Quasi-isometries, boundaries and JSJ-decompositions of relatively hyperbolic groups. (Q2870238)

Suppose \(\Gamma_1\) is a finitely generated group, hyperbolic relative to a finite collection \(\mathcal A_1\), such that no element in \(\mathcal A_1\) is properly hyperbolic. Let \(\Gamma_2\) be also a finitely generated group, quasi-isometric to \(\Gamma_1\). In this article, it is proved that there exists a collection \(\mathcal A_2\) of subgroups in \(\Gamma_2\), such that the cusped spaces of \((\Gamma_1,\mathcal A_1)\) and \((\Gamma_2,\mathcal A_2)\) are quasi-isometric. A similar result is also proved for cone spaces of \((\Gamma_1,\mathcal A_1)\) and \((\Gamma_2,\mathcal A_2)\). As a result, it is shown that the cusped spaces \(X(\Gamma_1,\mathcal A_1)\) and \(X(\Gamma_2,\mathcal A_2)\) have homeomorphic boundaries. The author applies the results to obtain a JSJ-decomposition for relatively hyperbolic groups which is invariant under quasi-isometry and outer automorphisms as well as a related splitting of the quasi-isometry groups of relatively hyperbolic groups.