GroupsWithPropertyT/AutFN_orbit.jl

include("Orb_AutFN.jl")

###############################################################################
#
#  Action of WreathProductElems on AutGroupElem
#
###############################################################################

function AutFG_emb(A::AutGroup, g::WreathProductElem)
    isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
    parent(g).P.n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(g)) into $A")
    powers = [(elt == parent(elt)() ? 0: 1) for elt in g.n.elts]
    elt = reduce(*, [A(Groups.flip_autsymbol(i))^pow for (i,pow) in enumerate(powers)])
    Groups.r_multiply!(elt, [Groups.perm_autsymbol(g.p)])
    return elt
end

function AutFG_emb(A::AutGroup, p::perm)
    isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
    parent(p).n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(g)) into $A")
    return A(Groups.perm_autsymbol(p))
end

function (g::WreathProductElem)(a::AutGroupElem)
    g = AutFG_emb(parent(a),g)
    return g*a*g^-1
end

function (p::perm)(a::AutGroupElem)
    g = AutFG_emb(parent(a),p)
    return g*a*g^-1
end

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

function SOutFN_generating_set(N::Int)

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

   return SOutFN, unique(S)
end

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

function parse_commandline()
    settings = ArgParseSettings()

    @add_arg_table settings begin
        "--tol"
            help = "set numerical tolerance for the SDP solver (default: 1e-5)"
            arg_type = Float64
            default = 1e-5
        "--iterations"
            help = "set maximal number of iterations for the SDP solver (default: 20000)"
            arg_type = Int
            default = 20000
        "--upper-bound"
            help = "Set an upper bound for the spectral gap (default: Inf)"
            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 (default: N=2)"
            arg_type = Int
            default = 2
        "--radius"
            help = "Find the decomposition over B_r(e,S)"
            arg_type = Int
            default = 2
    end

    return parse_args(settings)
end

###############################################################################
#
#  main
#
###############################################################################

function main()
   parsed_args = parse_commandline()

   N = parsed_args["N"]
   dirname = "oSOutF$(N)_r=$radius"
   radius = parsed_args["radius"]
   tol = parsed_args["tol"]
   iterations = parsed_args["iterations"]
   upper_bound = parsed_args["upper-bound"]

   isdir(dirname) || mkdir(dirname)
   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)
   AutS = WreathProduct(Nemo.FiniteField(2,1, "a")[1], PermutationGroup(N))
   # AutS = PermutationGroup(N)

   solver = SCS.SCSSolver(eps=tol, max_iters=iterations, verbose=true, linearsolver=SCS.Indirect)

   sett = Settings(dirname, N, G, S, AutS, radius, solver, upper_bound, tol)

   orbit_check_propertyT(logger, sett)
end

main()