Invariant Measures for Algebraic Actions, Zariski Dense Subgroups and Kazhdan’s Property (t )

Let k be any locally compact non-discrete field. We show that finite invariant measures for k-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 k-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 (T ) for algebraic groups, which we investigate and strengthen.