10.5281/zenodo.1913734
https://zenodo.org/records/1913734
oai:zenodo.org:1913734
Kaluba, Marek
Marek
Kaluba
0000-0002-8777-8223
Adam Mickiewicz University, Poznań, Poland
Nowak, Piotr W.
Piotr W.
Nowak
0000-0002-6519-004X
Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland
Ozawa, Narutaka
Narutaka
Ozawa
Research Institute for Mathematical Sciences, Kyoto, Japan
An approximation of the spectral gap for the Laplace operator on SAut(F₅)
Zenodo
2018
Aut(F_5)
property (T)
Laplace operator
spectral gap
sum of squares
semidefinite optimization
2018-01-26
arXiv:1712.07167
10.5281/zenodo.1133440
https://zenodo.org/communities/eu
1.3
Creative Commons Attribution Share Alike 4.0 International
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.
European Commission
10.13039/501100000780
677120
Rigidity of groups and higher index theory