Algebraic spectral gaps in group cohomology
In this project I am studying groups by means of their cohomologies. The focus is 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.
DURATION
24 months
HOSTING INSTITUTION
Institute of Mathematics of the Polish Academy of Sciences
PROJECT OUTCOME
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 $\mathrm{SL}_n(\mathbb{Z})$. The project is currently in the phase of writing it up in the article form. This is a joint work with M. Kaluba.
DISSEMINATION OF RESULTS
- Group ring aspects of cohomological properties, 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 $\mathrm{SAut}(F_n)$, Spanish+Polish Mathematical Meeting, University of Łódź, 2023.09.07
- Spectral gaps for the cohomological Laplacians of $\mathrm{SL}_n(\mathbb{Z})$, Two-Day Geometry Meeting, University of Bristol, 2023.05.22
