Non-Vanishing Unitary Cohomology of Low-Rank Integral Special Linear Groups
Abstract We construct explicit finite-dimensional orthogonal representations $\pi _N$ of $\textrm{SL}_{N}(\mathbb{Z})$ for ${N} \in \{3,4\}$ all of whose invariant vectors are trivial, and such that $H^{N - 1}(\textrm{SL}_{N}(\mathbb{Z}),\pi _N)$ is non-trivial. This implies that for $N$ as above, the group $\textrm{SL}_{N}(\mathbb{Z})$ does not have property $(T_{N-1})$ of Bader–Sauer and therefore is not $(N-1)$-Kazhdan in the sense of De Chiffre–Glebsky–Lubotzky–Thom, both being higher versions of Kazhdan’s property $T$.
Authors
- Piotr Mizerka (ORCID: https://orcid.org/0000-0001-5712-8513)
- Dawid Kielak (ORCID: https://orcid.org/0000-0002-5536-9070)
- Benjamin Brück (ORCID: https://orcid.org/0000-0001-9606-3783)
- Sam Hughes (ORCID: https://orcid.org/0000-0002-9992-4443)
Institutions
- University of Bonn (DE)
- GP Forschungsgruppe (DE)
- Mathematical Institute of the Slovak Academy of Sciences (SK)
- Institute of Mathematics (PL)
- Adam Mickiewicz University in Poznań (PL)
- Polish Academy of Sciences (PL)
Publication Details
- Journal
- International Mathematics Research Notices
- Published
- 2025-08-01
- DOI
- https://doi.org/10.1093/imrn/rnaf230
- Primary Topic
- Advanced Algebra and Geometry
- Type
- article
- Field-Weighted Citation Impact
- 0.00
Funders
- European Commission
- Deutsche Forschungsgemeinschaft
- Narodowym Centrum Nauki