GroupsWithPropertyT/README.md

This repository contains code for computations in [Certifying Numerical Estimates of Spectral Gaps](https://arxiv.org/abs/1703.09680).

# Installing
To run the code You need `julia-v0.5` (should work on `v0.6`, but with warnings).
You also need to install julia packages: `Nemo-v0.6.3`, `ArgParse`. To do so in `julia`'s REPL run:
```julia
Pkg.update()
Pkg.add("Nemo")
Pkg.add("ArgParse")
```
Then clone the main repository of `Groups.jl`, `GroupRings.jl` and `PropertyT.jl`:
```julia
Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/Groups.jl.git")
Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/GroupRings.jl.git")
Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/PropertyT.jl.git")
Pkg.resolve()
```
This should resolve all dependencies (e.g. install `JuMP`, `SCS`, `IntervalArithmetic`, `JLD`, `Memento`). Exit julia and finally clone this repository:
```shell
git clone https://git.wmi.amu.edu.pl/kalmar/GroupsWithPropertyT.git
cd GroupswithPropertyT
```

# Running

## Naive implementation

To check that $\Delta^2-\lambda\Delta$ is not decomposable to a sum of hermitian squares of elements in the ball of radius $2$ in $SL(2,7)$ run
```shell
julia SL.jl -N 2 -p 7 --radius 2 --iterations 100000
```
(~30 seconds, depending on hardware). The monotonous decreasing $\lambda$ during the optimisation is in column `pri obj` (or `dua obj`) of `solver.log`.

Compare this to
```shell
julia SL.jl -N 2 -p 7 --radius 3 --iterations 100000
```
which finds $\lambda \geq 0.5857$ and decomposes $\Delta^2-\lambda\Delta$ into sum of $47$ hermitian squares in less than 20 seconds (including certification).

If You see in the output (or in `full.log`) that the upper end of the interval where $\lVert\Delta^2 - \lambda\Delta - \sum{\xi_i}^*\xi_i\rVert_1$ belongs to is too large (resulting in positive `Floating point distance`, but negative `The Augmentation-projected actual distance`), decrease the `--tol` parameter, e.g.
```
julia SL.jl -N 2 -p 7 --radius 3 --iterations 100000 --tol 1e-9
```
to achieve a better estimate (the residuals $\ell_1$-norm should be around $\|B_d(e))\|\cdot tol$)

## Symmetrization enhanced implementation

A newer version of the software uses orbit and Wedderburn decomposition to effecitively find a (much) smaller optimisation problem to compute the spectral gap $\lambda$. In particular the solution to the original (naive) optimisation problem can be reconstructed from the solution of the symmetrised one.

E.g. Run
```shell
julia SL_orbit.jl -N 4 --radius 2 --upper-bound 1.3
```
to find (and certify) the spectral gap for $SL(4, \mathbb{Z})$ is at least `1.2999...` in just under $2$ minutes time (for comparison this result requires over `5` hours in the old implementation on the same hardware).

To replicate the results of _$\operatorname{Aut}(\textbf{F}_5)$ has property (T)_ article (currently not published) You neet to run (on a `4`-core CPU)
```shell
julia ../AutFN_orbit.jl -N 5 --upper-bound 1.2 --iterations 24000000 --cpus 4
```

Note that this computation took more than `12` days and required at least `32`GB of ram (and possible more).

# Help

```shell
julia SL.jl --help
usage: SL.jl [--tol TOL] [--iterations ITERATIONS]
             [--upper-bound UPPER-BOUND] [--cpus CPUS] [-N N] [-p P]
             [--radius RADIUS] [-h]

optional arguments:
  --tol TOL             set numerical tolerance for the SDP solver
                        (type: Float64, default: 1.0e-6)
  --iterations ITERATIONS
                        set maximal number of iterations for the SDP
                        solver (default: 20000) (type: Int64, default:
                        50000)
  --upper-bound UPPER-BOUND
                        Set an upper bound for the spectral gap (type:
                        Float64, default: Inf)
  --cpus CPUS           Set number of cpus used by solver (type:
                        Int64)
  -N N                  Consider elementary matrices EL(N) (type:
                        Int64, default: 2)
  -p P                  Matrices over field of p-elements (p=0 => over
                        ZZ) (type: Int64, default: 0)
  --radius RADIUS       Radius of ball B_r(e,S) to find solution over
                        (type: Int64, default: 2)
  -h, --help            show this help message and exit
```

# Specific version of [1703.09680](https://arxiv.org/abs/1703.09680)

To checkout the specific versions of packages used for [Certifying Numerical Estimates of Spectral Gaps](https://arxiv.org/abs/1703.09680) run (inside the cloned `GroupswithPropertyT`)
```shell
git checkout 1703.09680v1
```

Unfortunately: You need to link `~/.julia/v0.5/GroupRings` to `~/.julia/v0.5/GroupAlgebras` due to change in the name of the package. Then run in `julia`
```julia
Pkg.checkout("GroupRings", "1703.09680v1")
Pkg.checkout("PropertyT", "1703.09680v1")
Pkg.resolve()
```