On cocycle rigidity of Anosov actions
by
Alexander Starkov
All-Russian Electrotechnical Institute, Istra

All known Anosov multi-parameter actions preserving a smooth finite measure (except those arising from operations with one-parameter Anosov flows) are of the following form. Let G be a connected Lie group, A subset G a connected Abelian subgroup isomorphic to R^d, C subset G - a compact subgroup commuting with A, and \Gamma a lattice in G. Then the left action (C\G/\Gamma, A) of A on the double quotient space C\G/\Gamma is well defined. It is called an Anosov action if there exists a partially hyperbolic element a in A such that the neutral foliation for the a-action on C\G/\Gamma comprises of A-orbits.

Many examples of such actions were studied in papers of Katok and Spatzier. Two main of them are as follows. First, let \Sigma =~ Z^d be a commutative subgroup of SL(n, Z). Then the suspension over the \Sigma-action on Tⁿ is of the form (G/\Gamma, A), where G =~ R^d·Rⁿ is a solvable Lie group, \Gamma = \Sigma·Zⁿ is a lattice in G, and A =~ R^d. It is Anosov one if \Sigma contains an Anosov automorphism of Tⁿ.

Second, let G be a semisimple Lie group having no compact factors, \Gamma subset G a uniform irreducible lattice, A subset G the R-split component of a Cartan subgroup of G, and C subset G the compact part of the centralizer of A in G. Then the action (C\G/\Gamma, A) is Anosov.

While studying these examples Katok and Spatzier discovered the so called cocycle rigidity phenomenon. This means that under certain conditions the first C^\infty-smooth cohomology group over the A-action trivializes, i.e. every smooth cocycle is cohomologous to a constant one. Such examples are called the standard Anosov actions. In the first example one assumes that \Sigma contains a subgroup \Sigma' =~ Z² all of whose nontrivial elements are ergodic. In the second case one assumes that rank R G=d >= 2.

This phenomenon never occurs if d=1 because for one-parameter Anosov flows the first C^\infty-smooth cohomology group is of infinite dimension (and by Livshits theorem is parametrized by the countable family of periodic orbits).

We conjecture a general criterion for cocycle rigidity of Anosov actions. Basically, it says that the first cohomology group trivializes iff the action admits no one-parameter Anosov flows as quotient actions. Previously this conjecture was proved by the author for suspensions over algebraic Z^d-actions on tori.

To attack the conjecture in the general case we refine the structure of Anosov actions and give a general ``rectified'' construction of those. The criterion is proved in the case when a Levi subgroup of G has no factors locally isomorphic to SO(1, n) or SU(1, n) and acts ergodically on G/\Gamma.

This generalizes results on cocycle rigidity proved by Katok and Spatzier and presents a new class of rigid Anosov actions. An essential role is played by a general result on exponential mixing for such actions which is of independent interest.

Date received: July 5, 1999