Dataset Open Access
This is the dataset accompanying On property (T) for Aut(Fₙ) and SLₙ(ℤ) paper (https://arxiv.org/abs/1812.03456). See the appendix thereof and Section 4 of (Aut(F₅) has property (T)) for more details.
1812.03456-cf6dee7.zipcontains a julia environment specification (
Manifest.toml) as well as
1812.03456.jlscript used for automatic certification and jupyter noteboks in
SAutF5_r2.tar.xzcontains the precomputed solutions for expressing
Adj₅+3·Op₅-1.4Δ₅as sum of (hermitian) squares in the group ring of
SAut(F₅). The contents of this archive must be placed inside `
1812.03456`directory from the
The code needs to be run with
julia-1.4.0 or higher (tested versions include also versions
julia-1.5). In principle any version in
[1.4-2.0) should work due to the promise of forward compatibility.
While located in the main directory (
1812.03456) you should run the following code in
REPL console to instantiate the environment for computations:
using Pkg Pkg.activate(".") Pkg.instantiate()
(this needs to be done once per installation). Then the directory
SAutF5_r2 (from the
SAutF5_r2.tar.xz archive) needs to be placed in
Replication: Jupyter notebook
A jupyter server may be launched then within the directory
1812.03456 by issuing from julia command-line (
REPL) the following commands.
using Pkg Pkg.activate(".") using IJulia notebook(dir=".")
During the first run the user may be asked for installation of
Jupyter program (a server for running this notebook) within
miniconda environment, which will happen automatically after confirmation. To execute the commands in the notebook, one needs to navigate to
notebooks subdirectory of
1812.03456 and click either of the notebooks.
One can replicate the main computational results of the paper by executing all the cells in the
Positivity of Adj_n + kOp_n in ISAut(F_n) notebook.
To verify that (Adj₅ + 3.0·Op₅) - 1.4·Δ₅ admits an approximate sum of squares decomposition run in
julia --project=. --color=yes 1812.03456.jl -n 5 -k 3 -l 1.4
On a modern laptop computer this should finish in less than 2h.
At the end of computations you will see lines such as:
┌ Info: λ is certified to be >
└ λ_cert.lo = 1.3701131733828074
[ Info: i.e Adj_5 + 3.0·Op_5 - (1.3701131733828074)·Δ_5 ∈ Σ²₂ ISAut(F_5)
This means that Adj₅ + 3.0·Op₅ - λΔ₅ is a sum of Hermitian squares of elements from ISAut(F₅) for every
λ < 1.370....
A similar verification for Adj₅ + 2.0·Op₅ - 0.28·Δ₅ can be run by executing
julia --project=. --color=yes 1812.03456.jl -n 5 -k 2 -l 0.28
Generating the provided files
If you wish to produce the whole certificate on your own (including the generation of group ring and its multiplication table), delete all
*.jld files from the
SAutF5_r2 folder and run one of the above commands with the same (or different) parameters again. Note: To do this you need at least 16GB of RAM and spare 24h of your CPU.
This research was supported in part by National Science Center, Poland, grant 2017/26/D/ST1/00103.