kalmar/PropertyT.jl
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# Property(T) # Property(T)
[![Build Status](https://travis-ci.org/kalmarek/PropertyT.jl.svg?branch=master)](https://travis-ci.org/kalmarek/PropertyT.jl) [![CI](https://github.com/kalmarek/PropertyT.jl/actions/workflows/ci.yml/badge.svg)](https://github.com/kalmarek/PropertyT.jl/actions/workflows/ci.yml)
[![codecov](https://codecov.io/gh/kalmarek/PropertyT.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/kalmarek/PropertyT.jl) [![codecov](https://codecov.io/gh/kalmarek/PropertyT.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/kalmarek/PropertyT.jl)
This package is concerned with sum of squares decompositions in group rings of finitely presented groups. This package is concerned with sum of squares decompositions in group rings of finitely presented groups.
Please have a look into [test](https://github.com/kalmarek/GroupRings.jl/blob/master/test/runtests.jl) directory to see how to use this package. For an example applications have a look at our papers: Please have a look into e.g. this [test](https://github.com/kalmarek/PropertyT.jl/blob/master/test/1712.07167.jl#L87) to see how this package can be used to prove Kazdhan Property (T) for a finitely presented group. For an example applications have a look at our papers:
[1703.09680](https://arxiv.org/abs/1703.09680), [1712.07167](https://arxiv.org/abs/1712.07167) and [1812.03456](https://arxiv.org/abs/1812.03456). * M. Kaluba and P.W. Nowak _Certifying numerical estimates for spectral gaps_ [1703.09680](https://arxiv.org/abs/1703.09680)
* M. Kaluba, P.W. Nowak and N. Ozawa *$Aut(F₅)$ has property (T)* [1712.07167](https://arxiv.org/abs/1712.07167), and
* M. Kaluba, D. Kielak and P.W. Nowak *On property (T) for $Aut(Fₙ)$ and $SLₙ(Z)$* [1812.03456](https://arxiv.org/abs/1812.03456).
The package depends on The package depends on
* [AbstractAlgebra](https://github.com/Nemocas/AbstractAlgebra.jl), * [`Groups.jl`](https://github.com/kalmarek/Groups.jl) for computations with finitely presented groups,
* [Groups](https://github.com/kalmarek/Groups.jl) * [`SymbolicWedderburn.jl`](https://github.com/kalmarek/SymbolicWedderburn.jl) for symmetrizing the sum of squares relaxations of positivity problems,
* [GroupRings](https://github.com/kalmarek/GroupRings.jl) * [`JuMP.jl`](https://github.com/JuliaOpt/JuMP.jl) for formulating the optimization problems, and
* [JuMP](https://github.com/JuliaOpt/JuMP.jl) * [`SCS.jl`](https://github.com/JuliaOpt/SCS.jl) wrapper for the [`scs` solver](https://github.com/cvxgrp/scs), or
* [scs](https://github.com/JuliaOpt/SCS.jl) [solver](https://github.com/cvxgrp/scs) * [`COSMO.jl`](https://github.com/oxfordcontrol/COSMO.jl) solver to solve the problems.
Certification of the results is done via `ℓ₁`-convexity of the sum-of-squares cone and our knowledge of its interior points. The certified computations use
* [`IntervalArithmetic.jl`](https://github.com/JuliaIntervals/IntervalArithmetic.jl)