On Almost Representations of Groups $\pi\times{\mathbf Z}$

Various generalizations of group representations (like almost and asymptotic ones) are drawing attention due to their applications to classification theory of $C^*$-algebras and to the Novikov conjecture on higher signatures and the Baum–Connes conjecture. We study here almost representations of discrete groups $\pi\times\mathbf Z$ which can be viewed as finite-dimensional analogs of Fredholm representations. We give a construction of such almost representations and show that for some class of discrete groups (intermediate between commutative and nilpotent groups) these almost representations provide enough vector bundles over the classifying spaces $B\pi\times S^1$. In particular it gives a new proof of the well-known validity of the Novikov conjecture for these groups.