An approximation of the spectral gap for the Laplace operator on SAut(F₅)

This is the dataset accompanying Aut(𝔽₅) has property (T) paper (https://arxiv.org/abs/1712.07167). See Section 4 thereof for a detailed description of the content of the included files:

tar --list -f ./oSAutF5_r2.tar.xz

oSAutF5_r2/
oSAutF5_r2/1.3/
oSAutF5_r2/1.3/full_2018-01-26T12:29:58.143.log
oSAutF5_r2/1.3/solver_2018-01-26T12:29:58.143.log
oSAutF5_r2/1.3/SDPmatrix.jld
oSAutF5_r2/1.3/lambda.jld
oSAutF5_r2/U_pis.jld
oSAutF5_r2/pm.jld
oSAutF5_r2/delta.jld
oSAutF5_r2/orbits.jld
oSAutF5_r2/preps.jld

To replicate the computation of the spectral gap clone 1712.07167 repository first

git clone https://git.wmi.amu.edu.pl/kalmar/1712.07167.git

Then unpack the content of oSAutF5_r2.tar.xz into 1712.07167 folder.

You need julia-1.1.0 or above. In julias REPL run

using Pkg
Pkg.activate("1712.07167")
Pkg.instantiate()
Pkg.test("PropertyT")

Finally, to verify that the Laplace operator on SAut(𝔽₅) (associated to the standard generating set) has spectral gap of at least 1.3 run from within 1712.07167 folder

julia check_SAutF5.jl

If You want to generate the multiplication table and other files on Your own delete all *.jld files from the oSAutF5_r2 folder (but the ones in 1.3 folder) and run the same command again. Note: You need at least 20GB of RAM and spare a few hours of Your CPU.

We reproduce the content of check_SAutF5.jl script below.

using Pkg
Pkg.activate(".")
using Groups
using GroupRings
using PropertyT
using SparseArrays
using LinearAlgebra
using IntervalArithmetic
using JLD

@show Threads.nthreads()
BLAS.set_num_threads(Threads.nthreads());

G = SAut(FreeGroup(5))
pm = load("oSAutF5_r2/pm.jld", "pm");
RG = GroupRing(G, pm)
@info RG

S_size = 80
# due to technical problems we are no longer able to load delta.jl on julia-1.0
Δ_coeff = SparseVector(maximum(pm), collect(1:(1+S_size)), [S_size; -ones(S_size)])
Δ = GroupRingElem(Δ_coeff, RG);
Δ² = Δ^2;

@info "Loading solution"
λ₀ = load("oSAutF5_r2/1.3/lambda.jld", "λ")
P₀ = load("oSAutF5_r2/1.3/SDPmatrix.jld", "P");

@info "Taking square root of P"
@time Q = real(sqrt(P₀));

Q_aug, check_columns_augmentation = PropertyT.augIdproj(Interval, Q);
@show check_columns_augmentation
if !check_columns_augmentation
  @warn "Columns of Q are not guaranteed to represent elements of the augmentation ideal!"
end

@info "Computing SOS decomposition"
@time sos = PropertyT.compute_SOS(RG, Q_aug);

residual = Δ² - @interval(λ₀)*Δ - sos;
@show norm(residual, 1)

This research was supported in part by

• PL-Grid Infrastructure,
• grant 2015/19/B/ST1/01458, National Science Center, Poland
• grant 2017/26/D/ST1/00103, National Science Center, Poland.