GroupsWithPropertyT/AutFN.jl

using ArgParse

using Nemo
import SCS.SCSSolver
using PropertyT

using Groups
Nemo.setpermstyle(:cycles)

#=
Note that the element
    α(i,j,k) = ϱ(i,j)*ϱ(i,k)*inv(ϱ(i,j))*inv(ϱ(i,k)),
which surely belongs to ball of radius 4 in Aut(F₄) becomes trivial under the representation
    Aut(F₄) → GL₄(ℤ)⋉ℤ⁴ → GL₅(ℂ).
Moreover, due to work of Potapchik and Rapinchuk [1] every real representation of Aut(Fₙ) into GLₘ(ℂ) (for m ≤ 2n-2) factors through GLₙ(ℤ)⋉ℤⁿ, so will have the same problem.

We need a different approach: Here we actually compute in Aut(𝔽₄)
=#

###############################################################################
#
#  Generating set
#
###############################################################################

function SOutFN_generating_set(N::Int)

   SOutFN = AutGroup(FreeGroup(N), special=true, outer=true)
   S = gens(SOutFN);
   S = [S; [inv(s) for s in S]]

   return SOutFN, unique(S)
end

###############################################################################
#
#  Parsing command line
#
###############################################################################

function parse_commandline()
    s = ArgParseSettings()

    @add_arg_table s begin
        "--tol"
            help = "set numerical tolerance for the SDP solver"
            arg_type = Float64
            default = 1e-6
        "--iterations"
            help = "set maximal number of iterations for the SDP solver"
            arg_type = Int
            default = 50000
        "--upper-bound"
            help = "Set an upper bound for the spectral gap"
            arg_type = Float64
            default = Inf
        "--cpus"
            help = "Set number of cpus used by solver (default: auto)"
            arg_type = Int
            required = false
        "-N"
            help = "Consider automorphisms of free group on N generators"
            arg_type = Int
            default = 2
        "--radius"
            help = "Radius of ball B_r(e,S) to find solution over"
            arg_type = Int
            default = 2
        "--warmstart"
            help = "Use warmstart.jld as the initial guess for SCS"
            action = :store_true
    end

    return parse_args(s)
end

function main()
include("CPUselect.jl")
set_parallel_mthread(PARSEDARGS, workers=true)


    N = parsed_args["N"]
    radius = 2
    tol = parsed_args["tol"]
    iterations = parsed_args["iterations"]
    upper_bound = parsed_args["upper-bound"]

    dirname = "SOutF$(N)_$(upper_bound)_r=$radius"

    logger = PropertyT.setup_logging(dirname)

    info(logger, "Group:       $dirname")
    info(logger, "Iterations:  $iterations")
    info(logger, "Precision:   $tol")
    info(logger, "Upper bound: $upper_bound")

    G, S = SOutFN_generating_set(N)
    info(logger, G)
    info(logger, "Symmetric generating set of size $(length(S))")
    info(logger, S)
    Id = G()

    solver = SCSSolver(eps=tol, max_iters=iterations, linearsolver=SCS.Direct)

    @time PropertyT.check_property_T(dirname, S, Id, solver, upper_bound, tol, 2)
    return 0
end

main()