Groups, Geometry, and Dynamics


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Volume 8, Issue 3, 2014, pp. 933–983
DOI: 10.4171/GGD/253

Published online: 2014-10-02

$C$*-simple groups without free subgroups

Alexander Olshanskii[1] and Denis Osin[2]

(1) Vanderbilt University, Nashville, United States
(2) Vanderbilt University, Nashville, United States

We construct first examples of non-trivial groups without non-cyclic free subgroups whose reduced $C$*-algebra is simple and has unique trace. This answers a question of de la Harpe. Both torsion and torsion free examples are provided. In particular, we show that the reduced $C$*-algebra of the free Burnside group $B(m, n)$ of rank $m$ ≥ 2 and any sufficiently large odd exponent $n$ is simple and has unique trace.

Keywords: Simple $C$*algebra, $C$*simple group, unique trace, free Burnside group, free subgroup, relatively hyperbolic group, small cancellation theory

Olshanskii Alexander, Osin Denis: $C$*-simple groups without free subgroups. Groups Geom. Dyn. 8 (2014), 933-983. doi: 10.4171/GGD/253