Paradoxical decompositions and spectral gaps for linear group actions

By a celebrated theorem of Tarski, a (discrete) group action has no finitely additive invariant measure  (i.e. is "non-amenable") if and only if it exhibits a paradoxical decomposition, and if and only if it admits a spectral gap. We prove yet another equivalence by introducing the notion of "ping-pong with overlaps", which we then apply to characterise non-amenable algebraic actions of linear groups over any field and show that they are uniformly non-amenable uniformly over all fields. The proof makes key use of diophantine heights. Joint work with Oren Becker.