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
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\textit{E. Martínez-Pedroza}, Groups Geom. Dyn. 3, No. 2, 317--342 (2009; Zbl 1186.20029)

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