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

Abstract

We establish finiteness of low-dimensional actions of lattices in higher-rank semisimple Lie groups and establish Zimmer's conjecture for many such groups. This builds on previous work of the authors handling the case of actions by cocompact lattices and of actions by SL(n, Z). While the results are not sharp in all cases, they do dramatically improve all known results. The key difficulty overcome in this paper concerns escape of mass when taking limits of sequences of measures. Due to a need to control Lyapunov exponents for unbounded cocycles when taking such limits, quantitative controls on the concentration of mass at infinity are need and novel techniques are introduced to avoid "escape of Lyapunov exponent." Contents

C1 actions on manifolds by lattices in Lie groups

“…(2 The main result of this paper is the following generalization of the results in [BFH16,BFH21] to C 1 regularity.…”

“…A. Brown, D. Damjanović and Z. Zhang The purpose of the present paper is to extend the results in [BFH16,BFH20,BFH21] to C 1 actions, when the rank of the acting group is sufficiently large.…”

“…When α is a C 2 action, the conclusion of Theorem 1 is already obtained in [BFH16,BFH21]. Moreover, when G = Sp(2n, R), SO(n, n) or SO(n, n + 1), the dimension bound in Corollary A is not optimal.…”

“…Compared to the main result in [BFH16,BFH21] for almost-simple real Lie groups, in Theorem 1 we have posed a different requirement on the dimension of the manifold. Indeed, we can deduce from [BFH16,Theorem 2.7] that for a group homomorphism α : Γ → Diff 2 (M ), the conclusion of Theorem 1 is true if rank R G is replaced by the minimal resonant codimension r(G) (see [BFH16,Definition 2.1]).…”

“…We recommend the reader to have this paper close at hand as we make many references to these works, although we also repeat some of the main arguments for reader's convenience. Below we first describe the general strategy of the proofs in [BFH16,BFH20,BFH21], and then we point out the main new ideas and modifications we make here in order to obtain results in C 1 regularity.…”