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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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<oai_dc:dc xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:oai_dc="http://www.openarchives.org/OAI/2.0/oai_dc/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd">
  <dc:creator>Kaluba, Marek</dc:creator>
  <dc:creator>Nowak, Piotr W.</dc:creator>
  <dc:creator>Ozawa, Narutaka</dc:creator>
  <dc:date>2017-12-19</dc:date>
  <dc:description>This is the dataset accompanying Aut(𝔽₅) has property (T) paper (https://arxiv.org/abs/1712.07167). See Section 4 thereof for a detailed description of the content of the included files:


tar --list -f ./oSAutF5_r2.tar.xz

oSAutF5_r2/
oSAutF5_r2/delta.jld
oSAutF5_r2/orbits.jld
oSAutF5_r2/preps.jld
oSAutF5_r2/U_pis.jld
oSAutF5_r2/pm.jld
oSAutF5_r2/1.2/
oSAutF5_r2/1.2/full_2017-11-27T20:02:27.253.log
oSAutF5_r2/1.2/SDPmatrix.jld
oSAutF5_r2/1.2/solver_2017-11-27T20:05:39.134.log
oSAutF5_r2/1.2/lambda.jld


To replicate computation of the spectral gap clone GroupsWithPropertyT repository and checkout the 1712.07167 branch:


git clone https://git.wmi.amu.edu.pl/kalmar/GroupsWithPropertyT.git
cd ./GroupsWithPropertyT
git checkout 1712.07167


Then unpack the content of oSAutF5_r2.tar.xz in GroupsWithPropertyT folder.

You need julia-0.6. In julias REPL run


Pkg.add("ArgParse")
Pkg.add("Nemo")
Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/Groups.jl.git")
Pkg.checkout("Groups", "1712.07167")
Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/GroupRings.jl.git")
Pkg.checkout("GroupRings", "1712.07167")
Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/PropertyT.jl.git")
Pkg.checkout("PropertyT", "1712.07167")
Pkg.checkout("SCS")
Pkg.build("SCS")


Finally, to verify that the Laplace operator on SAut(𝔽₅) (associated to the standard generating set) has spectral gap of at least 1.2 run


julia AutFN_orbit.jl -N 5 --upper-bound=1.2


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

This research was supported in part by PL-Grid Infrastructure.</dc:description>
  <dc:identifier>https://zenodo.org/record/1133441</dc:identifier>
  <dc:identifier>10.5281/zenodo.1133441</dc:identifier>
  <dc:identifier>oai:zenodo.org:1133441</dc:identifier>
  <dc:relation>info:eu-repo/grantAgreement/EC/H2020/677120/</dc:relation>
  <dc:relation>arxiv:arXiv:1712.07167</dc:relation>
  <dc:relation>doi:10.5281/zenodo.1133440</dc:relation>
  <dc:rights>info:eu-repo/semantics/openAccess</dc:rights>
  <dc:rights>http://creativecommons.org/licenses/by-sa/4.0/legalcode</dc:rights>
  <dc:subject>spectral gap</dc:subject>
  <dc:subject>Laplace operator</dc:subject>
  <dc:subject>SAut(F5)</dc:subject>
  <dc:subject>property (T)</dc:subject>
  <dc:subject>semidefinite optimization</dc:subject>
  <dc:title>An approximation of the spectral gap for the Laplace operator on SAut(𝔽₅)</dc:title>
  <dc:type>info:eu-repo/semantics/other</dc:type>
  <dc:type>dataset</dc:type>
</oai_dc:dc>
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