Locally compact groups with every isometric action bounded or proper

A locally compact group G has property PL if every isometric G-action either has bounded orbits or is (metrically) proper. For p ≥ 1, say that G has property BP p if the same alternative holds for the smaller class of affine isometric actions on Lp-spaces. We explore properties PL and BP p and prove that they are equivalent for some interesting classes of groups: abelian groups, amenable almost connected Lie groups, amenable linear algebraic groups over a local field of characteristic 0. The appendix provides new examples of groups with property PL, including nonlinear ones.