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Cohomological obstructions to lifting properties for full group C$^*$-algebras

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