A spectral sequence for polynomially bounded cohomology

We construct an analogue of the Lyndon–Hochschild–Serre spectral sequence in the context of polynomially bounded cohomology. For G an extension of Q by H, this spectral sequences converges to the polynomially bounded cohomology of G, HP∗(G). If the extension is a polynomial extension in the sense of Noskov with H and Q isocohomological and Q of type HF∞, the spectral sequence has -term HPq(Q;HPp(H)), and G is isocohomological for C. By referencing results of Connes–Moscovici and Noskov if H and Q are both isocohomological and have the Rapid Decay property, then G satisfies the Novikov conjecture.