Lacunary hyperbolic groups with fast injectivity radius growth and enough loxodromic elements are selfless

We prove that a lacunary hyperbolic group $G=\mathrm{}{G}_i$ with sufficient generics is selfless in the sense of Amrutam–Gao–Kunnawalkam Elayavalli–Patchell, provided the hyperbolicity constants ${\delta }_i$ and injectivity radii $r_i$ satisfy ${\delta }_i(\log r_i{)}^7=o(r_i)$. The proof replaces the acylindricity-based machinery of that work with a direct geodesic $n$-gon criterion due to Arzhantseva, which applies in any $\delta$-hyperbolic space. As a consequence, combined with rapid decay, $G$ is $C^{\ast }$-selfless. The condition is mild: torsion-free Tarski monsters, Jacobson’s mixed-identity-free elementary amenable groups and Gromov monster groups satisfy it for appropriate parameter choices. The amenable examples are selfless but cannot be $C^{\ast }$-selfless, providing examples that separate these properties. Finally we remark that the Gromov monster group examples provide a potential avenue to a non-exact $C^{\ast }$-algebra with strict comparison.