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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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##### References:
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