Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan's property

Let be any locally compact non-discrete field. We show that finite invariant measures for -algebraic actions are obtained only via actions of compact groups. This extends both Borel's density and fixed point theorems over local fields (for semisimple/solvable groups, resp.). We then prove that for -algebraic actions, finitely additive finite invariant measures are obtained only via actions of amenable groups. This gives a new criterion for Zariski density of subgroups and is shown to have representation theoretic applications. The main one is to Kazhdan's property for algebraic groups, which we investigate and strengthen.