Ž. Journal of Algebra 27, 6572 999 Article ID jabr.998.7789, available online at http:www.idealibrary.com on Ping-Pong on Negatively Curved Groups Rita Gitik* Department of Mathematics, California Institute of Technology, Pasadena, California 925 E-mail: ritagtk@math.lsa.umich.edu Communicated by Efim Zelmano Received June 4, 997 INTRODUCTION The following result for classical Schottky groups of rank 2 goes back to Blaschke, Klein, Schottky, and Poincare. It can be viewed as a special case of the KleinMaskit combination theorem ŽMas, p. 35.. LEMMA. Let h and k be hyperbolic isometries of hyperbolic space such that their axes hae disjoint endpoints. There exists a constant C which depends only on h and k, such that if m C and n C, then the subgroup ² h m, k n : is isomorphic to the free product ² h m : ² k n :. The well-known proof of this result can be abstracted into the following simple form ŽCf. L-S, p. 68.. Let H and K be subgroups of a group G, and let ² H, K : be the smallest subgroup of G containing H and K. THE PING-PONG LEMMA. Let G be a group acting on a set S. Let H and K be subgroups of G such that K : G 2 or H : G 2, where G H K. If there exist disjoint nonempty subsets SK and SH of S such that K G maps SK into SH and H G maps SH into S K, then the subgroup ² H, K : is isomorphic to the amalgamated free product ² H, K : H K. Proof. Let hk h be an element of ² H, K : n G, where all hi and k are not in G. Then for any x S, Ž hk h.ž x. i H n S K,so Ž hk h.ž x. x; hence any element of odd syllable length in ² H, K : n is non-trivial. Consider an element of even length hk hnk n. Without loss of generality, H : G 2; hence there exists h H such that h G * Research partially supported by NSF Grant DMS 9224 at MSRI. Current address: A and H Consultants, Ann Arbor, MI. G 65 2-869399 $3. Copyright 999 by Academic Press All rights of reproduction in any form reserved.
66 RITA GITIK and h h G. But then the element h hk hnknh has odd syllable length; hence it is nontrivial. Therefore hk hnkn is nontrivial. Note that there exist counterexamples to the Ping-Pong lemma when K : G H : G 2, even if the action of G on S is free. The Ping-Pong Lemma allows us to describe the subgroup ² H, K :. In general, even in the simple case when H ² h: and K ² k: are infinite cyclic groups with trivial intersection, we do not have much information about the group ² h, k :. In particular, this group need not be isomorphic to the free group of rank 2. In this paper we describe the subgroup ² H, K : when H and K are quasiconvex subgroups of a negatively curved group G. We also give a condition for ² H, K : to be quasiconvex in G. In general, even if both H ² h: and K ² k: are infinite cyclic Žhence quasiconvex in G; Gr, p. 2., the group ² h, k: might be non-quasiconvex in G. This might happen even if ² h, k: is isomorphic to the free group of rank 2. We prove the following results. THEOREM. Let H and K be -quasiconex subgroups of a -negatiely cured group G. There exists a constant C, which depends only on G,, and, with the following property. For any subgroups H H and K K with H K H K G, if all the elements in H and in K which are shorter than C belong to G, then ² H, K : H K. If, in addition, G H and K are quasiconex in G, then the subgroup ² H, K : is quasiconex in G. Recall that a subgroup H is malnormal in G if for any g H the intersection of H and ghg is trivial. THEOREM 2. Let H and K be -quasiconex subgroups of a -negatiely cured group G. Assume that H is malnormal in G. There exists a constant C, which depends only on G,, and, with the following property. For any subgroup H H with H K H K G, if all the elements in H which are shorter than C belong to G, then ² H, K : HG K. If, in addition, H is quasiconex in G, then the subgroup ² H, K : is quasiconex in G. Of course, Theorems and 2 might be vacuously true, because there might be no subgroups H and K with the required properties. However, these theorems have interesting applications in the following frequently encountered case. Recall that a group K is residually finite if for any finite set of nontrivial elements in K, there exists a finite index subgroup of K which does not contain this set. The class of residually finite groups is very rich; it contains all finitely generated linear groups and all fundamental groups of geometric 3-manifolds.
PING-PONG ON NEGATIVELY CURVED GROUPS 67 If K is a quasiconvex subgroup of a negatively curved group G, then K is finitely generated, so for any positive integer n, K has only finitely many elements shorter than n. Hence if such K is infinite and residually finite, it has an infinite family of distinct finite index subgroups K n, such that K n does not contain nontrivial elements shorter than n. AsK n is a finite index subgroup of a quasiconvex subgroup K, it is quasiconvex in G: hence we have the following results. COROLLARY 3. Let H and K be infinite quasiconex subgroups of a negatiely cured group G. Assume that H and K are residually finite. If the intersection of H and K is triial, then there exist infinite families of distinct finite index subgroups Hm of H and K n of K, such that the subgroups ² H, K : are quasiconex in G and ² H, K : H K. m n m n m n COROLLARY 4. Let H and K be infinite quasiconex subgroups of a negatiely cured group G. Assume that H is malnormal in G and residually finite. If the intersection of H and K is triial, then there exists an infinite family of distinct finite index subgroups Hm of H such that the subgroup ² H, K : is quasiconex in G and ² H, K : H K. m m m The above corollaries apply when the intersection of K and H is trivial. In general, recall that a group K is LERF Žlocally extended residually finite. if for any finitely generated subgroup K of K and for any finite set S of elements in K, but not in K, there exists a finite index subgroup of K which contains K, but does not contain S. The class of LERF groups is much smaller than the class of residually finite groups, but it still contains many interesting groups. For example, free groups, surface groups, and nilpotent groups are LERF. Let K be an infinite quasiconvex subgroup of a negatively curved group G, and let G be a finitely generated subgroup of K. If K is LERF, then it has an infinite family of distinct finite index subgroups K containing G, n such that all the elements in K which are shorter than n belong to G. n Hence we have the following results. COROLLARY 5. Let H and K be infinite quasiconex subgroups of a negatiely cured group G with H K G. Assume that H and K are LERF. Then there exist infinite families of distinct finite index subgroups H m of H and K of K containing G, such that the subgroups ² H, K : n m n are quasiconex in G and ² H, K : H K. m n m G n COROLLARY 6. Let H and K be infinite quasiconex subgroups of a negatiely cured group G with H K G. Assume that H is malnormal in G and LERF. Then there exists an infinite family of distinct finite index subgroups H of H containing G, such that the subgroup ² H, K: m m is quasiconex in G and ² H, K : H K. m m G
68 RITA GITIK PROOFS OF THE RESULTS Let X be a set and let X* x, x x X 4, where Ž x. x for x X. Denote the set of all words in X* by WŽ X*., and denote the equality of two words by. Let G be a group generated by the set X*, and let CayleyŽ G. be the Cayley graph of G with respect to the generating set X*. The set of vertices of CayleyŽ G. is G, the set of edges of CayleyŽ G. is G X*, and the edge Ž g, x. joins the vertex g to gx. DEFINITION. The label of the path p Ž g, x.ž gx, x. Ž 2 gxx2 x, x. in CayleyŽ G. is the word LabŽ p. x x WŽ X*. n n n. As usual, we identify the word LabŽ p. with the corresponding element in G. We denote the initial and the terminal vertices of p by Ž p. and Ž p., respectively, and the inverse of p by p. Denote the length of the path p by p, where Ž g, x.ž gx, x. Ž gx x x, x. n. 2 2 n n A geodesic in the Cayley graph is a shortest path joining two vertices. A group G is -negatively curved if any side of any geodesic triangle in CayleyŽ G. belong to the -neighborhood of the union of two other sides Žsee Gr, C-D-P.. Let, L, and. A path p in the Cayley graph is a Ž,.-quasigeodesic if for any subpath p of p and for any geodesic with the same endpoints as p, p. A path p is a local Ž,, L. -quasigeodesic if for any subpath p of p that is shorter than L and for any geodesic with the same endpionts as p, p Žcf. C-D-P, p. 24.. Theorem.4 Ž p. 25. of C-D-P Žsee also Gr, p. 87. states that for any and for any there exist constants Ž L,,. that depend only on Ž,. and, such that any local Ž,, L. -quasigeodesic in G is a global Ž,.-quasigeodesic in G. Recall that H is a -quasiconvex subgroup of G if any geodesic in CayleyŽ G. which has its endpoints in H belongs to the -neighborhood of H. Proof of Theorem. Let H and K be subgroups of H and K, respectively, such that H K H K G. Consider an element l of ² H, K : such that l G. Then there exists a representation l hk kmh m, where hi H, ki K, k i, and hi do not belong to G, k i, and hi are geodesics in G, h is a shortest representative of the coset hg, hm is a shortest representative of the coset G h m, and for i m, hi is a shortest representative of the double coset G hig. Ž The elements h and h might be trivial.. Let p be the path in CayleyŽ G. m beginning at with the decomposition of the form p pq qm p m, where LabŽ p. h and LabŽ q. k. i i i i
PING-PONG ON NEGATIVELY CURVED GROUPS 69 Let A be the number of words in G which are shorter than 2. As mentioned above, there exist constants Ž L,,. which depend only on Ž, A,., such that any local Ž, Ž 4 A., L. 3 -quasigeodesic in G is a global Ž,.-quasigeodesic in G. Let C maxž L,.. We claim that if all the elements in H and in K which are shorter than C belong to G, then any path p, as above, is a Ž,.-quasigeodesic in G. Indeed, it is enough to show that p is a local Ž Ž.. 3, 4 A, L -quasigeodesic in G. As h G and k G, it follows that q C and p i i i i C.As L C, any subpath t of p with t L has a Ž unique. decomposition tt, 2 where without loss of generality, t is a subpath of some pi and t2 is a subpath of q i. Let t3 be a geodesic in G joining the endpoints of t. We will show that t t3 Ž 4 A. 3. Without loss of generality, assume that t t. If t t, then t t t t 2 3 3 2 3. So assume that t t.asttt is a geodesic triangle in CayleyŽ G. 3 2 3,itis-thin; hence there exists the decomposition t t t, where t is the maximal subpath of t which belongs to the -neighborhood of t, and t 2 belongs to the -neighborhood of t 3. It follows from Lemma 7 Ž below. that t A 4; hence t 3 t t t t A 4. Therefore, t t Ž A 4. 3 3 3, proving that p is a local Ž, Ž 4 A., L. 3 -quasigeodesic in G. So p is a global Ž,.-quasigeodesic in G. The definition of C implies that if a Ž,.-quasigeodesic p in G is longer than C, then LabŽ p.. As any element l ² H, K : that is not in G has a representative LabŽ p. in G, as above, with p C, it follows that l ; hence ² H, K : HG K. To prove the second part of the theorem, for any element l ² H, K : which is not in G, consider a geodesic in G joining the endpoints of the path p, as above. As G is negatively curved, there exists a constant which depends only on Ž,. and, such that belongs to the -neighborhood of p. Assume that H and K are -quasiconvex in G; then any vertex i on pi is in the -neighborhood of hk kih, and any vertex wi on qi is in the -neighborhood of hk hk, i so the path p belongs to the Ž.-neighborhood of ² H, K :. If l G, then l H, so any geodesic labeled with l with Ž. belongs to the -neighborhood of H. It follows that the subgroup ² H, K : is Ž.-quasiconvex in G. LEMMA 7. Using the notation of the proof of Theorem, t A 4. Proof. To simplify notation, we drop the subscript i, sot is a subpath of p, t is a subpath of q, LabŽ p. h, and LabŽ q. 2 k. As H H and K K, we consider h as an element of H and k as an element of K. Without loss of generality, assume that q begins at Ž so it ends at k.; then p begins at h and ends at. As K and H are -quasiconvex in G, any
7 RITA GITIK vertex i on p is in the -neighborhood of H, and any vertex wi on q is in the -neighborhood of K. Hence we can find vertices and in t 2, w and w in t, h and h in H, and k and k in K such that, w 2 2 i i,, Ž h. Ž.,, Ž h. Ž., w, k, and w, k 2 2. Then hk 2 and h k 2. Assume that t A 4. Then we can find vertices, as above, which, in Ž. Ž. addition, satisfy 2, 4 and hk h k. But then h h k Ž k. Ž., so both products are in G.As h is a shortest element in the double coset G hg, it follows that h hh Ž. Ž. h. Let r be a geodesic joining Ž h. Ž. to 2, let s be a subpath of p joining 2 to, and let s be a subpath of p joining h to. Then h p s s 2 and hh Ž. Ž. h hh Ž. Ž. h s r h ; hence s s s r h,sosr2rh.ashsr, and as r, it follows that h 2 h. However, as, 4, the triangle inequality implies that h 2 Ž h. s r,, r, 4, 2 2. Let a be a geodesic joining Ž h. to.asa, the triangle inequality implies that h Ž h., a,. Hence, h h 2, a contradiction. Therefore, t A 4. We use the following property of malnormal quasiconvex subgroups of negatively curved groups Žcf. Gi.. The original proof of this fact for the special case when G is a free group is due to Rips ŽG-R.. Let H be a subgroup of G, and let GH denote the set of right cosets of H in G. The relative Cayley graph of G with respect to H is an oriented graph whose vertices are the cosets GH and the set of edges is Ž GH. X*, such that an edge Ž Hg, x. begins at the vertex Hg and ends at the vertex Hgx. We denote it CayleyŽ G, H.. Note that for any path p in CayleyŽ G, H. if Ž p. H, then Ž p. H LabŽ p., so a path p beginning at H is closed, if and only if LabŽ p. H. LEMMA 8. Let H be a malnormal -quasiconex subgroup of a finitely generated group G, and let be a nonnegatie constant. ŽThe group G does not hae to be negatiely cured.. Let t be a path in CayleyŽ G. 2 such that and are geodesics in CayleyŽ G., LabŽ. H, LabŽ. 2 2 H, and LabŽ. t H. Let m be the number of ertices in the ball of radius 2 Ž. 2 around H in Cayley G, H, and let M m. Then any subpath of which belongs to the 2-neighborhood of in CayleyŽ G. 2 is shorter than M. Proof of Lemma 8. Assume that M. Without loss of generality, Ž.. Let be a geodesic in CayleyŽ G. 2 beginning at such that Ž Lab. LabŽ., and let be the subpath of such that LabŽ.
PING-PONG ON NEGATIVELY CURVED GROUPS 7 LabŽ.. Let : CayleyŽ G. CayleyŽ G, H. be the projection map Ž g. Hg and Ž g, x. Ž Hg, x.. Note that and are geodesics in CayleyŽ G. 2 beginning at, LabŽ. H and LabŽ. 2 H, so as H is -quasiconvex in G, the projection maps and 2 into a ball of radius around H in CayleyŽ G, H.. As N Ž. CayleyŽ G., it follows that Ž. 2 2 N Ž H. CayleyŽ G, H.. Denote the vertices of Ž. 2 by,...,n and the vertices of Ž. by w,...,w. As n M, and n n, w,...,w 4 N Ž H., there exist i j n such that Ž, w. n 2 i i Ž, w.. Let p and p be subpaths of Ž. connecting Ž Ž.. j j 2 and i and and, respectively. Let q and q be subpaths of Ž. i j 2 connecting H Ž Ž.. and w and w and w, respectively. Let s Ž t. i i j and let p denote the path p with the opposite orientation. Then spp2ps and qqq are closed paths in CayleyŽ G, H. 2 beginning at H, so LabŽ sp p p s. H and LabŽ qqq. 2 2 H. But LabŽ p. LabŽ q. and LabŽ p. LabŽ q., so LabŽ s. 2 2 Ž. Lab p p p Lab Ž s. H, and LabŽ p p f. 2 2 H. Therefore the malnormality of H in G implies that LabŽ s. H, contradicting the assumption that LabŽ s. LabŽ t. H. Hence a M. Proof of Theorem 2. Let H be a subgroup of H such that H K H K G. Let l be an element of ² H, K : such that l G. Then there exists a representation l hk kmh m, where ki and hi are as in the proof of Theorem. Let p be a path in CayleyŽ G. with a decomposition of the form p pq q p, where LabŽ p. m m i hi and LabŽ q. i k i. Let M be as in Lemma 8, and let A be as in the proof of Theorem. As was mentioned above, there exist constants Ž L,,. which depend only on Ž, A,., such that any local Ž, Ž 6 4 A M., L. -quasigeodesic in G is a global Ž,. -quasigeodesic in G. Let C maxž L,.. We claim that if all the elements in H which are shorter than C belong to G, then any path p, as above, is a Ž,. -quasigeodesic in G. Indeed, it is enough to show that p is a local Ž Ž.. 6, 4 A M, L -quasigeodesic in G. As h G, it follows that p i i C.As L C, any subpath t of p with t L has a Ž unique. decomposition ttt, 2 3 where t and t3 are subpaths of some p and p, and t is a subpath of q Ž i i 2 i some of ti might be empty.. Let t4 be a geodesic in G connecting the endpoints of t. If t 2t3, then t t t3, so t t Žt t. 2 3 4 2 3 2t3 t3 t3. So assume that t 2t 2 3. Without loss of generality, assume that t t ; hence t t 3 6. As tttt 2 3 4 is a geodesic 4-gon in a -negatively curved group G, there exists a decomposition t s s s such that 2 3 4
72 RITA GITIK s2 belongs to the -neighborhood of t 2, s3 belongs to the 2-neighborhood of t 3, and s4 belongs to the -neighborhood of t 4. According to Lemma 8, s M, and according to Lemma 7, s 4 A. But then t s 3 2 4 4 t s s t 4 A M t 2 3 6 4 A M. Hence t t6 Ž M 4 A., so the path p is a local Ž, Ž 4 6 M 4 A., L. -quasigeodesic in G; hence it is a Ž,. -quasigeodesic in G. Then we conclude the argument as in the proof of Theorem. ACKNOWLEDGMENTS The author thanks D. Cooper, S. Margolis, W. Neumann, and J. Koh for helpful conversations. C-D-P REFERENCES M. Coornaert, T. Delzant, and A. Papadopoulos, Geometrie et theorie des groupes, Lecture Notes in Mathematics, vol. 44, Springer-Verlag, BerlinNew York, 99. Gi R. Gitik, On quasiconvex subgroups of negative groups, J. Pure Appl. Algebra 9 Ž 997., 5569. G-R R. Gitik and E. Rips, On separability property of groups, Int. J. Algebra Comput. 5 Ž 995., 7377. Gr M. Gromov, Hyperbolic groups, in Essays in Group Theory Ž S. M. Gersten, Ed.., MSRI Series, Vol. 8, pp. 75263. Springer-Verlag, BerlinNew York, 987. L-S R. Lyndon and P. Schupp, Combinatorial Group Theory, Springer-Verlag, BerlinNew York, 977. Mac A. M. Macbeath, Packings, free products and residually finite groups, Proc. Cambridge Philos. Soc. 59 Ž 963., 555558. Mas B. Maskit, Kleinian Groups, Springer-Verlag, BerlinNew York, 987.