This repository provides sum of squares witnesses for the existence of the spectral gap for the first cohomological Laplacian
More precisely we are looking for a positive
For a finitely presented group
where
It has been shown by Lyndon in doi:10.2307/1969440 that for any
We focus on the vanishing of the first and the reducibility of the second cohomology with unitary coefficients (here reducibility means that the the image of differential
Replication details for 2207.02783
Note: replication for 2207.02783 has been moved to a separate branch 2207.02783.
For the computations we used julia in version 1.8.3
but in principle any later version should work.
To obtain the code for the replication, you can either download it directly from Zenodo, or use git for this. In the latter case, first clone this repository via
git clone https://github.com/piotrmizerka/LowCohomologySOS.git
and checkout to the correct branch
cd LowCohomologySOS
git checkout 2207.02783
First, run julia in the terminal in LowCohomologySOS
folder
julia --project=.
Next, to set up the proper environment for the replication run in julia REPL
julia> using Pkg; Pkg.instantiate()
This command installs and precompiles, if needed, all the necessary dependences, so it may take a while. Note that this step needs to be executed only once per installation.
We wish to prove that for for the Steinberg presentation of
We provide a script which performs the necessary optimization to find such sum of squares decomposition.
As before the following command needs to be executed in the terminal in LowCohomologySOS
folder:
julia --project=. ./scripts/SL_3_Z_Delta_1.jl
The running time of the script will be approximately 2
hours on a standard laptop computer.
Alternatively, you can run the script which uses the precomputed solution (stored in the file "sl_3_z_precomputed.sjl") and provides rigorous proof (certification) of the result. In order to do this, the following command needs to be executed in the terminal in LowCohomologySOS
folder:
julia --project=. ./scripts/SL_3_Z_Delta_1_precomputed.jl
The running time of the script will be approximately 2
minutes on a standard laptop computer.
If you use any code from this repository, or you find reading through the code enlightening please cite 2207.02783 as
@misc{https://doi.org/10.48550/arxiv.2207.02783,
doi = {10.48550/ARXIV.2207.02783},
url = {https://arxiv.org/abs/2207.02783},
author = {Kaluba, Marek and Mizerka, Piotr and Nowak, Piotr W.},
keywords = {Group Theory (math.GR), Operator Algebras (math.OA), FOS: Mathematics, FOS: Mathematics},
title = {Spectral gap for the cohomological Laplacian of $\operatorname{SL}_3(\mathbb{Z})$},
publisher = {arXiv},
year = {2022},
copyright = {arXiv.org perpetual, non-exclusive license}
}
Replication details for 2404.10287
Note: replication for 2404.10287 has been moved to a separate branch 2404.10287.
For the computations we used julia in version 1.8.3
but in principle any later version should work.
To obtain the code for the replication, you can either download it directly from Zenodo, or use git for this. In the latter case, first clone this repository via
git clone https://github.com/piotrmizerka/LowCohomologySOS.git
and checkout to the correct branch
cd LowCohomologySOS
git checkout 2404.10287
First, run julia in LowCohomologySOS
folder
julia --project=.
Next, to set up the proper environment for the replication run in julia REPL
julia> using Pkg; Pkg.instantiate()
This command installs and precompiles, if needed, all the necessary dependences, so it may take a while. Note that this step needs to be executed only once per installation.
We wish to prove that for the Steinberg presentation of
We provide a script which performs the necessary optimization to find such sum of squares decomposition.
As before the following command needs to be executed in LowCohomologySOS
folder:
julia --project=. ./scripts/SL_3_Z_adj.jl
The running time of the script will be approximately 2
hours on a standard laptop computer.
Instead of running the whole computation, one can use the precomputed solution instead. In order to run the script providing rigorous mathematical proof (see the Section 3.2 of 2207.02783) that LowCohomologySOS
folder:
julia --project=. ./scripts/sl3_adj_precom/SL_3_Z_adj_cert.jl
The running time of the script will be approximately 2
minutes on a standard laptop computer.