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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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    <subfield code="a">spectral gap</subfield>
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    <subfield code="a">Laplace operator</subfield>
  </datafield>
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    <subfield code="a">SAut(F5)</subfield>
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    <subfield code="a">property (T)</subfield>
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    <subfield code="a">semidefinite optimization</subfield>
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    <subfield code="u">Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland</subfield>
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    <subfield code="a">Nowak, Piotr W.</subfield>
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    <subfield code="u">Research Institute for Mathematical Sciences, Kyoto, Japan</subfield>
    <subfield code="a">Ozawa, Narutaka</subfield>
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    <subfield code="u">Adam Mickiewicz University, Poznań, Poland</subfield>
    <subfield code="0">(orcid)0000-0002-8777-8223</subfield>
    <subfield code="a">Kaluba, Marek</subfield>
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    <subfield code="a">An approximation of the spectral gap for the Laplace operator on SAut(𝔽₅)</subfield>
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    <subfield code="c">677120</subfield>
    <subfield code="a">Rigidity of groups and higher index theory</subfield>
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    <subfield code="a">&lt;p&gt;This is the dataset accompanying&amp;nbsp;&lt;em&gt;Aut(𝔽₅) has property (T)&lt;/em&gt; paper (https://arxiv.org/abs/1712.07167). See Section 4 thereof for a detailed description of the content of the included files:&lt;/p&gt;

&lt;blockquote&gt;
&lt;p&gt;tar --list -f ./oSAutF5_r2.tar.xz&lt;/p&gt;

&lt;p&gt;oSAutF5_r2/&lt;br&gt;
oSAutF5_r2/delta.jld&lt;br&gt;
oSAutF5_r2/orbits.jld&lt;br&gt;
oSAutF5_r2/preps.jld&lt;br&gt;
oSAutF5_r2/U_pis.jld&lt;br&gt;
oSAutF5_r2/pm.jld&lt;br&gt;
oSAutF5_r2/1.2/&lt;br&gt;
oSAutF5_r2/1.2/full_2017-11-27T20:02:27.253.log&lt;br&gt;
oSAutF5_r2/1.2/SDPmatrix.jld&lt;br&gt;
oSAutF5_r2/1.2/solver_2017-11-27T20:05:39.134.log&lt;br&gt;
oSAutF5_r2/1.2/lambda.jld&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;To replicate computation of the spectral gap clone &lt;em&gt;GroupsWithPropertyT&lt;/em&gt; repository and checkout the &lt;em&gt;1712.07167&lt;/em&gt; branch:&lt;/p&gt;

&lt;blockquote&gt;
&lt;p&gt;git clone https://git.wmi.amu.edu.pl/kalmar/GroupsWithPropertyT.git&lt;br&gt;
cd ./GroupsWithPropertyT&lt;br&gt;
git checkout 1712.07167&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;Then unpack the content of &lt;em&gt;oSAutF5_r2.tar.xz&lt;/em&gt; in &lt;em&gt;GroupsWithPropertyT&lt;/em&gt; folder.&lt;/p&gt;

&lt;p&gt;You need &lt;em&gt;julia-0.6. &lt;/em&gt;In &lt;em&gt;julia&lt;/em&gt;s REPL run&lt;/p&gt;

&lt;blockquote&gt;
&lt;p&gt;Pkg.add(&amp;quot;ArgParse&amp;quot;)&lt;br&gt;
Pkg.add(&amp;quot;Nemo&amp;quot;)&lt;br&gt;
Pkg.clone(&amp;quot;https://git.wmi.amu.edu.pl/kalmar/Groups.jl.git&amp;quot;)&lt;br&gt;
Pkg.checkout(&amp;quot;Groups&amp;quot;, &amp;quot;1712.07167&amp;quot;)&lt;br&gt;
Pkg.clone(&amp;quot;https://git.wmi.amu.edu.pl/kalmar/GroupRings.jl.git&amp;quot;)&lt;br&gt;
Pkg.checkout(&amp;quot;GroupRings&amp;quot;, &amp;quot;1712.07167&amp;quot;)&lt;br&gt;
Pkg.clone(&amp;quot;https://git.wmi.amu.edu.pl/kalmar/PropertyT.jl.git&amp;quot;)&lt;br&gt;
Pkg.checkout(&amp;quot;PropertyT&amp;quot;, &amp;quot;1712.07167&amp;quot;)&lt;br&gt;
Pkg.checkout(&amp;quot;SCS&amp;quot;)&lt;br&gt;
Pkg.build(&amp;quot;SCS&amp;quot;)&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;Finally, to verify that the Laplace operator on &lt;em&gt;SAut(𝔽₅)&lt;/em&gt; (associated to the standard generating set) has spectral gap of at least 1.2 run&lt;/p&gt;

&lt;blockquote&gt;
&lt;p&gt;julia AutFN_orbit.jl -N 5 --upper-bound=1.2&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;If You want to generate the multiplication table and other files on Your own delete all &lt;code&gt;*.jld&lt;/code&gt; files from the &lt;code&gt;oSAutF5_r2&lt;/code&gt; folder (but the ones in &lt;code&gt;1.2&lt;/code&gt; folder) and run the same command again. Note: You need at least &lt;code&gt;32&lt;/code&gt;GB of RAM and spare &lt;code&gt;24&lt;/code&gt;h of Your CPU.&lt;/p&gt;

&lt;p&gt;This research was supported in part by PL-Grid Infrastructure.&lt;/p&gt;</subfield>
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