. If (3) holds, we haveq⁰p¹:::psu¹:::um²Cas required. Thus Lemma 4 is established.

Lemma 5.

^Letz=u1:::um⁶²H be a strictly alternating product of elements of G^;C andG^1;C such thatum²G^1;C. Let y ²G be such that no nontrivial power ofy is conjugate inGto a power ofx=c. Then there is a constantK⁰>0 with the following property.

Letn^2Zand h²H. Let v be a d^M-geodesic from 1 to hand let u be a d^M- geodesic from 1 to zyⁿ. Take the vertex v(N) of v at the distance N from 1 and the vertexu(N) ofuat the distance N from1 (see Figure 3). Then

d^M(u(N)v(N))K⁰ N:

Proof.

Recall thatH =sgp(FF¹) =FCF¹M =GCG¹. Let >0 be such that any word from the automatic languageL on M denes a-quasigeodesic in the Cayley graph ofM.

Let ^ui be a d^G-geodesic (d^G1-geodesic) representative of ui, im. Let Y be a d^G-geodesic representative ofy. Since the elementzis xed and the cyclic subgroup

hyⁱ is quasiconvex inM, there is a constant¹ >0 such that for everyk^2Zthe word ^u¹:::u^m^;1u^mY^k is a ¹-quasigeodesic with respect to d^M. In particular, U = ^u¹:::u^m^;1u^mYⁿ is a ¹-quasigeodesic representative of zyⁿ with respect to d^M. Put ²=max(¹). Let >0 be such that any two²-quasigeodesics with common endpoints in the Cayley graph of M are-Hausdor-close. We will also assume that is such that for every k ^{2 Z}and every point x⁰ on a d^M-geodesic from 1 tozy^k there is k^02Z,k⁰²0k], such thatd^M(x⁰zy^k0).

Ifh²Candh^;1 =x^m, putp¹=h^;1 and letw¹be ad^G-geodesic representative ofp¹.

Ifh⁶²C leth^;1 =p¹p²:::ps be the strictly alternating product of elements of F^;C and F^1;C. Note thatp¹p²:::ps is also a strictly alternating product of elements ofG^;C and G^1;C. We then can nd the representativewof h^;1 in the automatic languageLonM which was described in Lemma 3.

Clearly,w=w¹w²:::ws, where

1. w¹C=p¹C andw¹is minimal in the cosetw¹C

2. forj < s CwjC=CpjC andwj is minimal in the cosetwjC 3. Cws=Cpsthat is ws=cpsfor some c²C

4. eachwi is adG ord^G1-geodesic word.

Note that sincepj ²FF¹ andCF,CF¹, the conditions above imply that wj ²FF¹forj = 1:::s.

Letv be ad^M geodesic from 1 tohand letube ad^M geodesic from 1 tozyⁿ. AssumeN is a positive number such thatN l(v) andN l(u).

Letv(N) be the point on the geodesicv at the distanceN from 1. Letu(N) be the point on the geodesicuat the distanceN from 1 (see Figure 3).

Recall thatz=u¹:::umandy are xed,um²G^1;C. Recall further thatY is ad^G-geodesic representative ofy. By the choice ofthere is a vertexV(N) ofwand a vertexU(N) =zy^k of U such that d^M(u(N)U(N)), d^M(v(N)V(N)) (see Figure 3). The segment S¹ of w from V(N) to 1 is a terminal segment of w=w¹:::w_s, and it has the form

S¹=qiwi⁺¹:::ws

v(N)

v V(N)

1

Y

Y

Y

Y

Y

Y u(N)

zyⁿ

s

i+1

1

h

u

U(N)=zy^k z

w

w

w

q_i

Figure 3

where is and qi is a nonempty terminal segment of wi. The segmentS² of U from 1 toU(N) is an initial segment of U = ^u¹:::u^m^;1u^mYⁿ of the form

S²= ^u¹:::u^m^;1u^mY^k

for some integer k ² 0n]. Notice that d^M(1u¹:::um^;1u^my^k) N ^; and therefore

jk^jl^M(y)l^A(y^k)N^;l(^u¹:::^um^;1u^m)^; and

jk^j(1=l^A(y))(N ^;l(^u¹:::u^_m^;1u^_m)^;): Thus, for some constantK¹>0 independent ofhn, we have

jk^jK¹ N:

By Lemma 4, either q_iwi⁺¹:::wsu¹:::um ² C orq_iwi⁺¹:::wsu¹:::um ends in the element of G^{1 ;}C, when rewritten in normal form with respect to (1).

Therefore, by Lemma 3, there is a constantD >0 independent ofh,nsuch that l^M(q_iw_i⁺¹:::w_su¹:::u_my^k) =l^M(S¹S²)D^jk^j

and hence l^M(q_iwi⁺¹:::wsu¹:::umy^k) K¹ D N. It remains to recall that

jd^M(u(N)v(N))^;l^M(q_iwi⁺¹:::wsu¹:::umy^k)^j 2to conclude that there is a constant K² > 0 independent of hn such that d^M(u(N)v(N)) K² N. This completes the proof of Lemma 5.

Proof of Theorem B.

^{Suppose z 2 Stab}M(K). We will show that z ² H by induction on the syllable length ofz with respect to presentation (1). When the syllable length of z is 0, that is z ² C, the statement is obvious. Suppose now that z ² StabM(K)^;H, the syllable length of z is m > 0 and the statement has been proved for elements ofStabM(K) of smaller syllable length. Writez as a strictly alternating productz =u¹:::um of elements fromG^;C andG^1;C. If um ² F F¹, then um ² H ^\StabM(K), and so u¹:::um^{;1 2} StabM(K).

Therefore, u¹:::u_m^{;1 2}H by the inductive hypothesis, u_m² H, and so z ² H. Thus,u_m²(G^;F)(G^1;F¹). Assume for deniteness thatu_m²G^1;F¹, that is,um=f¹t^j1 for somej⁶= 0, f¹²F¹.

Choosey ²F so that no power of y is conjugate in Gto a power of x. Fix a d^G-geodesic representativeY ofy.

Let y⁺ = lim_n

!1

y^{n 2} @M. By denition of K we have y^{+ 2} K and therefore zy⁺²K. This means that for anyN >0 there is an elementh²H and a positive poweryⁿ ofysuch that (hzyⁿ)¹> N, the Gromov inner product taken in thed^M- metric. This means thatl^M(h)N,l^M(zyⁿ)N andd^M(h(N)(zyⁿ)(N)) where h(N) and (zyⁿ)(N) are elements of M represented by initial segments of lengthN ofd^M-geodesic representatives ofhandzyⁿ.

Thenl^M(h(N)(zyⁿ)(N))K⁰ N whereK⁰is the constant independent ofh, nwhich is provided by Lemma 5. Thus,

l^M(h(N)(zyⁿ)(N))K⁰ N

and therefore N (1=K⁰) . This contradicts the fact that N can be chosen arbitrarily big.

Therefore,z⁶²StabM(K), which completes the proof of Theorem B.

Corollary 6.

^{LetM,G,G1,C andH} 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 normalizerV N_M(H) of H in M is equal to H.

Proof.

(a) Suppose there is a quasiconvex subgroup Q¹ of M such that @^M(H) =

@^M(Q¹) =K. Clearly,Q¹is innite since Kis nonempty. Set Q=StabM(K) =^fy²M^jyK=K^g:

Since Q¹ is innite and quasiconvex in M and Q = StabM(@^M(Q¹)), it follows from Lemma 3.9 of KS] that Q contains Q¹ as a subgroup of nite index and thereforeQis also quasiconvex inM. On the other hand, Theorem B implies that H =StabM(K), and soH =Q. This contradicts the fact thatH is not quasiconvex inM by Proposition A.

(b) It is not hard to see that A V NB(A) StabB(@B(A)) when A is an innite subgroup of a word hyperbolic group B. Indeed, if g ² V N_B(A), then A⁰ = A^\gAg^;1 has nite index n in A. Let A = A⁰ A⁰c¹ A⁰c_n^;1, and let D = max^fl^A(ci)^ji = 1:::n^;1^g. Suppose p ² @B(A). Then there is a sequence am ² A such that p = lim_m

!1

am. For each m there is b ² B with

lB(b)D+lB(g) such thatgamb =a_m²A⁰. Thereforegp ²@B(A⁰) =@B(A).

Since p² @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, AV NB(A)StabB(@B(A)).

For the subgroupH ofM we have H V NM(H) StabM(@^M(H)). On the other hand,StabM(@^M(H)) =H by Theorem B. Therefore,H =V NM(H).

References

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Math. Soc. (to appear).

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

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