On invariant finitely additive measures for automorphism groups acting on tori

Abstract

Consider the natural action of a subgroup H of GL(n, Z) on Tⁿ. We relate the H-invariant finitely additive measures on (Tⁿ, L) where L is the class of all Lebesgue measurable sets, to invariant subtori C such that the H-action on either C or Tⁿ/C factors to an action of an amenable group. In particular, we conclude that if H is a nonamenable group acting irreducibly on Tⁿ then the normalised Haar measure is the only H-invariant finitely additive probability measure on (Tⁿ, L) such that μ(R)=0, where R is the (countable) subgroup consisting of all elements of finite order; this answers a question raised by J. Rosenblatt. Along the way we analyse H-invariant finitely additive measures defined for all subsets of Tⁿ and deduce, in particular, that the Haar measure extends to an H-invariant finitely additive measure defined on all sets if and only if H is amenable.