# Locally compact groups with every isometric action bounded or proper

## TL;DR

This paper investigates properties of locally compact groups related to their isometric actions, showing equivalences for certain classes and providing new examples of groups with these properties, including non-linear groups.

## Contribution

It establishes the equivalence of properties PL and BP_{L^p} for specific classes of groups and introduces new examples of groups with property PL, expanding understanding of group actions.

## Key findings

- Properties PL and BP_{L^p} are equivalent for abelian, amenable almost connected Lie, and certain algebraic groups.
- New examples of groups with property PL, including non-linear groups, are provided.
- The paper clarifies the behavior of isometric actions on different classes of groups.

## Abstract

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 $BP_{L^p}$ if the same alternative holds for the smaller class of affine isometric actions on $L^p$-spaces. We explore properties PL and $BP_{L^p}$ 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 by Corina Ciobotaru provides new examples of groups with property PL, including non-linear ones.