On proper affine actions with better decaying cocycles

It is well-known that the study of proper affine isometric actions of countable discrete groups in various spaces plays a crucial role in geometric group theory, operator algebras, and high-dimensional topology. In this article, we shall construct a proper affine isometric action of countable discrete group $\mathrm{\Gamma }$ on the Banach space ${\oplus }_{n=1}^{\mathrm{\infty }}{\mathrm{\ell }}^{2n}(\mathrm{\Gamma })$ such that the cocycles have decay rates that are better than those of Haagerup and Przybyszewska in.