Local rigidity of partially hyperbolic actions

On the rigidity of Weyl chamber flows and Schur multipliers as topological groups

“…A (partially hyperbolic) twisted Weyl chamber flow is the action of a Cartan action (ie, a generic restriction of the action of a Cartan subalgebra in S) on M = G/Γ. Zhenqi Wang has shown that generic restrictions of such actions (in the sense of (G)) are C ∞,k,∞ -rigid for sufficiently large k in the case when S is (semi)split and rigid in the universal cover in the case when S is any semisimple Lie group [14]. Her method relies on choosing nice G-orbits inside M which are exactly Weyl chamber flows and correcting the cocycle in each such orbit using methods presented here.…”

“…It may be the case that no such ergodic actions exist when G is genuinely higher-rank 2 Such actions have been shown to be rigid in the universal cover[14] …”

Mixing properties of commuting nilmanifold automorphisms

“…For certain actions by partially hyperbolic left translations on homogeneous spaces G/Γ, where G is a semisimple Lie groups and Γ is a lattice in G, a similar theorem was proved by D. Damjanovic and A. Katok [4,5,3] and Z. J. Wang [24]. We note that these authors also prove Hölder versions of this result which are not amenable to our techniques.…”

“…We note that these authors also prove Hölder versions of this result which are not amenable to our techniques. Furthermore, cocycle rigidity results are proven for small perturbations of these actions on G/Γ in [5,24]. Again we cannot obtain these results by our methods.…”

New Cases of Differential Rigidity for Non-Generic Partially Hyperbolic Actions

“…Note that for Sp(2n, R) generating relations are available in [17], but to get enough information for reducible classes, further calculations are carried out on the Schur multiplier. The proof resembles Theorem 2 in [24] on dealing with infinite homotopic classes.…”