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.