Cohomological obstructions to lifting properties for full group C${}^{\ast }$-algebras

We develop a new method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C${}^{\ast }$-algebras of countable groups with (relative) property (T). We derive that the full C${}^{\ast }$-algebras of the groups ${\mathbb{Z}}^2⋊{\text{SL}}_2(\mathbb{Z})$ and ${\text{SL}}_n(\mathbb{Z})$, for $n\ge 3$, do not have the local lifting property (LLP). We also prove that the full C${}^{\ast }$-algebras of a large class of groups $\mathrm{\Gamma }$ with property (T), including those such that ${\text{H}}^2(\mathrm{\Gamma },\mathbb{R})≠0$ or $$\text{H}$^2(\Gamma,\mathbb Z\Gamma)\not=0$, do not have the lifting property (LP). More generally, we show that the same holds if $\mathrm{\Gamma }$ admits a probability measure preserving action with non-vanishing second $\mathbb{R}$-valued cohomology. Finally, we prove that the full C${}^{\ast }$-algebra of any non-finitely presented property (T) group fails the LP.