Zimmer's conjecture for lattice actions: the ${\rm SL}(n, \mathbb C)$-case

TL;DR

This paper proves Zimmer's conjecture for co-compact lattices in ${\rm SL}(n, \mathbb{C})$, showing that low-dimensional actions on compact manifolds are essentially finite, extending understanding of group actions in complex Lie groups.

Contribution

It establishes Zimmer's conjecture for ${\rm SL}(n, \mathbb{C})$ lattices, demonstrating that low-dimensional smooth actions are finite, a significant advancement in the field.

Findings

Actions of co-compact lattices in ${\rm SL}(n, \mathbb{C})$ on low-dimensional manifolds are finite.

The dimension thresholds depend on whether $n=4$ or not.

The result applies to $C^{1+\epsilon}$ diffeomorphisms.

Abstract

We prove Zimmer's conjecture for co-compact lattices in ${\rm SL}(n, \mathbb C)$: for any co-compact lattice in ${\rm SL}(n, \mathbb C)$, $n \geq 3$, any $\Gamma$-action on a compact manifold $M$ with dimension: (I) less than $2n-2$ if $n \neq 4$, (II) less than $5$ if $n = 4$, by $C^{1+\epsilon}$ diffeomorphisms factors through a finite action.