Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T)

Abstract

We prove several cases of Zimmer's conjecture for actions of higher-rank, cocompact lattices on low-dimensional manifolds. For example, if Γ is a cocompact lattice in SL(n, R), M is a compact manifold, and ω a volume form on M we show that any homomorphism α : Γ → Diff(M ) has finite image if the dimension of M is less than n − 1 and that any homomorphism α : Γ → Diff(M, ω) has finite image if the dimension of M is less than n. The key step in the proof is to show that any such action has uniform subexponential growth of derivatives. This is established using ideas from the smooth ergodic theory of higher-rank abelian groups, structure theory of semisimple groups, and results from homogeneous dynamics. Having established uniform subexponential growth of derivatives, we apply Lafforgue's strong property (T) to establish the existence of an invariant Riemannian metric.

Zimmer's conjecture for non-uniform lattices and escape of mass

“…Step 2: Strong property (T). We apply [16,Theorem 2.4] and de la Salle's recent result establishing strong property (T ) for nonuniform lattices [22,Theorem 1.1] to conclude that any action α satisfying the conclusions of Theorem C preserves a continuous Riemannian metric g. In particular, the image α(Γ) is contained in the compact Lie group K = Isom g (M ). For completeness, we recall these two results.…”

“…A special case of Theorem A (and of Theorem B below) for cocompact Γ was proved by the authors in [16]. This paper primarily concerns the case that Γ is not cocompact and develops many new arguments to overcome the lack of compactness of the homogeneous space G/Γ.…”

“…We end the introduction with the proof of Theorem B, introducing the first technical result of the paper, Theorem C below. The proof of Theorem B follows the same outline as in [16].…”

“…(See [16,Lemma 3.7].) Let d(G) denote the minimal dimension of all non-trivial homogenous spaces K/C as K varies over all compact real-forms of all simple factors of the complexification of G. Given the numbers n(G), d(G), and v(G) the full conjecture is the following.…”

“…The number r(g) was defined in terms of combinatorics of root systems in [16]; that definition is reviewed in Definition 3.11 below. For R-split Lie groups G, we always have r(G) = v(G).…”