Almost-representations and asymptotic representations of discrete groups

We define a new property of finitely presented groups connected with their asymptotic representations. Namely, we say that a group is AGA if each of its almost-representations generates an asymptotic representation. We give examples of groups with and without this property. In particular, free groups, finite groups and free Abelian groups are AGA. In our example of a group $\Gamma$ that is not AGA, the group $K^0(\mathrm B\Gamma)$ contains elements that are not covered by asymptotic representations of $\Gamma$.