Let $G$ be a countable discrete group, $C_r^*(G)$ its reduced $C^*$-algebra. We say that $G$ has stable rank 1 if $C_r^*(G)$ has stable rank one, that is, the set of invertible elements is dense in $C_r^*(G)$.
Finite groups and non-cyclic torsion free hyperbolic groups have stable rank 1. For motivation and more examples see [Dykema, Kenneth J.; de la Harpe, Pierre Some groups whose reduced $C^*$-algebras have stable rank one. J. Math. Pures Appl. (9) 78 (1999), no. 6, 591–608.]
Here is a basic question, which I do not know how to answer.
Question. Suppose that $Q$ is a group of stable rank 1 and let $G$ be an extension $$ 1\to K\to G\to Q\to 1, $$ where $K$ is finite. Is it true that $G$ also has stable rank 1?
For example, if one wants to show that all non-elementary hyperbolic groups (not necessarily torsion free) have stable rank 1, then the affirmative answer to the above question would be a natural first step.
PS. Just to add some motivation: I think that the affirmative answer to the question will likely imply that every acylindrically hyperbolic group in the sense of arXiv:1304.1246 has stable rank one. This would cover all hyperbolic and relatively hyperbolic groups, mapping class groups, $Out(F_n)$ and many other examples. To be precise, I think I know how to prove this, but I have not checked all details.
PPS. For details about stable rank of $C^*$-algebras and applications to K-theory, see [M.A. Rieffel, Dimension and stable rank in the K-theory of C*-algebras, Proc. London Math. Soc. (1983) 46 (2): 301-333.]
