Local spectral gap and the Stuck--Zimmer conjecture

We prove two new cases of the Stuck–Zimmer conjecture. First, we settle the Stuck–Zimmer conjecture for ergodic probability preserving actions of irreducible lattices (both uniform and non-uniform) in connected semisimple real Lie groups with finite center, no compact factor and real rank at least two. We furthermore extend the Stuck–Zimmer theorem to all irreducible probability preserving actions of such groups whenever one simple factor is locally isomorphic to ${\mathrm{S}\mathrm{L}}_2(\mathbb{R})$. In particular, this settles the Stuck–Zimmer conjecture for irreducible actions of ${\mathrm{S}\mathrm{L}}_2(\mathbb{R})\times {\mathrm{S}\mathrm{L}}_2(\mathbb{R})$. The proof relies on a new stabilizer rigidity criterion using local spectral gap of projected stabilizers in a single simple factor. Here local spectral gap is understood in the sense of Boutonnet, Ioana and Salehi Golsefidy.