January 26, 2018 Dataset Open Access

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

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

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_r2/1.3/
oSAutF5_r2/1.3/full_2018-01-26T12:29:58.143.log
oSAutF5_r2/1.3/solver_2018-01-26T12:29:58.143.log
oSAutF5_r2/1.3/SDPmatrix.jld
oSAutF5_r2/1.3/lambda.jld
oSAutF5_r2/U_pis.jld
oSAutF5_r2/pm.jld
oSAutF5_r2/delta.jld
oSAutF5_r2/orbits.jld
oSAutF5_r2/preps.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.build("SCS")

Finally, to verify that the Laplace operator on SAut(𝔽₅) (associated to the standard generating set) has spectral gap of at least 1.3 run

julia AutFN_orbit.jl -N 5 --upper-bound=1.3

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 1.3 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.