Pureness and stable rank one for reduced twisted group ${\mathrm{C}}^{\ast }$-algebras of certain group extensions

The purpose of this note is to prove two results. First, we observe that discrete groups with property ${\mathrm{P}}_{\mathrm{P}\mathrm{H}\mathrm{P}}$ in the sense of Ozawa give rise to completely selfless reduced twisted group ${\mathrm{C}}^{\ast }$-algebras, thereby extending a theorem of Ozawa from the untwisted to the twisted case. We also observe that an adaptation of property ${\mathrm{P}}_{\mathrm{P}\mathrm{H}\mathrm{P}}$ for an inclusion of groups implies that the associated inclusion of reduced twisted group ${\mathrm{C}}^{\ast }$-algebras is selfless in the sense of Hayes-Kunnawalkam Elayavalli-Patchell-Robert. Second, we show that reduced (twisted) ${\mathrm{C}}^{\ast }$-algebras of some group extensions of the form finite-by-$G$, with $G$ having the property ${\mathrm{P}}_{\mathrm{P}\mathrm{H}\mathrm{P}}$, have stable rank one and are pure, which implies strict comparison. Our results do not assume rapid decay, and extend a theorem of Raum-Thiel-Vilalta. Examples covered by our results include reduced twisted group ${\mathrm{C}}^{\ast }$-algebras of all acylindrically hyperbolic groups and all lattices in $\mathrm{S}\mathrm{L}(n,\mathbb{R})$ for $n\ge 2$.