LowCohomologySOS

This repository provides sum of squares witnesses for the existence of the spectral gap for the first cohomological Laplacian $\Delta_1$ for finitely presented groups.

More precisely we are looking for a positive $\lambda$ such that $\Delta_1-\lambda I_n$ admits a sum of squares decomposition, which proves that $\lambda$ is a lower bound on the spectral gap of $\Delta_1$.

For a finitely presented group $G=\langle s_1,\ldots,s_n|r_1,\ldots,r_m\rangle$ the first Laplacian $\Delta_1$ is given by the formula

$$\Delta_1=d_0d_0^*+d_1^*d_1\in\mathbb M_{n,n}(\mathbb ZG)\subseteq M_{n,n}(\mathbb RG),$$

where $d_0=\left[1-s_i\right]\in\mathbb M_{n,1}(\mathbb ZG)$ and $d_1$, known as Jacobian, is given by the $m\times n$ matrix of Fox derivatives of the relations (for the definition see the original papers of Fox, doi:10.2307/1969736 and doi:10.2307/1969686). The involution $*$ is given by the composition of the matrix transposition and the standard involution on the group ring $\mathbb RG$.

Group cohomology

It has been shown by Lyndon in doi:10.2307/1969440 that for any $G$-module $V$ the cohomology $H^*(G,V)$ can be computed from the following complex

$$0\longrightarrow\mathbb ZG\xrightarrow{d_0}(\mathbb ZG)^n\xrightarrow{d_1}(\mathbb ZG)^m\longrightarrow\cdots$$

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 $d_1$ is closed for every unitary $G$-module $V$). By the recent work of Bader and Nowak doi:10.1016/j.jfa.2020.108730 these two conditions are equivalent to the existence of a positive $\lambda>0$ such that $\Delta_1-\lambda I_n$ admits a sum of squares decomposition, that is, there exist matrices $M_1,\ldots,M_l$ such that

$$ \Delta_1-\lambda I_n=M_1^*M_1+\ldots+M_l^*M_l. $$

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.

Obtaining code

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

Setting up the environment

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.

Running actual replication

We wish to prove that for for the Steinberg presentation of $\text{SL}_3(\mathbb{Z})$ on six generators (as defined in Section 2 of 2207.02783) $\Delta_1-\lambda I_6$ is a sum of squares for some $\lambda\geq 0.32$.

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.

Citing

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.

Obtaining code

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

Setting up the environment

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.

Running actual replication

We wish to prove that for the Steinberg presentation of $\text{SL}_3(\mathbb{Z})$ on six generators (as defined in Section 3 of 2404.10287) $\text{Adj}_3-\lambda I_6$ is a sum of squares for some $\lambda\geq 0.217$.

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 $\text{Adj}_3-0.217 I_6$ is a sum of squares, execute the following command in 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.