The inner amenability conjecture for strong 1-boundedness

A countable discrete group $G$ is inner amenable if the conjugation action of $G$ on $G\setminus\{1\}$ admits an invariant mean, and it is strongly 1-bounded if its group von Neumann algebra $L(G)$ is strongly 1-bounded. Inner amenability conjecture. Every inner amenable group is strongly 1-bounded. The theorem established in the paper proves this for inner amenable groups that are either finitely presented and sofic or i.c.c. and not sofic; the general case remains open.