The first cohomology group, mixing, and minimal sets of the commutative group of algebraic actions on a torus

Abstract

This elementary paper originated in the process of reading Katok and Spatzier's paper [3] concerning the first cohomology group of Z kand Rk-Anosov actions. It was proved in [3] that for the so-called standard Anosov actions any 1-cocycle (in C ~ and HSlder categories) with coefficients in R was cohomologous to a constant one. We will consider the .simplest class of actions, namely, the action of the discrete commutative group C SL (n, Z) by automorphisms of the n-toms T ~. We say that E is cohomologicalIy rigid if any ~ -cocyc l e : E ---* R is C~-cohomologous to a homomorphism ~ e Hom (E, R), i.e., Hi(Z, R) = H o m (~, R). It is proved in [3], in particular, that Z is cohomologically rigid provided that it is Anosov (i.e., contains an element a that has no eigenvalue of absolute value 1) and standard, i.e., contains a subgroup E' "-~ Z 2 that consists of ergodic elements (i.e., any nontrivial b E E' is ergodic on T~). Note that this result implies cohomological rigidity in the HSlder category [3], which is a special case of Katok and Schmidt's result [4] for mixing and expanding actions of Z k by automorphisms on a compact abetian group (see also [5]). On the other hand, higher cohomology groups for the Anosov ~ C SL (n, Z) were calculated in [2]. We will study more thoroughly the first cohomology group Hi(Z, R) of C~-cocycles with coefficients in R over the ergodic action of the commutative group Z C SL (n, Z) on T n by automorphisms. Our purpose here is to find a criterion for cohomological rigidity of ~. In fact, as Katok mentioned to the author, the result of [3] holds without the assumption that Z is Anosov. This requires involving the Lipschitz-type theorem of Veech [6] for ergodic automorphisms of a toms; we discuss this in Sec. 1. On the other hand, the cohomological rigidity easily implies that T" has no ~..-invariant proper subtorus T C T ~ such that the quotient action of ~ on T ~ / T is of rank I, i.e., contains a subgroup of finite index isomorphic to Z (see Sec. 1 for more details). Henceforth by a subtorus we mean a connected closed subgroup T C T~; thus T " / T ~_ T ~ .for some integer m and the quotient action of ~ lies in SL (m, Z). We will prove that the absence of rank-1 quotients is, in fact, sufficient for cohomological rigidity. The main result of the paper is as follows.