If a group is such that every its $\e$-almost representation for small enough $\e$ can be included into an asymptotic representation we call this group asymptotically stable. Free, finite, abelian and some other groups are proved to be asymptotically stable. The main point of the proof are some properties of almost commuting operators.
We give also an example of a group without asymptotical stability. This example provides also an example of a group without sufficiently many asymptotic representations, namely there are elements in the $K$-group of its classifying space which cannot be obtained from asymptotic representations.
We discuss also relations between asymptotic representations of $\Gamma$ and representations of $\Gamma\times{\bf Z}$ into the Calkin algebra (Fredholm representations). An approach developed in \cite{mm} helps in better understanding of relations between the Kasparov's $KK$-bifunctor and the $E$ functor of Connes--Higson on the category of $C^*$-algebras.
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\bibitem{mm} {\sc V. M. Manuilov, A. S. Mishchenko}: Asymptotic and Fredholm representations of discrete groups. {\it Matem. Sb.} {\bf 189} (1998), No 10, 53--72.
\bibitem{mish-noor} {\sc A.~S.~Mishchenko, Noor Mohammad}: Asymptotic representations of discrete groups. ``Lie Groups and Lie Algebras. Their Representations, Generalizations and Applications.'' Mathematics and its Applications {\bf 433}. Kluver Acad. Publ.: Dordrecht, 1998, 299--312.