On proper affine actions with better decaying cocycles

Abstract

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 $\hat{I}“$ on the Banach space ${\oplus }_{n=1}^{\infty }{\ell }^{2n}(\hat{I}“)$ such that the cocycles have decay rates that are better than those of Haagerup and Przybyszewska in.

We will rewrite the paper and submit a replacement version later