PropertyT.jl/README.md

Property(T)

This package is concerned with sum of squares decompositions in group rings of finitely presented groups. Please have a look into e.g. this test 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:

• M. Kaluba and P.W. Nowak Certifying numerical estimates for spectral gaps 1703.09680
• M. Kaluba, P.W. Nowak and N. Ozawa Aut(F₅) has property (T) 1712.07167, and
• M. Kaluba, D. Kielak and P.W. Nowak On property (T) for Aut(Fₙ) and $SLₙ(Z)$ 1812.03456.

The package depends on

• Groups.jl for computations with finitely presented groups,
• SymbolicWedderburn.jl for symmetrizing the sum of squares relaxations of positivity problems,
• JuMP.jl for formulating the optimization problems, and
• SCS.jl wrapper for the scs solver, or
• 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