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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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