# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Arzhantseva, G.; Minasyan, A.; Osin, D., The SQ-universality and residual properties of relatively hyperbolic groups, preprint, 2006, arxiv: · Zbl 1132.20022  Bekka, B.; Cowling, M.; de la Harpe, P., Some groups whose reduced $$C^\ast$$-algebra is simple, Publ. math. inst. hautes études sci., 80, 117-134, (1994) · Zbl 0827.22001  Bekka, B.; de la Harpe, P., Groups with simple reduced $$C^\ast$$-algebras, Expo. math., 18, 3, 215-230, (2000) · Zbl 0966.46032  B.H. Bowditch, Relatively hyperbolic groups, preprint, Southampton, 1998 · Zbl 0918.20027  Bridson, M.R.; de la Harpe, P., Mapping class groups and outer automorphism groups of free groups are $$C^\ast$$-simple, J. funct. anal., 212, 1, 195-205, (2004) · Zbl 1064.46052  Champetier, C.; Guirardel, V., Limit groups as limits of free groups, Israel J. math., 146, 1-75, (2005) · Zbl 1103.20026  Dahmani, F., Combination of convergence groups, Geom. topol., 7, 933-963, (2003) · Zbl 1037.20042  Delzant, T., Sous-groupes distingués quotients des groupes hyperboliques, Duke math. J., 83, 3, 661-682, (1996) · Zbl 0852.20032  Drutu, C.; Sapir, M., Tree graded spaces and asymptotic cones, Topology, 44, 5, 959-1058, (2005), with appendix by D. Osin and M. Sapir · Zbl 1101.20025  Farb, B., Relatively hyperbolic groups, Geom. funct. anal., 8, 810-840, (1998) · Zbl 0985.20027  Fendler, G., Simplicity of the reduced $$C^\ast$$-algebras of certain Coxeter groups, Illinois J. math., 47, 3, 883-897, (2003) · Zbl 1035.43006  Gromov, M., Hyperbolic groups, (), 75-263  de la Harpe, P., Reduced $$C^\ast$$-algebras of discrete groups which are simple with a unique trace, (), 230-253  de la Harpe, P., Groupes hyperboliques, algèbres d’opérateurs et un théorème de jolissaint, C. R. acad. sci. Paris Sér. I math., 307, 14, 771-774, (1988) · Zbl 0653.46059  de la Harpe, P., On simplicity of reduced $$C^\ast$$-algebras of groups, preprint, 2005, arxiv:  Hruska, G.C.; Kleiner, B., Hadamard spaces with isolated flats, Geom. topol., 9, 1501-1538, (2005) · Zbl 1087.20034  Murray, F.J.; von Neumann, J., On rings of operators. IV, Ann. of math. (2), 44, 716-808, (1943) · Zbl 0060.26903  Osin, D.V., Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. amer. math. soc., 179, 843, vi+100, (2006) · Zbl 1093.20025  Osin, D.V., Elementary subgroups of relatively hyperbolic groups and bounded generation, Internat. J. algebra comput., 16, 1, 99-118, (2006) · Zbl 1100.20033  Paschke, W.L.; Salinas, N., $$C^\ast$$-algebras associated with free products of groups, Pacific J. math., 82, 1, 211-221, (1979) · Zbl 0413.46049  Powers, R.T., Simplicity of the $$C^\ast$$-algebra associated with the free group on two generators, Duke math. J., 42, 151-156, (1975) · Zbl 0342.46046  Valette, A., Introduction to the baum – connes conjecture, (2002), Birkhäuser Basel · Zbl 1136.58013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.