The action of a solvable group on an infinite set never has a unique invariant mean

Abstract

Theorem 1 of the paper proves a conjecture of J. Rosenblatt on nonuniqueness of invariant means for the action of a solvable group G G on an infinite set X X . The same methods used in this proof yield even a more general result: Nonuniqueness still holds if G G is an amenable group containing a solvable subgroup H H such that card ⁡ ( G / H ) ⩽ card ⁡ ( H ) \operatorname {card} (G/H) \leqslant \operatorname {card} (H) .