On proper affine actions with better decaying cocycles
Chunliu Feng, Geng Tian, Ran Yi
TL;DR
This paper advances the theory of proper affine isometric actions by showing that every countable group $\Gamma$ admits a proper affine isometric action on the Banach space $\bigoplus_{n\in\mathbb{N}} \ell^{2n}(\Gamma)$ with cocycles decaying faster than previously known. The construction uses a sequence of Lipschitz functions $\phi^n_e$ and associated $1$-cocycles $b_n(\gamma)=\pi(\gamma)\phi^n_e-\phi^n_e$ to form $b(\gamma)=\bigoplus_n b_n(\gamma)$, with each component satisfying $\|b_n(\gamma)\|_{\ell^{2n}}=O\big(\frac{d(\gamma,e)}{n\sqrt{n\eth_{1}(n)\cdots\eth_{k}(n)}}\big)$. The resulting $b$ yields a proper action due to disjoint supports outside large balls and the divergence of $\|b(\gamma)\|$ as $\gamma$ tends to infinity. The main theorem thus provides a broad, quantitative improvement over the decay rate previously achieved by Haagerup–Przybyszewska, with potential implications for the Baum–Connes and Novikov conjectures and related areas in geometric group theory and operator algebras.
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 $Γ$ on the Banach space $\oplus_{n=1}^\infty\ell^{2n}(Γ)$ such that the cocycles have decay rates that are better than those of Haagerup and Przybyszewska in.