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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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  <identifier identifierType="DOI">10.5281/zenodo.1133441</identifier>
  <creators>
    <creator>
      <creatorName>Kaluba, Marek</creatorName>
      <givenName>Marek</givenName>
      <familyName>Kaluba</familyName>
      <nameIdentifier nameIdentifierScheme="ORCID" schemeURI="http://orcid.org/">0000-0002-8777-8223</nameIdentifier>
      <affiliation>Adam Mickiewicz University, Poznań, Poland</affiliation>
    </creator>
    <creator>
      <creatorName>Nowak, Piotr W.</creatorName>
      <givenName>Piotr W.</givenName>
      <familyName>Nowak</familyName>
      <nameIdentifier nameIdentifierScheme="ORCID" schemeURI="http://orcid.org/">0000-0002-6519-004X</nameIdentifier>
      <affiliation>Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland</affiliation>
    </creator>
    <creator>
      <creatorName>Ozawa, Narutaka</creatorName>
      <givenName>Narutaka</givenName>
      <familyName>Ozawa</familyName>
      <affiliation>Research Institute for Mathematical Sciences, Kyoto, Japan</affiliation>
    </creator>
  </creators>
  <titles>
    <title>An approximation of the spectral gap for the Laplace operator on SAut(𝔽₅)</title>
  </titles>
  <publisher>Zenodo</publisher>
  <publicationYear>2017</publicationYear>
  <subjects>
    <subject>spectral gap</subject>
    <subject>Laplace operator</subject>
    <subject>SAut(F5)</subject>
    <subject>property (T)</subject>
    <subject>semidefinite optimization</subject>
  </subjects>
  <dates>
    <date dateType="Issued">2017-12-19</date>
  </dates>
  <resourceType resourceTypeGeneral="Dataset"/>
  <alternateIdentifiers>
    <alternateIdentifier alternateIdentifierType="url">https://zenodo.org/record/1133441</alternateIdentifier>
  </alternateIdentifiers>
  <relatedIdentifiers>
    <relatedIdentifier relatedIdentifierType="arXiv" relationType="IsPartOf">arXiv:1712.07167</relatedIdentifier>
    <relatedIdentifier relatedIdentifierType="DOI" relationType="IsVersionOf">10.5281/zenodo.1133440</relatedIdentifier>
  </relatedIdentifiers>
  <version>1.2</version>
  <rightsList>
    <rights rightsURI="http://creativecommons.org/licenses/by-sa/4.0/legalcode">Creative Commons Attribution Share Alike 4.0 International</rights>
    <rights rightsURI="info:eu-repo/semantics/openAccess">Open Access</rights>
  </rightsList>
  <descriptions>
    <description descriptionType="Abstract">&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;</description>
  </descriptions>
  <fundingReferences>
    <fundingReference>
      <funderName>European Commission</funderName>
      <funderIdentifier funderIdentifierType="Crossref Funder ID">10.13039/501100000780</funderIdentifier>
      <awardNumber awardURI="info:eu-repo/grantAgreement/EC/H2020/677120/">677120</awardNumber>
      <awardTitle>Rigidity of groups and higher index theory</awardTitle>
    </fundingReference>
  </fundingReferences>
</resource>
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