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Height in splittings of hyperbolic groups. (English) Zbl 1059.20040
Let \(H\) be a hyperbolic subgroup of the hyperbolic group \(G\). Call the elements \(\{g_i\mid i=1,\dots,n\}\) essentially distinct if \(Hg_i\neq Hg_j\) whenever \(i\neq j\). Conjugates of \(H\) by essentially distinct elements are called essentially distinct conjugates. The height of an infinite subgroup \(H\) in \(G\) is \(n\) if there is a collection of \(n\) essentially distinct conjugates of \(H\) such that the intersection of the collection is infinite and \(n\) is maximal with respect to this property. Finite groups have height \(0\) by definition.
The main result of R. Gitik, M. Mitra, E. Rips, and M. Sageev [Trans. Am. Math. Soc. 350, No. 1, 321-329 (1998; Zbl 0897.20030)] is the following: if \(H\) is a quasiconvex subgroup of a hyperbolic group \(G\), then \(H\) has finite height. The paper under review proves the converse of that theorem for hyperbolic groups \(G\) which split as \(G=G_1*_HG_2\) or \(G=G_1*_H\) with hyperbolic edge and vertex groups where the inclusions are quasi-isometric embeddings, giving an affirmative answer to a question of Swarup.
The main result states: Let \(G\) be a hyperbolic group splitting over \(H\) with hyperbolic edge and vertex groups; assume the inclusions of \(H\) are quasi-isometric embeddings; then \(H\) is of finite height in \(G\) if and only if it is quasiconvex in \(G\).
The paper includes details regarding consequences and questions related to the main theorem including malnormality, more general graphs of hyperbolic groups as well as more geometric applications.

20F67 Hyperbolic groups and nonpositively curved groups
20F65 Geometric group theory
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
57M05 Fundamental group, presentations, free differential calculus
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[1] Anderson J and Maskit B, Local connectivity of limit sets of Kleinian groups, preprint · Zbl 0869.30034
[2] Alonso J, Brady T, Cooper D, Ferlini V, Lustig M, Mihalik M, Shapiro M and Short H, Notes on word hyperbolic groups, Group theory from a geometrical viewpoint (eds) E Ghys, A Haefliger, A Verjovsky (1991) pp. 3–63 · Zbl 0849.20023
[3] Bestvina M and Feighn M, A combination theorem for negatively curved groups,J. Diff. Geom. 35 (1992) 85–101 · Zbl 0724.57029
[4] Bestvina M and Handel M, Train tracks and automorphisms of free groups,Ann. Math. 135 (1992) 1–51 · Zbl 0757.57004 · doi:10.2307/2946562
[5] Bestvina M, Feighn M and Handel M, Laminations, trees and irreducible automorphisms of free groups,GAFA 7(2) (1997) 215–244 · Zbl 0884.57002 · doi:10.1007/PL00001618
[6] Bestvina M, Feighn M and Handel M, The Tits’ alternative for Out(Fn) I: Dynamics of exponentially growing automorphisms, preprint · Zbl 0984.20025
[7] Bestvina M, Feighn M and Handel M, The Tits’ alternative for Out(Fn) II: A Kolchin Theorem, preprint · Zbl 1139.20026
[8] Bestvina M, Feighn M and Handel M, The Tits’ alternative for Out(Fn) III: Solvable subgroups, preprint · Zbl 1139.20026
[9] Bestvina M, Geometric group theory problem list, M Bestvina’s home page: http:math.utah.edu
[10] Bonahon F, Geodesic currents on negatively curved groups, in: Arboreal group theory (ed.) R C Alperin (Springer Verlag: MSRI Publ.) (1991) vol. 19, pp. 143–168 · Zbl 0772.57004
[11] Bonahon F, Bouts de varietes hyperboliques de dimension 3,Ann. Math. 124 (1986) 71–158 · Zbl 0671.57008 · doi:10.2307/1971388
[12] Bonahon F and Otal J P, Varietes hyperboliques a geodesiques arbitrairement courtes,Bull. LMS 20 (1988) 255–261 · Zbl 0648.53027
[13] Brady N, Branched coverings of cubical complexes and subgroups of hyperbolic groups, preprint · Zbl 0940.20048
[14] Canary R D, Epstein DBA and Green P, Notes on notes of Thurston, in: Analytical and geometric aspects of hyperbolic spaces (1987) (Conventry/Durham, 1984)Lond. Math. Soc. Lecture Notes Ser. III, pp. 3–92
[15] Cannon J and Thurston W P, Group invariant Peano curves, preprint · Zbl 1136.57009
[16] Coornaert M, Delzant T and Papadopoulos A, Geometrie et theorie des groupes,Lecture Notes in Math. (Springer Verlag) (1990) vol. 1441 · Zbl 0727.20018
[17] Farb B, The extrinsic geometry of subgroups and the generalized word problem,Proc. LMS 68(3) (1994) 577–593 · Zbl 0816.20032
[18] Farb B and Schwartz R E, Quasi-isometric rigidity of Hilbert modular groups, preprint · Zbl 0871.11035
[19] Fathi A, Laudenbach M and Poenaru V, Travaux de Thurston sur les surfaces,Asterisque 66–67 (1979) 1–284
[20] Floyd W J, Group completions and limit sets of Kleinian groups,Invent. Math. 57 (1980) 205–218 · Zbl 0428.20022 · doi:10.1007/BF01418926
[21] Gersten S, Cohomological lower bounds on isoperimetric functions of groups, preprint · Zbl 0933.20026
[22] Ghys E and de la Harpe P (eds), Sur les groupes hyperboliques d’apres Mikhael Gromov,Prog. Math. (Birkhauser, Boston, Ma.) (1990) vol. 83 · Zbl 0731.20025
[23] Gitik R, Mitra M, Rips E and Sageev M, Widths of subgroups,Trans. AMS (Jan. 1997) 321–329 · Zbl 0897.20030
[24] Gromov M, Hyperbolic groups, in: Essays in group theory, (ed.) Gersten (Springer Verlag: MSRI Publ.) (1985) vol. 8, 75–263
[25] Gromov M, Asymptotic invariants of infinite groups, in: Geometric group theory vol. 2;Lond. Math. Soc. Lecture Notes (Cambridge University Press) (1993) 182 · Zbl 0841.20039
[26] Hocking J G and Young G S,Topology (Addison Wesley) (1961)
[27] Klarreich E, Semiconjugacies between Kleinian group actions on the Riemann sphere, Ph.D. Thesis (SUNY, Stonybrook) (1997) · Zbl 1011.30035
[28] Lyndon R C and Schupp P E, Combinatorial group theory (Springer) (1977) · Zbl 0368.20023
[29] Masur H, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential,Duke Math. J. 66 (1992) 387–442 · Zbl 0780.30032 · doi:10.1215/S0012-7094-92-06613-0
[30] McMullen C, Iteration on Teichmuller space,Invent. Math. 9 (1990) 425–454 · Zbl 0695.57012 · doi:10.1007/BF01234427
[31] McMullen C, Amenability PoincarĂ© series and quasiconformal maps,Invent. Math. 97 (1989) 95–127 · Zbl 0672.30017 · doi:10.1007/BF01850656
[32] Minsky Y N, On rigidity, limit sets, and end invariants of hyperbolic 3-manifolds,J.A.M.S. 7 (1994) 539–588 · Zbl 0808.30027
[33] Minsky Y N, Teichmuller geodesics and ends of 3-manifolds,Topology 35 (1992) 1–25
[34] Minsky Y N, The classification of punctured torus groups, preprint · Zbl 0939.30034
[35] Mitra M, Maps on boundaries of hyperbolic metric states, Ph.D. Thesis (U.C. Berkeley) (1997)
[36] Mitra M, Cannon-Thurston maps for hyperbolic group extensions,Topology 137(3) (1998) 527–538 · Zbl 0907.20038 · doi:10.1016/S0040-9383(97)00036-0
[37] Mitra M, Ending laminations for hyperbolic group extensions,GAFA 7(2) (1997) 379–402 · Zbl 0880.57001 · doi:10.1007/PL00001624
[38] Mitra M, Cannon-Thurston maps for trees of hyperbolic metric spaces,J. Diff. Geom. 48(1) (1998) 135–164 · Zbl 0906.20023
[39] Mitra M, On a theorem of Scott and Swamp,Proc. AMS 127(6) (1999) 1625–1631 · Zbl 0918.20028 · doi:10.1090/S0002-9939-99-04935-7
[40] Mosher L, Hyperbolic extensions of groups,J. Pure Appl. Algebra 110(3) (1996) 305–314 · Zbl 0851.20037 · doi:10.1016/0022-4049(95)00081-X
[41] Mosher L, A hyperbolic-by-hyperbolic hyperbolic group, to appear inProc. AMS 125(12) (1997) 3447–3455 · Zbl 0895.20028 · doi:10.1090/S0002-9939-97-04249-4
[42] Paulin F, Outer automorphisms of hyperbolic groups and small actions on R-trees, in: Arboreal group theory (ed.) R C Alperin (MSRI Publ., Springer Verlag) (1991) vol. 19, pp.331–344
[43] Rips E and Sela Z, Structure and rigidity in hyperbolic groups,GAFA 4(3) (1994) 337–371 · Zbl 0818.20042 · doi:10.1007/BF01896245
[44] Scott P, There are no fake Seifert fibered spaces with infinite \(\pi\)1,Ann. Math. 117 (1983) 35–70 · Zbl 0516.57006 · doi:10.2307/2006970
[45] Scott P, Compact submanifolds of 3-manifolds,J. L.M.S. 7(2) (1973) 246–250 · Zbl 0266.57001
[46] Scott P, Subgroups of surface groups are almost geometric,J. L.M.S. 17 (1978) 555–65 · Zbl 0412.57006
[47] Scott P and Wall C T C, Topological methods in group theory, homological group theory (ed.) C T C Wall,London Math. Soc. Lecture Notes Series (Cambridge Univ. Press) (1979) vol. 36 · Zbl 0423.20023
[48] Scott P and Swamp G, Geometric finiteness of certain Kleinian groups,Proc. AMS 109 (1990) 765–768 · Zbl 0699.30040 · doi:10.1090/S0002-9939-1990-1013981-6
[49] Sela Z, Structure and rigidity in (Gromov) hyperbolic groups and discrete subgroups in rank 1 Lie groups, preprint · Zbl 0884.20025
[50] Short H, Quasiconvexity and a theorem of Howson’s group theory from a geometrical viewpoint (eds) E Ghys, A Haefliger and A Verjovsky (1991) · Zbl 0869.20023
[51] Sullivan D, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and related topics: Proceedings of the 1978 Stonybrook Conference,Ann. Math. Stud. (Princeton) (1981) vol. 97 · Zbl 0567.58015
[52] Thurston W P, The geometry and topology of 3-manifolds (Princeton University Notes) (1980)
[53] Thurston W P, Hyperbolic structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle, preprint
[54] Thurston W P, On the geometry and dynamics of diffeomorphisms of surfaces,Bull. AMS 19 (1987) 417–431 · Zbl 0674.57008 · doi:10.1090/S0273-0979-1988-15685-6
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