On invariant finitely additive measures for automorphism groups acting on tori

Abstract:

Consider the natural action of a subgroup $H$ of ${\text {GL}}(n,{\mathbf {Z}})$ on ${{\mathbf {T}}^n}$. We relate the $H$-invariant finitely additive measures on $({{\mathbf {T}}^n},\mathcal {L})$ where $\mathcal {L}$ is the class of all Lebesgue measurable sets, to invariant subtori $C$ such that the $H$-action on either $C$ or ${{\mathbf {T}}^n}/C$ factors to an action of an amenable group. In particular, we conclude that if $H$ is a nonamenable group acting irreducibly on ${{\mathbf {T}}^n}$ then the normalised Haar measure is the only $H$-invariant finitely additive probability measure on $({{\mathbf {T}}^n},\mathcal {L})$ such that $\mu (R) = 0$, 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 ${{\mathbf {T}}^n}$ 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.