Warped cones and proper affine isometric actions of discrete groups on Banach spaces

Abstract: Warped cones are metric spaces introduced by John Roe from discrete group actions on compact metric spaces to produce interesting examples in coarse geometry. We show that a certain class of warped cones $\mathcal{O}_\Gamma (M)$ admit a fibred coarse embedding into a $L_p$-space ($1\leq p<\infty$) if and only if the discrete group $\Gamma$ admits a proper affine isometric action on a $L_p$-space. This actually holds for any class of Banach spaces stable under taking Lebesgue-Bochner $L_p$-spaces and ultraproducts, e.g., uniformly convex Banach spaces or Banach spaces with nontrivial type. It follows that the maximal coarse Baum-Connes conjecture and the coarse Novikov conjecture hold for a certain class of warped cones which do not coarsely embed into any $L_p$-space for any $1\leq p<\infty$.