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

Example: SL_3(\mathbb{Z}) has property (T)

Lets prove that SL_3(\mathbb{Z}) has property (T). We start with

julia> using Groups

julia> SL = MatrixGroups.SpecialLinearGroup{3}(Int8)
special linear group of 3×3 matrices over Int8

To define the sum of squares problem we need a group algebra:

julia> using PropertyT

julia> RSL, S, sizes = PropertyT.group_algebra(SL, gens(SL), halfradius=2, twisted=true)
[ Info: generating wl-metric ball of radius 4
  0.011135 seconds (279.78 k allocations: 9.858 MiB)
[ Info: sizes = [13, 121, 883, 5455]
[ Info: computing the *-algebra structure for G
  0.049325 seconds (158.86 k allocations: 7.218 MiB, 92.13% compilation time)
(*-algebra of special linear group of 3×3 matrices over Int8, FPGroupElement{Groups.MatrixGroups.SpecialLinearGroup{3, Int8, DataType, Alphabet{Groups.MatrixGroups.ElementaryMatrix{3, Int8}}, Vector{Groups.MatrixGroups.ElementaryMatrix{3, Int8}}}, …}[E₁₂, E₁₃, E₂₁, E₂₃, E₃₁, E₃₂, E₁₂^-1, E₁₃^-1, E₂₁^-1, E₂₃^-1, E₃₁^-1, E₃₂^-1], [13, 121, 883, 5455])

We supplied halfradius=2 to be able to multiply in RSL algebra elements supported in the ball of radius 2 of SL (with the word-length metric).

The Laplacian

We define the (symmetric) group Laplacian Δ using the familiar formula

julia> Δ = RSL(length(S)) - sum(RSL(s) for s in S)
12·(id) -1·E₁₂ -1·E₁₃ -1·E₂₁ -1·E₂₃ -1·E₃₁ -1·E₃₂ -1·E₁₂^-1 -1·E₁₃^-1 -1·E₂₁^-1 -1·E₂₃^-1 -1·E₃₁^-1 -1·E₃₂^-1

julia> Δ² = Δ^2
156·(id) -24·E₁₂ -24·E₁₃ -24·E₂₁ -24·E₂₃ -24·E₃₁ -24·E₃₂ -24·E₁₂^-1 -24·E₁₃^-1 -24·E₂₁^-1 -24·E₂₃^-1 -24·E₃₁^-1 -24·E₃₂^-1 +1·E₁₂^2 +2·E₁₂*E₁₃ +1·E₁₂*E₂₁ +1·E₁₂*E₂₃ +1·E₁₂*E₃₁ +2·E₁₂*E₃₂ +2·E₁₂*E₁₃^-1 +1·E₁₂*E₂₁^-1 +1·E₁₂*E₂₃^-1 +1·E₁₂*E₃₁^-1 +2·E₁₂*E₃₂^-1 +1·E₁₃^2 +1·E₁₃*E₂₁ +2·E₁₃*E₂₃ +1·E₁₃*E₃₁ +1·E₁₃*E₃₂ +2·E₁₃*E₁₂^-1 +1·E₁₃*E₂₁^-1 +2·E₁₃*E₂₃^-1 +1·E₁₃*E₃₁^-1 +1·E₁₃*E₃₂^-1 +1·E₂₁*E₁₂ +1·E₂₁*E₁₃ +1·E₂₁^2 +2·E₂₁*E₂₃ +2·E₂₁*E₃₁ +1·E₂₁*E₃₂ +1·E₂₁*E₁₂^-1 +1·E₂₁*E₁₃^-1 +2·E₂₁*E₂₃^-1 +2·E₂₁*E₃₁^-1 +1·E₂₁*E₃₂^-1 +1·E₂₃*E₁₂ +1·E₂₃^2 +1·E₂₃*E₃₁ +1·E₂₃*E₃₂ +1·E₂₃*E₁₂^-1 +2·E₂₃*E₁₃^-1 +2·E₂₃*E₂₁^-1 +1·E₂₃*E₃₁^-1 +1·E₂₃*E₃₂^-1 +1·E₃₁*E₁₂ +1·E₃₁*E₁₃ +1·E₃₁*E₂₃ +1·E₃₁^2 +2·E₃₁*E₃₂ +1·E₃₁*E₁₂^-1 +1·E₃₁*E₁₃^-1 +2·E₃₁*E₂₁^-1 +1·E₃₁*E₂₃^-1 +2·E₃₁*E₃₂^-1 +1·E₃₂*E₁₃ +1·E₃₂*E₂₁ +1·E₃₂*E₂₃ +1·E₃₂^2 +2·E₃₂*E₁₂^-1 +1·E₃₂*E₁₃^-1 +1·E₃₂*E₂₁^-1 +1·E₃₂*E₂₃^-1 +2·E₃₂*E₃₁^-1 +1·E₁₂^-1*E₂₁ +1·E₁₂^-1*E₂₃ +1·E₁₂^-1*E₃₁ +1·E₁₂^-2 +2·E₁₂^-1*E₁₃^-1 +1·E₁₂^-1*E₂₁^-1 +1·E₁₂^-1*E₂₃^-1 +1·E₁₂^-1*E₃₁^-1 +2·E₁₂^-1*E₃₂^-1 +1·E₁₃^-1*E₂₁ +1·E₁₃^-1*E₃₁ +1·E₁₃^-1*E₃₂ +1·E₁₃^-2 +1·E₁₃^-1*E₂₁^-1 +2·E₁₃^-1*E₂₃^-1 +1·E₁₃^-1*E₃₁^-1 +1·E₁₃^-1*E₃₂^-1 +1·E₂₁^-1*E₁₂ +1·E₂₁^-1*E₁₃ +1·E₂₁^-1*E₃₂ +1·E₂₁^-1*E₁₂^-1 +1·E₂₁^-1*E₁₃^-1 +1·E₂₁^-2 +2·E₂₁^-1*E₂₃^-1 +2·E₂₁^-1*E₃₁^-1 +1·E₂₁^-1*E₃₂^-1 +1·E₂₃^-1*E₁₂ +1·E₂₃^-1*E₃₁ +1·E₂₃^-1*E₃₂ +1·E₂₃^-1*E₁₂^-1 +1·E₂₃^-2 +1·E₂₃^-1*E₃₁^-1 +1·E₂₃^-1*E₃₂^-1 +1·E₃₁^-1*E₁₂ +1·E₃₁^-1*E₁₃ +1·E₃₁^-1*E₂₃ +1·E₃₁^-1*E₁₂^-1 +1·E₃₁^-1*E₁₃^-1 +1·E₃₁^-1*E₂₃^-1 +1·E₃₁^-2 +2·E₃₁^-1*E₃₂^-1 +1·E₃₂^-1*E₁₃ +1·E₃₂^-1*E₂₁ +1·E₃₂^-1*E₂₃ +1·E₃₂^-1*E₁₃^-1 +1·E₃₂^-1*E₂₁^-1 +1·E₃₂^-1*E₂₃^-1 +1·E₃₂^-2

Formulating the optimization problem

As proven by N. Ozawa here (Main Theorem), property (T) for SL is equivalent to a sum of (hermitian) squares decomposition for \Delta^2 - \lambda\Delta for some \lambda > 0. Let's find such decomposition using semi-definite optimization:

julia> opt_problem = PropertyT.sos_problem_primal(Δ², Δ)
A JuMP Model
Maximization problem with:
Variables: 7382
Objective function type: JuMP.VariableRef
`JuMP.AffExpr`-in-`MathOptInterface.EqualTo{Float64}`: 5455 constraints
`Vector{JuMP.VariableRef}`-in-`MathOptInterface.PositiveSemidefiniteConeTriangle`: 1 constraint
Model mode: AUTOMATIC
CachingOptimizer state: NO_OPTIMIZER
Solver name: No optimizer attached.
Names registered in the model: P, psd, λ

This problem tries to find maximal λ as long as an internal matrix P defines a sum of squares decomposition for Δ² - λΔ (you may consult the docstring of sos_problem_primal for more information).

Solving the optimization problem

To solve the problem we need a solver/optimizer - a software to numerically find a solution using e.g. iterative procedures. There are two solvers predefined in test/optimizers.jl:

• scs_optimizer and
• cosmo_optimizer. These are just thin wrappers around JuMP (or actually MathOptInterface) optimizers.

julia> include("test/optimizers.jl");

Now we have everything what we need to solve the problem!

julia> status, warmstart = PropertyT.solve(
    opt_problem,
    scs_optimizer(max_iters=5_000, accel=50, alpha=1.9),
);
------------------------------------------------------------------
               SCS v3.2.1 - Splitting Conic Solver
        (c) Brendan O'Donoghue, Stanford University, 2012
------------------------------------------------------------------
problem:  variables n: 7382, constraints m: 12836
cones:    z: primal zero / dual free vars: 5455
          s: psd vars: 7381, ssize: 1
settings: eps_abs: 1.0e-09, eps_rel: 1.0e-09, eps_infeas: 1.0e-07
          alpha: 1.90, scale: 1.00e-01, adaptive_scale: 1
          max_iters: 5000, normalize: 1, rho_x: 1.00e-06
          acceleration_lookback: 50, acceleration_interval: 10
lin-sys:  sparse-direct-amd-qdldl
          nnz(A): 57566, nnz(P): 0
------------------------------------------------------------------
 iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
------------------------------------------------------------------
     0| 1.57e+02  9.96e-01  2.61e+02 -1.69e+02  1.00e-01  6.54e-02
   250| 1.08e-02  2.03e-04  1.69e-01 -1.02e+00  1.00e-01  8.05e-01
   500| 2.94e-03  2.46e-04  5.78e-02 -5.24e-01  1.00e-01  1.55e+00
   [...]
  4500| 4.62e-06  4.05e-09  1.71e-06 -2.80e-01  6.45e-03  1.40e+01
  4750| 3.74e-06  2.92e-09  1.72e-06 -2.80e-01  6.45e-03  1.48e+01
  5000| 3.06e-06  2.03e-09  1.63e-06 -2.80e-01  6.45e-03  1.56e+01
------------------------------------------------------------------
status:  solved (inaccurate - reached max_iters)
timings: total: 1.56e+01s = setup: 6.09e-02s + solve: 1.55e+01s
         lin-sys: 1.90e+00s, cones: 1.26e+01s, accel: 1.87e-01s
------------------------------------------------------------------
objective = -0.280408 (inaccurate)
------------------------------------------------------------------

julia> status
ALMOST_OPTIMAL::TerminationStatusCode = 7

The solver didn't manage to solve the problem but it got quite close! (duality gap is ~1.63e-6). Let's try once again this time warmstarting the solver:

julia> status, warmstart = PropertyT.solve(
    opt_problem,
    scs_optimizer(max_iters=10_000, accel=50, alpha=1.9),
    warmstart,
);
------------------------------------------------------------------
               SCS v3.2.1 - Splitting Conic Solver
        (c) Brendan O'Donoghue, Stanford University, 2012
------------------------------------------------------------------
problem:  variables n: 7382, constraints m: 12836
cones:    z: primal zero / dual free vars: 5455
          s: psd vars: 7381, ssize: 1
settings: eps_abs: 1.0e-09, eps_rel: 1.0e-09, eps_infeas: 1.0e-07
          alpha: 1.90, scale: 1.00e-01, adaptive_scale: 1
          max_iters: 10000, normalize: 1, rho_x: 1.00e-06
          acceleration_lookback: 50, acceleration_interval: 10
lin-sys:  sparse-direct-amd-qdldl
          nnz(A): 57566, nnz(P): 0
------------------------------------------------------------------
 iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
------------------------------------------------------------------
     0| 3.00e-06  2.96e-08  1.37e-05 -2.80e-01  1.00e-01  4.32e-02
   250| 5.01e-07  6.75e-08  6.62e-06 -2.80e-01  1.00e-01  7.78e-01
   500| 7.79e-07  1.92e-08  1.19e-06 -2.80e-01  3.10e-02  1.58e+00
   750| 7.80e-07  7.20e-09  4.80e-07 -2.80e-01  3.10e-02  2.33e+00
  1000| 7.74e-07  5.22e-10  2.06e-07 -2.80e-01  3.10e-02  3.15e+00
  1250| 7.74e-07  3.84e-10  2.40e-07 -2.80e-01  3.10e-02  4.01e+00
  1500| 7.71e-07  1.38e-10  2.48e-07 -2.80e-01  3.10e-02  4.88e+00
  1750| 7.72e-07  2.63e-11  2.48e-07 -2.80e-01  3.10e-02  5.77e+00
  2000| 7.60e-07  1.31e-11  2.48e-07 -2.80e-01  3.10e-02  6.68e+00
  2200| 2.70e-09  1.46e-10  8.93e-10 -2.80e-01  5.37e-01  7.37e+00
------------------------------------------------------------------
status:  solved
timings: total: 7.37e+00s = setup: 3.89e-02s + solve: 7.33e+00s
         lin-sys: 8.18e-01s, cones: 6.07e+00s, accel: 6.05e-02s
------------------------------------------------------------------
objective = -0.280408
------------------------------------------------------------------

julia> status
OPTIMAL::TerminationStatusCode = 1

This time solver was successful in reaching the desired accuracy (1e-9). Lets query the solution:

julia> λ = JuMP.value(opt_problem[:λ])
0.28040750495076683

julia> P = JuMP.value.(opt_problem[:P]); size(P)
(121, 121)

julia> Q = real.(sqrt(P));

julia> maximum(abs, Q'*Q - P)
7.951418690144152e-11

Certifying the result

Thus we obtained a matrix Q which defines elements ξᵢ ∈ ℝSL (coefficents read by columns of Q) whose sum of squares is close to Δ²-λΔ in ℓ₁-norm. Let's check it out.

julia> sos = PropertyT.compute_sos(RSL, Q, augmented=true);

julia> using LinearAlgebra

julia> norm(Δ²-λ*Δ - sos, 1)
2.948008083982383e-7

We'd like to conclude from this that since the norm of the residual is much larger than λ we obtain (by ℓ₁-convexity of sum of squares cone in ℝSL) a proof of the existence of an exact sum of squares decomposition of Δ² - λ₀Δ for some λ₀ not far from the numerical λ above. To be able to do so we'd need to provide a certified bound on the magnitude of the norm. Here's how to do it in an automated fashion:

julia> _, λ_cert = PropertyT.certify_solution(Δ², Δ, λ, Q, halfradius=2, augmented=true)
  0.070032 seconds (4.11 k allocations: 400.047 KiB)
┌ Info: Checking in Float64 arithmetic with
└   λ = 0.28040750495076683
┌ Info: Numerical metrics of the obtained SOS:
│ ɛ(elt - λu - ∑ξᵢ*ξᵢ) ≈ -3.3119650146808302e-12
│ ‖elt - λu - ∑ξᵢ*ξᵢ‖₁ ≈ 2.948008083982383e-7
└  λ ≈ 0.2804063257475332
  5.393452 seconds (15.30 M allocations: 581.181 MiB, 3.08% gc time, 83.70% compilation time)
┌ Info: Checking in IntervalArithmetic.Interval{Float64} arithmetic with
└   λ = 0.28040750495076683
┌ Info: Numerical metrics of the obtained SOS:
│ ɛ(elt - λu - ∑ξᵢ*ξᵢ) ∈ [-1.92655e-10, 2.00372e-10]
│ ‖elt - λu - ∑ξᵢ*ξᵢ‖₁ ∈ [2.94597e-07, 2.94991e-07]
└  λ ∈ [0.280406, 0.280407]
(true, 0.28040632499038354)

The true returned means that the result is certified to be correct and the value 0.28040632499038354 is the certified lower bound on the spectral gap of Δ.

Lower bound on the Kazhdan constant

Together with estimate for the Kazhdan constant of (G,S):

\sqrt{\frac{2\lambda(G,S)}{\vert S\vert}} \leqslant \kappa(G,S)

we obtain

julia> sqrt(2λ/length(S))
0.21618183124041931

hence 0.2161... \leqslant \kappa(SL_3(\mathbb{Z}), S_3), i.e. SL_3(\mathbb{Z}) has property (T).