Algebraic spectral gaps in group cohomology
In this project I studied groups by means of their cohomologies. The focus was put to investigating group cohomology by its translation into group rings. In the group ring setting, one can distinguish specific Laplacians whose spectral properties are closely related to the cohomological ones.
The short description for the general public can be found here.
You can familiarize yourself with the project on the NSC Poland webpage as well.
TEAM MEMBERS
- Piotr Mizerka – PI
- Jakub Szymański – 2023-2024
DURATION
24 months
HOSTING INSTITUTION
Institute of Mathematics of the Polish Academy of Sciences
PROJECT OUTCOME
We construct explicit finite-dimensional orthogonal representations $\pi_N$ of $\operatorname{SL}_n(\mathbb{Z})$ for $N=3,4$ all of whose invariant vectors are trivial, and such that $H^{N-1}(\operatorname{SL}_N(\mathbb{Z}),\pi_N)$ is non-trivial. This implies that for $N$ as above, the group $\operatorname{SL}_N(\mathbb{Z})$ 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 $T$.
Joint work of Benjamin Brück, Sam Hughes, Dawid Kielak and PI.
This paper descibes induction of spectral gaps of the Laplacian $\Delta_1=d_0d_0^*+d_1^*d_1$, where the differentials $d_i$ are given by Fox calculus. The induction can be applied to the cases of $\mathrm{SL}_n(\mathbb{Z})$, the group of $n\times n$ integer matrices with determinant one, and $\mathrm{SAut}(F_n)$, the group of special automorphisms of the free group on $n$ generators. In the former case, due to a direct computation for $\mathrm{SL}_3(\mathbb{Z})$, we are able to show the existence of such spectral gaps for any $\mathrm{SL}_n(\mathbb{Z})$, where $n\geq 3$. This constitutes, in particular, an alternative proof of Kazhdan’s property (T) for $\mathrm{SL}_n(\mathbb{Z})$.
Single-authored paper by PI.
Recently, PI and Jakub Szymański developed an analogous induction technique for $\mathrm{Sp}_2$$_n(\mathbb{Z})$. This leads to the existence of spectral gaps for $\Delta_1$ for these groups. We are currently writing down our findings and expect to send them for publication by the end of April 2025.
DISSEMINATION OF RESULTS
- Nonvanishing of group cohomology of SL(n,ℤ), a lightning talk of PI at LiT II: Lost in Topology, Pisa, 2024.09.17
- Nieznikanie kohomologii grupy SL(n,ℤ), a talk of PI (in Polish) at 9. Forum Matematyków Polskich, Katowice, 2024.09.10
- Spectral gaps for property (T) groups, a lightning talk of Jakub Szymański at Dynamical Group Theory III, Seoul, 2024.08.14
- Induction of spectral gaps for the cohomological Laplacians of $\mathrm{SL}_n(\mathbb{Z})$ and $\mathrm{SAut}(F_n)$, a talk of PI at Seminarium z Algebry, Geometrii i Arytmetyki, Poznań, 2024.07.11
- Induction of spectral gaps for the cohomological Laplacians of $\mathrm{SL}_n(\mathbb{Z})$ and $\mathrm{SAut}(F_n)$, a talk of PI at Groups and topological Groups, Plentzia, 2024.06.28
- Group ring aspects of cohomological properties, a talk of PI at Geometry Seminar of the University of Wrocław, 2024.01.29
- Induction of spectral gaps for the cohomological Laplacians of $\mathrm{SL}_n(\mathbb{Z})$ and $\mathrm{SAut}(F_n)$, a talk of PI at Spanish+Polish Mathematical Meeting, University of Łódź, 2023.09.07
- Spectral gaps for the cohomological Laplacians of $\mathrm{SL}_n(\mathbb{Z})$, a lightning talk of PI at Two-Day Geometry Meeting, University of Bristol, 2023.05.22
