Skip to content
DraconiticResearch
Search Sources API
201,683 studies31 sourcesUpdated 17 min ago
PreprintarXiv licencearXiv

Local spectral gap and the Stuck--Zimmer conjecture

  • Simon Machado
  • Yuval Yifrach
Open at sourcePDFDOICite

Abstract

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 SL2(ℝ). In particular, this settles the Stuck--Zimmer conjecture for irreducible actions of SL2(ℝ) × SL2(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.

Related

  1. Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and Stück-Zimmer Theorem
    Ilya Gekhtman, Simon Machado, Omri Solan +1
    arXivPreprintarXiv licenceFull textDynamical systemsarXiv
  2. Adams Spectral Sequences and Franke’s Algebraicity Conjecture
    Irakli Patchkoria, Piotr Pstrągowski
    Memoirs of the American Mathematical SocietyPeer-reviewedLicence not statedFull text27 citedMathematical physicsOpenAlex
  3. BMV conjecture and positivity of spectral shift for relatively Schatten perturbations
    Chandan Pradhan, Anna Skripka
    arXivPreprintarXiv licenceFull textAnalysis & PDEsMathematical physicsarXiv
  4. Asymptotics of local height pairing without Hodge conjecture
    Yuta Nakayama
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. A counterexample to the generalized Viterbo conjecture on the spectral distance in a disc cotangent bundle
    Filip Broćić, Baptiste Serraille, Ibrahim Trifa
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{machado2026local,
  title = {{Local spectral gap and the Stuck--Zimmer conjecture}},
  author = {Simon Machado and Yuval Yifrach},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39424},
  eprint = {2609.39424},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39424},
}
RIS
TY  - UNPB
TI  - Local spectral gap and the Stuck--Zimmer conjecture
AU  - Simon Machado
AU  - Yuval Yifrach
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39424
UR  - https://arxiv.org/abs/2609.39424
AB  - 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 SL_2(ℝ). In particular, this settles the Stuck--Zimmer conjecture for irreducible actions of SL_2(ℝ) × SL_2(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.
ER  -