Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 10 min ago
PreprintarXiv licencearXiv

Fixed point properties and cohomology of Banach representations of arithmetic groups

  • Saar Bader
  • Shaked Bader
  • Uri Bader
  • Roman Sauer
Open at sourcePDFDOICite

Abstract

We study fixed point theorems for actions of lattices of semisimple groups. They are deduced from vanishing results for the group cohomology of Lp-representations. We show that for lattices in simple groups of higher rank, the cohomology with Lp-coefficients vanishes below the rank whenever there are no invariant vectors. As a corollary of the vanishing for L1-coefficients, we obtain that every action on an acyclic simplicial complex of dimension lower than the rank has a finite orbit. This in particular proves a conjecture by Farb. The Lp-vanishing below the rank proves a conjecture by Gromov regarding Lp-cohomology of symmetric spaces.

Related

  1. Steinberg module and top-dimensional cohomology for arithmetic Chevalley groups
    Miguel Gentili
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. Computations of Cohomology of Arithmetic Groups, Part 1
    Ivan Horozov
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Tate cohomology of unipotent cuspidal representations of finite groups of Lie type
    Sabyasachi Dhar
    arXivPreprintCC0Full textAlgebra & number theoryarXiv
  4. Matrix representations and arithmetic properties of jacobsthal numbers via binary 3x3 matrices
    Wilson Arley Martinez, Samin Ingrith Ceron
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. Arithmetic duality for p -adic pro-étale cohomology of analytic curves
    Pierre Colmez, Sally Gilles, Wiesława Nizioł
    Compositio MathematicaPeer-reviewedLicence not statedFull text16 citedGeometry & topologyOpenAlex

Cite

BibTeXRIS
BibTeX
@misc{bader2026fixed,
  title = {{Fixed point properties and cohomology of Banach representations of arithmetic groups}},
  author = {Saar Bader and Shaked Bader and Uri Bader and Roman Sauer},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24951},
  eprint = {2609.24951},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24951},
}
RIS
TY  - UNPB
TI  - Fixed point properties and cohomology of Banach representations of arithmetic groups
AU  - Saar Bader
AU  - Shaked Bader
AU  - Uri Bader
AU  - Roman Sauer
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24951
UR  - https://arxiv.org/abs/2609.24951
AB  - We study fixed point theorems for actions of lattices of semisimple groups. They are deduced from vanishing results for the group cohomology of L^p-representations. We show that for lattices in simple groups of higher rank, the cohomology with L^p-coefficients vanishes below the rank whenever there are no invariant vectors. As a corollary of the vanishing for L^1-coefficients, we obtain that every action on an acyclic simplicial complex of dimension lower than the rank has a finite orbit. This in particular proves a conjecture by Farb. The L^p-vanishing below the rank proves a conjecture by Gromov regarding L^p-cohomology of symmetric spaces.
ER  -