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On quasiconvexity and relatively hyperbolic structures on groups. (English) Zbl 1276.20055
From the text: Let $$G$$ be a group which is hyperbolic relative to a collection of subgroups $$\mathcal H_1$$, and it is also hyperbolic relative to a collection of subgroups $$\mathcal H_2$$. Suppose that $$\mathcal H_1\subset\mathcal H_2$$. We characterize when a relative quasiconvex subgroup of $$(G,\mathcal H_2)$$ is still relatively quasiconvex in $$(G,\mathcal H_1)$$. We also show that relative quasiconvexity is preserved when passing from $$(G,\mathcal H_1)$$ to $$(G,\mathcal H_2)$$. Applications are discussed.
Our main result is the following: Theorem 1.1. Let $$G$$ be a countable group, and let $$(G,\mathcal H_1)$$ and $$(G,\mathcal H_2)$$ be relatively hyperbolic structures with $$\mathcal H_1\subset\mathcal H_2$$ and $$\mathcal H_2$$ finite. (1) If $$Q$$ is a quasiconvex subgroup of $$(G,\mathcal H_1)$$, then $$Q$$ is quasiconvex in $$(G,\mathcal H_2)$$. (2) If $$Q$$ is a quasiconvex subgroup of $$(G,\mathcal H_2)$$, and for each $$H\in\mathcal H_2\setminus\mathcal H_1$$ and each $$g\in G$$ the subgroup $$Q\cap gHg^{-1}$$ is quasiconvex in $$(G,\mathcal H_1)$$, then $$Q$$ is quasiconvex in $$(G,\mathcal H_1)$$.

##### MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 20F65 Geometric group theory
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