Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds

Abstract

We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers r(G) and m(G) associated with the roots system of the Lie algebra of a Lie group G. If the dimension of the manifold is smaller than r(G), then we show the action preserves a Borel probability measure. If the dimension of the manifold is at most m(G), we show there is a quasi-invariant measure on the manifold such that the action is measurable isomorphic to a relatively measure preserving action over a standard boundary action.

Rigidity, lattices and invariant measures beyond homogeneous dynamics

“…There are two obstructions that arise, each of which is serious. For non-split groups, including even SL(𝑛, C) and SL(𝑛, H), there is an issue arising where we employ the work of Brown, Rodriguez Hertz and Wang [23]. That work only "sees" the number of roots of a simple Lie group and not the dimensions of the root subspaces.…”

“…The difficulty here is to do the averaging while retaining that 𝜇 is 𝐴 invariant and that some 𝑎 in 𝐴 has positive first Lyapunov exponent for 𝜇 . After this step, we can use a result of Brown, Rodriguez Hertz and Wang together with some algebraic computations to show that 𝜇 is in fact 𝐺 invariant [23]. This contradiction shows that 𝜌 does in fact have subexponential growth of derivatives.…”

Superrigidity, arithmeticity, normal subgroups: results, ramifications and directions

“…In the proof of Theorem 5.1 on Zimmer's conjecture we use a different method of detecting invariant measures and projective factors for Γ-actions on compact manifolds M that is more effective for applications. This is developed by Brown, Rodriguez-Hertz and Wang in [BRHW1]. Where Nevo and Zimmer follow Margulis and study G-invariant σ-algebras of measurable sets on X to find the projective factor, Brown, Rodriguez Hertz and Wang study invariant measures instead.…”

“…If one can find enough such non-resonant roots, the object O is automatically G-invariant. We will illustrate this philosphy by sketching the proof of the following theorem from [BRHW1].…”

Projective and conformal closed manifolds with a higher-rank lattice action

“…This is due to the fact that these structures do not naturally define a finite Γ-invariant measure, making more difficult the use of celebrated results such as Zimmer's cocycle super-rigidity. The new powerful tools about invariant measures, introduced in [BRHW16] and used in [BFH16] for proving Zimmer's conjectures, invite us to pay attention to these non-volume preserving dynamics.…”