## Combination of quasiconvex subgroups of relatively hyperbolic groups.(English)Zbl 1186.20029

Let $$G$$ 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 $$G$$.

### 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
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