Current status:

This repository contains notebooks and standalone code to replicate the computational results of Kazhdan constants for Chevalley groups over the integers by Marek Kaluba and Dawid Kielak.

Before exploring the notebooks you need to clone the main repository:

git clone https://github.com/kalmarek/2306.12358.git

The replicating notebooks are located in 2306.12358/notebooks subdirectory.

You should install julia from the official repository. Then while located in 2306/12358 directory run julia in terminal and execute the following commands in julias command-line (REPL) to instantiate the environment for computations.

using Pkg
Pkg.activate(".")
Pkg.instantiate()

Instantiation should install (among others):

• JuMP package for mathematical programming,
• Splitting conic solver,
• COSMO solver, and
• IntervalArithmetic.jl package from ValidatedNumerics (https://juliaintervals.github.io/).

The environment uses Groups.jl, StarAlgebras.jl, SymbolicWedderburn.jl and PropertyT.jl (unregistered) packages.

After instantiating the environment a jupyter server should be launched form 2306.12358 directory by issuing from julia command-line (REPL).

using Pkg
Pkg.activate(".")
using IJulia
notebook(dir=pwd())

During the first run, the user may be asked for installation of Jupyter program (a server for running this notebook) within miniconda environment, which will happen automatically after confirmation. To execute the commands in the notebook, one needs to navigate to notebooks subdirectory and click either of the notebooks.

The following scripts are included in scripts subdirectory:

• SLnZ_AdjA2.jl can be used to check that $\operatorname{Adj}{A_2} - \lambda \Delta$ is positive in $\mathbb{R} \operatorname{SL}{n}(\mathbb{Z})$. Note: this script, using the new language of gradings by root systems, reproves that $\operatorname{Adj}_n - \lambda \Delta \geqslant 0$ in $\mathbb{R} \operatorname{SL}_n(\mathbb{Z})$ (a result of Kaluba-Kielak-Nowak KKN).
• Sp2nZ_AdjC2.jl can be used to check that $\operatorname{Adj}{C_2} - \lambda \Delta$ is positive in $\mathbb{R} \operatorname{Sp}{2n}(\mathbb{Z})$.
• G2_Adj.jl can be used to check that $\operatorname{Adj}_{G_2} - \lambda \Delta$ is positive in $\mathbb{R} G_2$.

Additionally the two next scripts can be used to prove property T by computing sum of squares decomposition for $\Delta^2 - \lambda \Delta$:

• Sp2nZ_has_T.jl for group $\operatorname{Sp}_{2n}(\mathbb{Z})$.
• G2_has_T.jl for group $G_2$.

All of these scripts accept the command-line optional arguments:

  -N N                  the degree/genus/etc. parameter for a group
                        (type: Int64, default: 3)
  -R, --halfradius HALFRADIUS
                        the halfradius on which perform the sum of
                        squares decomposition (type: Int64, default: 2)
  -u, --upper_bound UPPER_BOUND
                        set upper bound for the optimization problem to
                        speed-up the convergence (type: Float64, default: Inf)
  -h, --help            show this help message and exit

Thus the following invocations can be used to reprove the computational results in the paper.

• Theorem 3.6:
  • julia --project=@. scripts/SLnZ_AdjA2.jl -N 3 -R 2
  • julia --project=@. scripts/SLnZ_AdjA2.jl -N 3 -R 3 (Note: the default solvers parameters in the script are not suitable for -R 3; in particular one has to increase max_iters considerably.)
• Theorem 3.10:
  • julia --project=@. scripts/Sp2nZ_AdjC2.jl -N 2 -R 3 (Note: this requires plenty of memory and computation time.)
• Theorem 3.12:
  • julia --project=@. scripts/Sp2nZ_has_T.jl -N 2 -R 2
  • julia --project=@. scripts/Sp2nZ_has_T.jl -N 2 -R 3
• Theorem 3.15:
  • julia --project=@. scripts/Sp2nZ_Level.jl -N 3 -R 2
• Theorem 3.17:
  • julia --project=@. scripts/G2_has_T.jl -R 2 (option -N is ignored)
• Theorem 3.18:
  • julia --project=@. scripts/G2_Adj.jl -R 3 (option -N is ignored)

If you find yourself using or studying code in this repository please cite

@misc{kaluba2023kazhdan,
      title={Kazhdan constants for Chevalley groups over the integers},
      author={Marek Kaluba and Dawid Kielak},
      year={2023},
      eprint={2306.12358},
      archivePrefix={arXiv},
      primaryClass={math.GR}
}