Amenable actions of real and p -adic algebraic groups

Let K be a locally compact field of characteristic 0. Let G be a linear algebraic group defined over K , acting algebraically on an algebraic variety V . We prove that the action of G ( K ) (the group of K -rational points of G ) on V ( K ) is topologically amenable, if and only if all point stabilizers in G ( K ) are solvable-by-compact. This follows by combining a result by Borel–Serre 5with the following fact: let G be a second countable locally compact group acting continuously on a second countable locally compact space Y . If the action G ↷ Y is smooth (i.e. the Borel structure on G ∖ Y is countably separated), then topological amenability of G ↷ Y is equivalent to amenability of all point stabilizers in G .