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

Kaluba, Marek; Nowak, Piotr W.; Ozawa, Narutaka


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    "title": "An approximation of the spectral gap for the Laplace operator on SAut(\ud835\udd3d\u2085)", 
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    "keywords": [
      "spectral gap", 
      "Laplace operator", 
      "SAut(F5)", 
      "property (T)", 
      "semidefinite optimization"
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    "publication_date": "2017-12-19", 
    "creators": [
      {
        "orcid": "0000-0002-8777-8223", 
        "affiliation": "Adam Mickiewicz University, Pozna\u0144, Poland", 
        "name": "Kaluba, Marek"
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        "orcid": "0000-0002-6519-004X", 
        "affiliation": "Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland", 
        "name": "Nowak, Piotr W."
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        "affiliation": "Research Institute for Mathematical Sciences, Kyoto, Japan", 
        "name": "Ozawa, Narutaka"
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    "description": "<p>This is the dataset accompanying&nbsp;<em>Aut(\ud835\udd3d\u2085) 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>\n\n<blockquote>\n<p>tar --list -f ./oSAutF5_r2.tar.xz</p>\n\n<p>oSAutF5_r2/<br>\noSAutF5_r2/delta.jld<br>\noSAutF5_r2/orbits.jld<br>\noSAutF5_r2/preps.jld<br>\noSAutF5_r2/U_pis.jld<br>\noSAutF5_r2/pm.jld<br>\noSAutF5_r2/1.2/<br>\noSAutF5_r2/1.2/full_2017-11-27T20:02:27.253.log<br>\noSAutF5_r2/1.2/SDPmatrix.jld<br>\noSAutF5_r2/1.2/solver_2017-11-27T20:05:39.134.log<br>\noSAutF5_r2/1.2/lambda.jld</p>\n</blockquote>\n\n<p>To replicate computation of the spectral gap clone <em>GroupsWithPropertyT</em> repository and checkout the <em>1712.07167</em> branch:</p>\n\n<blockquote>\n<p>git clone https://git.wmi.amu.edu.pl/kalmar/GroupsWithPropertyT.git<br>\ncd ./GroupsWithPropertyT<br>\ngit checkout 1712.07167</p>\n</blockquote>\n\n<p>Then unpack the content of <em>oSAutF5_r2.tar.xz</em> in <em>GroupsWithPropertyT</em> folder.</p>\n\n<p>You need <em>julia-0.6. </em>In <em>julia</em>s REPL run</p>\n\n<blockquote>\n<p>Pkg.add(&quot;ArgParse&quot;)<br>\nPkg.add(&quot;Nemo&quot;)<br>\nPkg.clone(&quot;https://git.wmi.amu.edu.pl/kalmar/Groups.jl.git&quot;)<br>\nPkg.checkout(&quot;Groups&quot;, &quot;1712.07167&quot;)<br>\nPkg.clone(&quot;https://git.wmi.amu.edu.pl/kalmar/GroupRings.jl.git&quot;)<br>\nPkg.checkout(&quot;GroupRings&quot;, &quot;1712.07167&quot;)<br>\nPkg.clone(&quot;https://git.wmi.amu.edu.pl/kalmar/PropertyT.jl.git&quot;)<br>\nPkg.checkout(&quot;PropertyT&quot;, &quot;1712.07167&quot;)<br>\nPkg.checkout(&quot;SCS&quot;)<br>\nPkg.build(&quot;SCS&quot;)</p>\n</blockquote>\n\n<p>Finally, to verify that the Laplace operator on <em>SAut(\ud835\udd3d\u2085)</em> (associated to the standard generating set) has spectral gap of at least 1.2 run</p>\n\n<blockquote>\n<p>julia AutFN_orbit.jl -N 5 --upper-bound=1.2</p>\n</blockquote>\n\n<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>\n\n<p>This research was supported in part by PL-Grid Infrastructure.</p>"
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