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Relatively hyperbolic groups are \(C^*\)-simple. (English) Zbl 1115.20034
A countable discrete group is said to be \(C^*\)-simple if its reduced \(C^*\)-algebra is simple, that is, if it has no non-trivial two-sided ideals. \(C^*\)-simplicity is a form of non-amenability.
In the paper under review, the authors give a characterization of \(C^*\)-simple relatively hyperbolic groups. This characterization uses the notion of maximal finite normal subgroup, which was introduced in the paper by G. Arzhantseva, A. Minasyan and D. Osin [The SQ-universality and residual properties of relatively hyperbolic groups, preprint (2006)], in which the authors prove that every non-elementary group \(G\) which is hyperbolic relative to a collection of proper subgroups has a maximal finite normal subgroup.
The main result of the paper under review is then the following Theorem: Let \(G\) be a non-elementary group which is hyperbolic with respect to a collection of proper subgroups. If the maximal finite normal subgroup of \(G\) is reduced to the identity, then, for any finite subset \(F\) of \(G\setminus\{\text{Id}\}\) there exists an element \(g_0\in G\) of infinite order such that for each \(f\in F\), the subgroup of \(G\) generated by \(f\) and \(g_0\) is canonically isomorphic to the free product \(\langle f\rangle*\langle g_0\rangle\).
The last condition in the statement of the theorem was highlighted in the paper [M. Bekka, M. Cowling and P. de la Harpe, Publ. Math., Inst. Hautes Étud. Sci. 80, 117-134 (1995; Zbl 0827.22001)], where it was called property \({\mathbf P}_{\text{nai}}\).

MSC:
20F67 Hyperbolic groups and nonpositively curved groups
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
20E07 Subgroup theorems; subgroup growth
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