This repository contains code for computations in Certifying Numerical Estimates of Spectral Gaps.

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:

Pkg.update()
Pkg.add("Nemo")
Pkg.add("ArgParse")

Then clone the main repository of Groups.jl, GroupRings.jl and PropertyT.jl:

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:

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

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

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A newer version of the software uses orbit and Wedderburn decomposition to effecitively find a (much) smaller optimisation problem to compute the spectral gap \lambda.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

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)

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 32GB of ram (and possible more).

Help

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

To checkout the specific versions of packages used for Certifying Numerical Estimates of Spectral Gaps run (inside the cloned GroupswithPropertyT)

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

Pkg.checkout("GroupRings", "1703.09680v1")
Pkg.checkout("PropertyT", "1703.09680v1")
Pkg.resolve()