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

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Kaluba, Marek ; Nowak, Piotr W. ; Ozawa, Narutaka
  • Publisher: Zenodo
  • Related identifiers: doi: 10.5281/zenodo.1133441
  • Subject: spectral gap | Laplace operator | SAut(F5) | property (T) | semidefinite optimization

<p>This is the dataset accompanying&nbsp;<em>Aut(𝔽₅) has property (T)</em> paper ( See Section 4 thereof for a detailed description of the content of the included files:</p> <blockquote> <p>tar --list -f ./oSAutF5_r2.tar.xz</p> <p>oSAutF5_r2/<br> oSAutF5_r2/delta.jld<br> oSAutF5_r2/orbits.jld<br> oSAutF5_r2/preps.jld<br> oSAutF5_r2/U_pis.jld<br> oSAutF5_r2/pm.jld<br> oSAutF5_r2/1.2/<br> oSAutF5_r2/1.2/full_2017-11-27T20:02:27.253.log<br> oSAutF5_r2/1.2/SDPmatrix.jld<br> oSAutF5_r2/1.2/solver_2017-11-27T20:05:39.134.log<br> oSAutF5_r2/1.2/lambda.jld</p> </blockquote> <p>To replicate computation of the spectral gap clone <em>GroupsWithPropertyT</em> repository and checkout the <em>1712.07167</em> branch:</p> <blockquote> <p>git clone<br> cd ./GroupsWithPropertyT<br> git checkout 1712.07167</p> </blockquote> <p>Then unpack the content of <em>oSAutF5_r2.tar.xz</em> in <em>GroupsWithPropertyT</em> folder.</p> <p>You need <em>julia-0.6. </em>In <em>julia</em>s REPL run</p> <blockquote> <p>Pkg.add(&quot;ArgParse&quot;)<br> Pkg.add(&quot;Nemo&quot;)<br> Pkg.clone(&quot;;)<br> Pkg.checkout(&quot;Groups&quot;, &quot;1712.07167&quot;)<br> Pkg.clone(&quot;;)<br> Pkg.checkout(&quot;GroupRings&quot;, &quot;1712.07167&quot;)<br> Pkg.clone(&quot;;)<br> Pkg.checkout(&quot;PropertyT&quot;, &quot;1712.07167&quot;)<br> Pkg.checkout(&quot;SCS&quot;)<br>;SCS&quot;)</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 1.2 run</p> <blockquote> <p>julia AutFN_orbit.jl -N 5 --upper-bound=1.2</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>1.2</code> folder) and run the same command again. Note: You need at least <code>32</code>GB of RAM and spare <code>24</code>h of Your CPU.</p> <p>This research was supported in part by PL-Grid Infrastructure.</p>
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