Polynomially weighted ${\mathrm{\ell }}^p$-completions and group homology

We introduce polynomially weighted ${\mathrm{\ell }}^p$-norms on the bar complex of a finitely generated group. We prove that, for groups of polynomial or exponential growth, the homology of the completed complex does not depend on the value of $p$ in the range $(1,\mathrm{\infty })$.