Combination of quasiconvex subgroups of relatively hyperbolic groups.

*(English)*Zbl 1186.20029Let \(G\) be a group generated by a finite set \(X\) and hyperbolic relative to a collection of subgroups \(\mathcal H\). A subgroup of \(G\) is called parabolic if it can be conjugated into one of the subgroups in \(\mathcal H\). Moreover, a subgroup of \(G\) is called a relatively quasiconvex subgroup if it is a quasiconvex subgroup of the coned-off Cayley graph of \((G,X,\mathcal H)\).

In the paper under review are proved the following main theorems. 1. For any relatively quasiconvex subgroup \(Q\) and any maximal parabolic subgroup \(P\) of \(G\), there is a constant \(C=C(Q,P)\geq 0\) with the following property. If \(R\) is a subgroup of \(P\) such that (a) \(Q\cap P\subset R\), and (b) \(d_X(g,1)\geq C\) for any \(g\in R\setminus Q\), then the natural homomorphism \(Q*_{Q\cap R}R\to G\) is injective with image a relatively quasiconvex subgroup. Moreover, every parabolic subgroup of \(\langle Q\cup R\rangle\subset G\) is either conjugate to a subgroup of \(Q\) or a subgroup of \(R\) in \(\langle Q\cup R\rangle\).

2. For any pair of relatively quasiconvex subgroups \(Q_1\) and \(Q_2\), and any maximal parabolic subgroup \(P\) such that \(R=Q_1\cap P=Q_2\cap P\), there is a constant \(C=C(Q_1, Q_2,P)\geq 0\) with the following property. If \(h\in P\) is such that (a) \(hRh^{-1}=R\), and (b) \(d_X(q,1)\geq C\) for any \(g\in RhR\), then the natural homomorphism \(Q_1*_RhQ_2h^{-1}\to G\) is injective and its image is a relatively quasiconvex subgroup. Moreover, every parabolic subgroup of \(\langle Q_1\cup hQ_2h^{-1}\rangle\subset G\) is either conjugate to a subgroup of \(Q_1\) or \(hQ_2h^{-1}\) in \(\langle Q_1\cup hQ_2h^{-1}\rangle\). Here \(d_X\) denotes a word metric induced by \(X\) on \(G\).

In the paper under review are proved the following main theorems. 1. For any relatively quasiconvex subgroup \(Q\) and any maximal parabolic subgroup \(P\) of \(G\), there is a constant \(C=C(Q,P)\geq 0\) with the following property. If \(R\) is a subgroup of \(P\) such that (a) \(Q\cap P\subset R\), and (b) \(d_X(g,1)\geq C\) for any \(g\in R\setminus Q\), then the natural homomorphism \(Q*_{Q\cap R}R\to G\) is injective with image a relatively quasiconvex subgroup. Moreover, every parabolic subgroup of \(\langle Q\cup R\rangle\subset G\) is either conjugate to a subgroup of \(Q\) or a subgroup of \(R\) in \(\langle Q\cup R\rangle\).

2. For any pair of relatively quasiconvex subgroups \(Q_1\) and \(Q_2\), and any maximal parabolic subgroup \(P\) such that \(R=Q_1\cap P=Q_2\cap P\), there is a constant \(C=C(Q_1, Q_2,P)\geq 0\) with the following property. If \(h\in P\) is such that (a) \(hRh^{-1}=R\), and (b) \(d_X(q,1)\geq C\) for any \(g\in RhR\), then the natural homomorphism \(Q_1*_RhQ_2h^{-1}\to G\) is injective and its image is a relatively quasiconvex subgroup. Moreover, every parabolic subgroup of \(\langle Q_1\cup hQ_2h^{-1}\rangle\subset G\) is either conjugate to a subgroup of \(Q_1\) or \(hQ_2h^{-1}\) in \(\langle Q_1\cup hQ_2h^{-1}\rangle\). Here \(d_X\) denotes a word metric induced by \(X\) on \(G\).

Reviewer: Andrzej Szczepański (Gdańsk)

##### MSC:

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 |

20F05 | Generators, relations, and presentations of groups |

57M07 | Topological methods in group theory |

20E07 | Subgroup theorems; subgroup growth |

##### Keywords:

coned-off Cayley graphs; relative hyperbolicity; quasiconvex subgroups; combination theorems; parabolic subgroups##### References:

[1] | I. Agol, D. Groves, J. Manning, Residual finiteness, QCERF, and fillings of hyperbolic groups. Geom. Topol. 13 (2009), 1043-1073. · Zbl 1229.20037 |

[2] | E. Alibegović, A combination theorem for relatively hyperbolic groups. Bull. London Math. Soc. 37 (2005), 459-466. · Zbl 1074.57001 |

[3] | J. W. Anderson, J. Aramayona, and K. J. Shackleton, An obstruction to the strong rel- ative hyperbolicity of a group. J. Group Theory 10 (2007), 749-756. · Zbl 1188.20041 |

[4] | G. N. Arzhantseva, On quasiconvex subgroups of word hyperbolic groups. Geom. Dedi- cata 87 (2001), 191-208. · Zbl 0994.20036 |

[5] | G. Arzhantseva and A. Minasyan, Relatively hyperbolic groups are C -simple. J. Funct. Anal. 243 (2007), 345-351. · Zbl 1115.20034 |

[6] | M. Baker and D. Cooper, A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds. J. Topology 1 (2008), 603-642. · Zbl 1151.57014 |

[7] | J. Behrstock, C. Drutu, and L. Mosher, Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity. Preprint 2005. |

[8] | M. Bestvina, Questions in geometric group theory. Preprint 2004. |

[9] | B. H. Bowditch, Relatively hyperbolic groups. Preprint 1999. · Zbl 1259.20052 |

[10] | N. Brady, M. Forester, and E. Martinez-Pedroza, Surface subgroups of hyperbolization of groups. In preparation. |

[11] | D. Calegari, Surface subgroups from homology. Geom. Topol. 12 (2008), 1995-2007. · Zbl 1185.20046 |

[12] | D. Cooper, D. D. Long, and A. W. Reid, Essential closed surfaces in bounded 3-manifolds. J. Amer. Math. Soc. 10 (1997), 553-563. · Zbl 0896.57009 |

[13] | D. Cooper and D. D. Long, Some surface subgroups survive surgery. Geom. Topol. 5 (2001), 347-367. · Zbl 1009.57017 |

[14] | F. Dahmani, Combination of convergence groups. Geom. Topol. 7 (2003), 933-963. · Zbl 1037.20042 |

[15] | C. Dru\?tu and M. Sapir, Tree-graded spaces and asymptotic cones of groups. Topology 44 (2005), 959-1058. · Zbl 1101.20025 |

[16] | B. Farb, Relatively hyperbolic groups. Geom. Funct. Anal. 8 (1998), 810-840. · Zbl 0985.20027 |

[17] | R. Gitik, Ping-pong on negatively curved groups. J. Algebra 217 (1999), 65-72. · Zbl 0936.20019 |

[18] | M. Gromov, Hyperbolic groups. In Essays in group theory , Math. Sci. Res. Inst. Publ. 8, Springer-Verlag, New York 1987, 75-263. · Zbl 0634.20015 |

[19] | D. Groves and J. F. Manning, Dehn filling in relatively hyperbolic groups. Israel J. Math. 168 (2008), 317-429. · Zbl 1211.20038 |

[20] | G. C. Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups. Preprint 2008. · Zbl 1202.20046 |

[21] | C. J. Leininger and A. W. Reid, A combination theorem for Veech subgroups of the map- ping class group. Geom. Funct. Anal. 16 (2006), 403-436. · Zbl 1099.57002 |

[22] | J. F. Manning and E. Martinez-Pedroza, Separation of relatively quasiconvex subgroups. Preprint 2008. · Zbl 1201.20024 |

[23] | B. Maskit, Kleinian groups . Grundlehren Math. Wiss. 287, Springer-Verlag, Berlin 1988. · Zbl 0627.30039 |

[24] | H. A. Masur and Y. N. Minsky, Geometry of the complex of curves I: hyperbolicity. Hyperbolicity. Invent. Math. 138 (1999), 103-149. · Zbl 0941.32012 |

[25] | D. V. Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems. Mem. Amer. Math. Soc. 179 (2006), no. 843. · Zbl 1093.20025 |

[26] | D. V. Osin, Peripheral fillings of relatively hyperbolic groups. Invent. Math. 167 (2007), 295-326. · Zbl 1116.20031 |

[27] | M. V. Sapir, Some group theory problems. Internat. J. Algebra Comput. 17 (2007), 1189-1214. · Zbl 1172.20032 |

[28] | J. Stallings, A finitely presented group whose 3-dimensional integral homology is not finitely generated. Amer. J. Math. 85 (1963), 541-543. · Zbl 0122.27301 |

[29] | A. Szczepański, Relatively hyperbolic groups. Michigan Math. J. 45 (1998), 611-618. · Zbl 0962.20031 |

[30] | A. Yaman, A topological characterisation of relatively hyperbolic groups. J. Reine Angew. Math. 566 (2004), 41-89. · Zbl 1043.20020 |

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