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

<p>This is the dataset accompanying&nbsp;<em>Aut(𝔽₅) has property (T)</em> paper (https://arxiv.org/abs/1712.07167). See Section 4 thereof for a detailed description of the content of the included files:</p> <blockquote> <p><code>tar --list -f ./oSAutF5_r2.tar.xz</code></p> <p><br> <code>oSAutF5_r2/<br> oSAutF5_r2/1.3/<br> oSAutF5_r2/1.3/full_2018-01-26T12:29:58.143.log<br> oSAutF5_r2/1.3/solver_2018-01-26T12:29:58.143.log<br> oSAutF5_r2/1.3/SDPmatrix.jld<br> oSAutF5_r2/1.3/lambda.jld<br> oSAutF5_r2/U_pis.jld<br> oSAutF5_r2/pm.jld<br> oSAutF5_r2/delta.jld<br> oSAutF5_r2/orbits.jld<br> oSAutF5_r2/preps.jld</code></p> </blockquote> <p>To replicate the computation of the spectral gap clone <code>1712.07167</code> repository first</p> <blockquote> <p><code>git clone https://git.wmi.amu.edu.pl/kalmar/1712.07167.git</code></p> </blockquote> <p>Then unpack the content of <code>oSAutF5_r2.tar.xz</code> into <code>1712.07167</code> folder.</p> <p>You need <code>julia-1.1.0</code> or above. In <code>julia</code>s REPL run</p> <blockquote> <p><code>using Pkg<br> Pkg.activate(&quot;1712.07167&quot;)<br> Pkg.instantiate()<br> Pkg.test(&quot;PropertyT&quot;)</code></p> </blockquote> <p>Finally, to verify that the Laplace operator on <em>SAut(𝔽₅)</em> (associated to the standard generating set) has spectral gap of at least <code>1.3</code> run from within <code>1712.07167</code> folder</p> <blockquote> <p><code>julia check_SAutF5.jl</code></p> </blockquote> <p>If You want to generate the multiplication table and other files on Your own delete all <code>*.jld</code> files from the <code>oSAutF5_r2</code> folder (but the ones in <code>1.3</code> folder) and run the same command again. Note: You need at least <code>20</code>GB of RAM and spare a few hours of Your CPU.</p> <p>We reproduce the content of <code>check_SAutF5.jl</code> script below.</p> <blockquote> <p><code>using Pkg<br> Pkg.activate(&quot;.&quot;)<br> using Groups<br> using GroupRings<br> using PropertyT<br> using SparseArrays<br> using LinearAlgebra<br> using IntervalArithmetic<br> using JLD</code></p> <p><code>@show Threads.nthreads()<br> BLAS.set_num_threads(Threads.nthreads());</code></p> <p><code>G = SAut(FreeGroup(5))<br> pm = load(&quot;oSAutF5_r2/pm.jld&quot;, &quot;pm&quot;);<br> RG = GroupRing(G, pm)<br> @info RG</code></p> <p><code>S_size = 80<br> # due to technical problems we are no longer able to load delta.jl on julia-1.0<br> &Delta;_coeff = SparseVector(maximum(pm), collect(1:(1+S_size)), [S_size; -ones(S_size)])<br> &Delta; = GroupRingElem(&Delta;_coeff, RG);<br> &Delta;&sup2; = &Delta;^2;</code></p> <p><code>@info &quot;Loading solution&quot;<br> &lambda;₀ = load(&quot;oSAutF5_r2/1.3/lambda.jld&quot;, &quot;&lambda;&quot;)<br> P₀ = load(&quot;oSAutF5_r2/1.3/SDPmatrix.jld&quot;, &quot;P&quot;);</code></p> <p><code>@info &quot;Taking square root of P&quot;<br> @time Q = real(sqrt(P₀));</code></p> <p><code>Q_aug, check_columns_augmentation = PropertyT.augIdproj(Interval, Q);<br> @show check_columns_augmentation<br> if !check_columns_augmentation<br> &nbsp; @warn &quot;Columns of Q are not guaranteed to represent elements of the augmentation ideal!&quot;<br> end</code></p> <p><code>@info &quot;Computing SOS decomposition&quot;<br> @time sos = PropertyT.compute_SOS(RG, Q_aug);</code></p> <p><code>residual = &Delta;&sup2; - @interval(&lambda;₀)*&Delta; - sos;<br> @show norm(residual, 1)</code></p> </blockquote> <p>&nbsp;</p> <p>This research was supported in part by</p> <ul> <li>PL-Grid Infrastructure,</li> <li>grant 2015/19/B/ST1/01458, National Science Center, Poland</li> <li>grant 2017/26/D/ST1/00103, National Science Center, Poland.</li> </ul>