Isometric group actions on Banach spaces and representations vanishing at infinity (Q941083)

The main result of the paper under review is Theorem 1.1: Let \(k\) be a local field and \(G\) a simple algebraic group of rank 1 over \(k\). Let \(p_0\) be the Haudorff dimension of the visual boundary of \(G\). Then, for every \(p > \max\{1,p_0\}\), there exists a metrically proper affine action of \(G\) on the space \(L^p(G)\) with linear part \(\lambda_{G,p}\).