add deprecation warning

everything is handled by run.jl

.gitignore

@@ -1,4 +1,15 @@
+Articles
+Higman
+MCG*
+notebooks
+Oldies
+oSAutF*
+oSL*
+SAutF*
+SL*_*
 *ipynb*
 *.gws
-*/*.jld
-*/*.log
+.*
+tests*
+*.py
+*.pyc

AutF4.jl

@@ -1,120 +0,0 @@
-using Combinatorics
-
-using JuMP
-import SCS: SCSSolver
-import Mosek: MosekSolver
-
-push!(LOAD_PATH, "./")
-using SemiDirectProduct
-using GroupAlgebras
-include("property(T).jl")
-
-const N = 4
-
-const VERBOSE = true
-
-function permutation_matrix(p::Vector{Int})
-    n = length(p)
-    sort(p) == collect(1:n) || throw(ArgumentError("Input array must be a permutation of 1:n"))
-    A = eye(n)
-    return A[p,:]
-end
-
-SymmetricGroup(n) = [nthperm(collect(1:n), k) for k in 1:factorial(n)]
-
-# const SymmetricGroup = [permutation_matrix(x) for x in SymmetricGroup_perms]
-
-function E(i, j; dim::Int=N)
-    @assert i≠j
-    k = eye(dim)
-    k[i,j] = 1
-    return k
-end
-
-function eltary_basis_vector(i; dim::Int=N)
-    result = zeros(dim)
-    if 0 < i ≤ dim
-        result[i] = 1
-    end
-    return result
-end
-
-v(i; dim=N) = eltary_basis_vector(i,dim=dim)
-
-ϱ(i,j::Int,n=N) = SemiDirectProductElement(E(i,j,dim=n), v(j,dim=n))
-λ(i,j::Int,n=N) = SemiDirectProductElement(E(i,j,dim=n), -v(j,dim=n))
-
-function ɛ(i, n::Int=N)
-    result = eye(n)
-    result[i,i] = -1
-    return SemiDirectProductElement(result)
-end
-
-σ(permutation::Vector{Int}) =
-     SemiDirectProductElement(permutation_matrix(permutation))
-
-# Standard generating set: 103 elements
-
-function generatingset_ofAutF(n::Int=N)
-    indexing = [[i,j] for i in 1:n for j in 1:n if i≠j]
-    ϱs = [ϱ(ij...) for ij in indexing]
-    λs = [λ(ij...) for ij in indexing]
-    ɛs = [ɛ(i) for i in 1:N]
-    σs = [σ(perm) for perm in SymmetricGroup(n)]
-    S = vcat(ϱs, λs, ɛs, σs);
-    S = unique(vcat(S, [inv(x) for x in S]));
-    return S
-end
-
-#=
-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!
-=#
-
-const ID = eye(N+1)
-
-const S₁ = generatingset_ofAutF(N)
-
-matrix_S₁ = [matrix_repr(x) for x in S₁]
-
-const TOL=10.0^-7
-
-matrix_S₁[1:10,:][:,1]
-
-Δ, cm = prepare_Laplacian_and_constraints(matrix_S₁)
-
-#solver = SCSSolver(eps=TOL, max_iters=ITERATIONS, verbose=true);
-solver = MosekSolver(MSK_DPAR_INTPNT_CO_TOL_REL_GAP=TOL,
-#                      MSK_DPAR_INTPNT_CO_TOL_PFEAS=1e-15,
-#                      MSK_DPAR_INTPNT_CO_TOL_DFEAS=1e-15,
-#                      MSK_IPAR_PRESOLVE_USE=0,
-                  QUIET=!VERBOSE)
-
-# κ, A = solve_for_property_T(S₁, solver, verbose=VERBOSE)
-
-product_matrix = readdlm("SL₃Z.product_matrix", Int)
-L = readdlm("SL₃Z.Δ.coefficients")[:, 1]
-Δ = GroupAlgebraElement(L, product_matrix)
-
-A = readdlm("matrix.A.Mosek")
-κ = readdlm("kappa.Mosek")[1]
-
-# @show eigvals(A)
-@assert isapprox(eigvals(A), abs(eigvals(A)), atol=TOL)
-@assert A == Symmetric(A)
-
-
-const A_sqrt = real(sqrtm(A))
-
-SOS_EOI_fp_L₁, Ω_fp_dist = check_solution(κ, A_sqrt, Δ)
-
-κ_rational = rationalize(BigInt, κ;)
-A_sqrt_rational = rationalize(BigInt, A_sqrt)
-Δ_rational = rationalize(BigInt, Δ)
-
-SOS_EOI_rat_L₁, Ω_rat_dist = check_solution(κ_rational, A_sqrt_rational, Δ_rational)

AutFN.jl

@@ -1,232 +0,0 @@
-using ArgParse
-
-using Groups
-using GroupAlgebras
-using PropertyT
-
-import SCS.SCSSolver
-
-#=
-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)
-=#
-
-import Combinatorics.nthperm
-SymmetricGroup(n) = [nthperm(collect(1:n), k) for k in 1:factorial(n)]
-
-function generating_set_of_AutF(N::Int)
-
-    indexing = [[i,j] for i in 1:N for j in 1:N if i≠j]
-    σs = [symmetric_AutSymbol(perm) for perm in SymmetricGroup(N)[2:end]];
-    ϱs = [rmul_AutSymbol(i,j) for (i,j) in indexing]
-    λs = [lmul_AutSymbol(i,j) for (i,j) in indexing]
-    ɛs = [flip_AutSymbol(i) for i in 1:N];
-
-    S = vcat(ϱs,λs)
-    S = vcat(S..., σs..., ɛs)
-    S = vcat(S..., [inv(g) for g in S])
-    return Vector{AutWord}(unique(S)), one(AutWord)
-end
-
-function generating_set_of_OutF(N::Int)
-    indexing = [[i,j] for i in 1:N for j in 1:N if i≠j]
-    ϱs = [rmul_AutSymbol(i,j) for (i,j) in indexing]
-    λs = [lmul_AutSymbol(i,j) for (i,j) in indexing]
-    ɛs = [flip_AutSymbol(i) for i in 1:N];
-
-    S = ϱs
-    push!(S, λs..., ɛs...)
-    push!(S,[inv(g) for g in S]...)
-
-    return Vector{AutWord}(unique(S)), one(AutWord)
-end
-
-function generating_set_of_Sym(N::Int)
-    σs = [symmetric_AutSymbol(perm) for perm in SymmetricGroup(N)[2:end]];
-
-    S = σs
-    push!(S, [inv(s) for s in S]...)
-    return Vector{AutWord}(unique(S)), one(AutWord)
-end
-
-
-function products(S1::Vector{AutWord}, S2::Vector{AutWord})
-    result = Vector{AutWord}()
-    seen = Set{Vector{FGWord}}()
-    n = length(S1)
-    for (i,x) in enumerate(S1)
-        for y in S2
-            z::AutWord = x*y
-            v::Vector{FGWord} = z(domain)
-            if !in(v, seen)
-                push!(seen, v)
-                push!(result, z)
-            end
-        end
-    end
-    return result
-end
-
-function products_images(S1::Vector{AutWord}, S2::Vector{AutWord})
-    result = Vector{Vector{FGWord}}()
-    seen = Set{Vector{FGWord}}()
-    n = length(S1)
-    for (i,x) in enumerate(S1)
-        z = x(domain)
-        for y in S2
-            v = y(z)
-            if !in(v, seen)
-                push!(seen, v)
-                push!(result, v)
-            end
-        end
-    end
-    return result
-end
-
-function hashed_product{T}(image::T, B, images_dict::Dict{T, Int})
-    n = size(B,1)
-    column = zeros(Int,n)
-    Threads.@threads for j in 1:n
-        w = (B[j])(image)
-        k = images_dict[w]
-        k ≠ 0 || throw(ArgumentError(
-            "($i,$j): $(x^-1)*$y don't seem to be supported on basis!"))
-        column[j] = k
-    end
-    return column
-end
-
-function create_product_matrix(basis::Vector{AutWord}, images)
-    n = length(basis)
-    product_matrix = zeros(Int, (n, n));
-    print("Creating hashtable of images...")
-    @time images_dict = Dict{Vector{FGWord}, Int}(x => i
-        for (i,x) in enumerate(images))
-
-    for i in 1:n
-        z = (inv(basis[i]))(domain)
-        product_matrix[i,:] = hashed_product(z, basis, images_dict)
-    end
-    return product_matrix
-end
-
-function ΔandSDPconstraints(identity::AutWord, S::Vector{AutWord})
-
-    println("Generating Balls of increasing radius...")
-    @time B₁ = vcat([identity], S)
-    @time B₂ = products(B₁,B₁);
-    @show length(B₂)
-    if length(B₂) != length(B₁)
-        @time B₃ = products(B₁, B₂)
-        @show length(B₃)
-        if length(B₃) != length(B₂)
-            @time B₄_images = products_images(B₁, B₃)
-        else
-            B₄_images = unique([f(domain) for f in B₃])
-        end
-    else
-        B₃ = B₂
-        B₄ = B₂
-        B₄_images = unique([f(domain) for f in B₃])
-    end
-
-    @show length(B₄_images)
-    # @assert length(B₄_images) == 3425657
-
-    println("Creating product matrix...")
-    @time pm = create_product_matrix(B₂, B₄_images)
-    println("Creating sdp_constratints...")
-    @time sdp_constraints = PropertyT.constraints_from_pm(pm)
-
-    L_coeff = PropertyT.splaplacian_coeff(S, B₂, length(B₄_images))
-    Δ = PropertyT.GroupAlgebraElement(L_coeff, Array{Int,2}(pm))
-
-    return Δ, sdp_constraints
-end
-
-const symbols =  [FGSymbol("x₁",1), FGSymbol("x₂",1), FGSymbol("x₃",1), FGSymbol("x₄",1), FGSymbol("x₅",1), FGSymbol("x₆",1)]
-
-const TOL=1e-8
-const N = 4
-const domain = Vector{FGWord}(symbols[1:N])
-
-function cpuinfo_physicalcores()
-    maxcore = -1
-    for line in eachline("/proc/cpuinfo")
-        if startswith(line, "core id")
-            maxcore = max(maxcore, parse(Int, split(line, ':')[2]))
-        end
-    end
-    maxcore < 0 && error("failure to read core ids from /proc/cpuinfo")
-    return maxcore + 1
-end
-
-function parse_commandline()
-    s = ArgParseSettings()
-
-    @add_arg_table s 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=3)"
-            arg_type = Int
-            default = 3
-    end
-
-    return parse_args(s)
-end
-
-
-# const name = "SYM$N"
-# const upper_bound=factorial(N)-TOL^(1/5)
-# S() = generating_set_of_Sym(N)
-
-# name = "AutF$N"
-# S() = generating_set_of_AutF(N)
-
-function main()
-    parsed_args = parse_commandline()
-
-    tol = parsed_args["tol"]
-    iterations = parsed_args["iterations"]
-
-    solver = SCSSolver(eps=tol, max_iters=iterations, verbose=true, linearsolver=SCS.Indirect)
-
-    N = parsed_args["N"]
-    upper_bound = parsed_args["upper-bound"]
-
-    name = "OutF$N"
-    name = name*"-$(string(upper_bound))"
-    S() = generating_set_of_OutF(N)
-
-    if parsed_args["cpus"] ≠ nothing
-        if parsed_args["cpus"] > cpuinfo_physicalcores()
-            warn("Number of specified cores exceeds the physical core cound. Performance will suffer.")
-        end
-        Blas.set_num_threads(parsed_args["cpus"])
-    end
-    @time PropertyT.check_property_T(name, S, solver, upper_bound, tol)
-    return 0
-end
-
-main()

CPUselect.jl

@@ -0,0 +1,35 @@
+function cpuinfo_physicalcores()
+    maxcore = -1
+    for line in eachline("/proc/cpuinfo")
+        if startswith(line, "core id")
+            maxcore = max(maxcore, parse(Int, split(line, ':')[2]))
+        end
+    end
+    maxcore < 0 && error("failure to read core ids from /proc/cpuinfo")
+    return maxcore + 1
+end
+
+function set_parallel_mthread(N::Int, workers::Bool)
+    if N > cpuinfo_physicalcores()
+        warn("Number of specified cores exceeds the physical core count. Performance may suffer.")
+    end
+
+    if workers
+        addprocs(N)
+        info("Using $N cpus in @parallel code.")
+    end
+    info("Using $(Threads.nthreads()) threads in @threads code.")
+    BLAS.set_num_threads(N)
+    info("Using $N threads in BLAS.")
+
+end
+
+function set_parallel_mthread(parsed_args::Dict; workers=false)
+    if parsed_args["cpus"] == nothing
+        N = cpuinfo_physicalcores()
+    else
+        N = parsed_args["cpus"]
+    end
+
+    set_parallel_mthread(N, workers)
+end

FPGroups_GAP.jl

@@ -0,0 +1,157 @@
+using JLD
+
+function GAP_code(group_code, dir, R; maxeqns=10_000, infolevel=2)
+    code = """
+LogTo("$(dir)/GAP.log");
+RequirePackage("kbmag");
+SetInfoLevel(InfoRWS, $infolevel);
+
+
+MetricBalls := function(rws, R)
+    local l, basis, sizes, i;
+    l := EnumerateReducedWords(rws, 0, R);;
+    SortBy(l, Length);
+    sizes := [1..R];
+    Apply(sizes, i -> Number(l, w -> Length(w) <= i));
+    return [l, sizes];
+end;;
+
+ProductMatrix := function(rws, basis, len)
+    local result, dict, g, tmpList, t;
+    result := [];
+    dict := NewDictionary(basis[1], true);
+    t := Runtime();
+    for g in [1..Length(basis)] do;
+        AddDictionary(dict, basis[g], g);
+    od;
+    Print("Creating dictionary: \t\t", StringTime(Runtime()-t), "\\n");
+    for g in basis{[1..len]} do;
+        tmpList := List(Inverse(g)*basis{[1..len]}, w->ReducedForm(rws, w));
+        #t := Runtime();
+        tmpList := List(tmpList, x -> LookupDictionary(dict, x));
+        #Print(Runtime()-t, "\\n");
+        Assert(1, ForAll(tmpList, x -> x <> fail));
+        Add(result, tmpList);
+    od;
+    return result;
+end;;
+
+SaveCSV := function(fname, pm)
+   local file, i, j, k;
+   file := OutputTextFile(fname, false);;
+   for i in pm do;
+      k := 1;
+      for j in i do;
+         if k < Length(i) then
+            AppendTo(file, j, ", ");
+         else
+            AppendTo(file, j, "\\n");
+         fi;
+         k := k+1;
+      od;
+   od;
+   CloseStream(file);
+end;;
+
+$group_code
+
+# G:= SimplifiedFpGroup(G);
+RWS := KBMAGRewritingSystem(G);
+# ResetRewritingSystem(RWS);
+O:=OptionsRecordOfKBMAGRewritingSystem(RWS);;
+O.maxeqns := $maxeqns;
+O.maxstates := 1000*$maxeqns;
+#O.maxstoredlen := [100,100];
+
+before := Runtimes();;
+KnuthBendix(RWS);
+after := Runtimes();;
+delta := after.user_time_children - before.user_time_children;;
+Print("Knuth-Bendix completion: \t", StringTime(delta), "\\n");
+
+t := Runtime();
+res := MetricBalls(RWS,$(2*R));;
+Print("Metric-Balls generation: \t", StringTime(Runtime()-t), "\\n");
+B := res[1];; sizes := res[2];;
+Print("Sizes of generated Balls: \t", sizes, "\\n");
+
+t := Runtime();
+pm := ProductMatrix(RWS, B, sizes[$R]);;
+Print("Computing ProductMatrix: \t", StringTime(Runtime()-t), "\\n");
+
+S := EnumerateReducedWords(RWS, 1, 1);
+S := List(S, s -> Position(B,s));
+
+SaveCSV("$(dir)/pm.csv", pm);
+SaveCSV("$(dir)/S.csv", [S]);
+SaveCSV("$(dir)/sizes.csv", [sizes]);
+
+Print("DONE!\\n");
+
+quit;""";
+    return code
+end
+
+function GAP_groupcode(S, rels)
+    F = "FreeGroup("*join(["\"$v\""for v in S], ", ") *");"
+    m = match(r".*(\[.*\])$", string(rels))
+    rels = replace(m.captures[1], " ", "\n")
+    code = """
+F := $F;
+AssignGeneratorVariables(F);;
+relations := $rels;;
+G := F/relations;
+    """
+    return code
+end
+
+function GAP_execute(gap_code, dir)
+   isdir(dir) || mkdir(dir)
+   GAP_file = joinpath(dir, "GAP_code.g")
+   @show dir
+   @show GAP_file;
+
+   open(GAP_file, "w") do io
+      write(io, gap_code)
+   end
+   run(pipeline(`cat $(GAP_file)`, `gap -q`))
+end
+
+function prepare_pm_delta_csv(name, group_code, R; maxeqns=10_000, infolevel=2)
+   info("Preparing multiplication table using GAP (via kbmag)")
+   gap_code = GAP_code(group_code, name, R, maxeqns=maxeqns, infolevel=infolevel)
+   GAP_execute(gap_code, name)
+end
+
+function prepare_pm_delta(name, group_code, R; maxeqns=100_000, infolevel=2)
+
+   pm_fname = joinpath(name, "pm.csv")
+   S_fname = joinpath(name, "S.csv")
+   sizes_fname = joinpath(name, "sizes.csv")
+   delta_fname = joinpath(name, "delta.jld")
+
+   if !isfile(pm_fname) || !isfile(S_fname) || !isfile(sizes_fname)
+      prepare_pm_delta_csv(name, group_code, R, maxeqns=maxeqns, infolevel=infolevel)
+   end
+
+   if isfile(sizes_fname)
+      sizes = readcsv(sizes_fname, Int)[1,:]
+      if 2R > length(sizes)
+         prepare_pm_delta_csv(name, group_code, R, maxeqns=maxeqns, infolevel=infolevel)
+      end
+   end
+
+   pm = readcsv(pm_fname, Int)
+   S = readcsv(S_fname, Int)[1,:]
+   sizes = readcsv(sizes_fname, Int)[1,:]
+
+   Δ = spzeros(sizes[2R])
+   Δ[S] .= -1
+   Δ[1] = length(S)
+
+   pm = pm[1:sizes[R], 1:sizes[R]]
+
+   save(joinpath(name, "pm.jld"), "pm", pm)
+   save(joinpath(name, "delta.jld"), "Δ", Δ)
+
+end

LICENSE.md

@@ -0,0 +1,634 @@
+> Copyright (c) 2017: Marek Kaluba.
+> This program is free software: you can redistribute it and/or modify
+> it under the terms of the GNU General Public License as published by
+> the Free Software Foundation, either version 3 of the License, or
+> (at your option) any later version.
+>
+> This program is distributed in the hope that it will be useful,
+> but WITHOUT ANY WARRANTY; without even the implied warranty of
+> MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+> GNU General Public License for more details.
+>
+>
+>                     GNU GENERAL PUBLIC LICENSE
+>                        Version 3, 29 June 2007
+>
+>  Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/>
+>  Everyone is permitted to copy and distribute verbatim copies
+>  of this license document, but changing it is not allowed.
+>
+>                             Preamble
+>
+>   The GNU General Public License is a free, copyleft license for
+> software and other kinds of works.
+>
+>   The licenses for most software and other practical works are designed
+> to take away your freedom to share and change the works.  By contrast,
+> the GNU General Public License is intended to guarantee your freedom to
+> share and change all versions of a program--to make sure it remains free
+> software for all its users.  We, the Free Software Foundation, use the
+> GNU General Public License for most of our software; it applies also to
+> any other work released this way by its authors.  You can apply it to
+> your programs, too.
+>
+>   When we speak of free software, we are referring to freedom, not
+> price.  Our General Public Licenses are designed to make sure that you
+> have the freedom to distribute copies of free software (and charge for
+> them if you wish), that you receive source code or can get it if you
+> want it, that you can change the software or use pieces of it in new
+> free programs, and that you know you can do these things.
+>
+>   To protect your rights, we need to prevent others from denying you
+> these rights or asking you to surrender the rights.  Therefore, you have
+> certain responsibilities if you distribute copies of the software, or if
+> you modify it: responsibilities to respect the freedom of others.
+>
+>   For example, if you distribute copies of such a program, whether
+> gratis or for a fee, you must pass on to the recipients the same
+> freedoms that you received.  You must make sure that they, too, receive
+> or can get the source code.  And you must show them these terms so they
+> know their rights.
+>
+>   Developers that use the GNU GPL protect your rights with two steps:
+> (1) assert copyright on the software, and (2) offer you this License
+> giving you legal permission to copy, distribute and/or modify it.
+>
+>   For the developers' and authors' protection, the GPL clearly explains
+> that there is no warranty for this free software.  For both users' and
+> authors' sake, the GPL requires that modified versions be marked as
+> changed, so that their problems will not be attributed erroneously to
+> authors of previous versions.
+>
+>   Some devices are designed to deny users access to install or run
+> modified versions of the software inside them, although the manufacturer
+> can do so.  This is fundamentally incompatible with the aim of
+> protecting users' freedom to change the software.  The systematic
+> pattern of such abuse occurs in the area of products for individuals to
+> use, which is precisely where it is most unacceptable.  Therefore, we
+> have designed this version of the GPL to prohibit the practice for those
+> products.  If such problems arise substantially in other domains, we
+> stand ready to extend this provision to those domains in future versions
+> of the GPL, as needed to protect the freedom of users.
+>
+>   Finally, every program is threatened constantly by software patents.
+> States should not allow patents to restrict development and use of
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+>                      END OF TERMS AND CONDITIONS
+>

README.md

@@ -0,0 +1,143 @@
+# DEPRECATED!
+
+This repository has not been updated for a while!
+If You are interested in replicating results for [1712.07167](https://arxiv.org/abs/1712.07167) please check [these instruction](https://kalmar.faculty.wmi.amu.edu.pl/post/1712.07176/)
+Also [this notebook](https://nbviewer.jupyter.org/gist/kalmarek/03510181bc1e7c98615e86e1ec580b2a) could be of some help. If everything else fails the [zenodo dataset](https://zenodo.org/record/1133440) should contain the last-resort instructions.
+
+This repository contains some legacy code for computations in [Certifying Numerical Estimates of Spectral Gaps](https://arxiv.org/abs/1703.09680).
+
+# Installing
+To run the code You need `julia-v0.5` (should work on `v0.6`, but with warnings).
+You also need to install julia packages: `Nemo-v0.6.3`, `ArgParse`. To do so in `julia`'s REPL run:
+```julia
+Pkg.update()
+Pkg.add("Nemo")
+Pkg.add("ArgParse")
+```
+Then clone the main repository of `Groups.jl`, `GroupRings.jl` and `PropertyT.jl`:
+```julia
+Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/Groups.jl.git")
+Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/GroupRings.jl.git")
+Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/PropertyT.jl.git")
+Pkg.resolve()
+```
+This should resolve all dependencies (e.g. install `JuMP`, `SCS`, `IntervalArithmetic`, `JLD`, `Memento`). Exit julia and finally clone this repository:
+```shell
+git clone https://git.wmi.amu.edu.pl/kalmar/GroupsWithPropertyT.git
+cd GroupswithPropertyT
+```
+
+# Running
+
+## Naive implementation
+
+To check that $\Delta^2-\lambda\Delta$ is not decomposable to a sum of hermitian squares of elements in the ball of radius $2$ in $SL(2,7)$ run
+```shell
+julia SL.jl -N 2 -p 7 --radius 2 --iterations 100000
+```
+(~30 seconds, depending on hardware). The monotonous decreasing $\lambda$ during the optimisation is in column `pri obj` (or `dua obj`) of `solver.log`.
+
+Compare this to
+```shell
+julia SL.jl -N 2 -p 7 --radius 3 --iterations 100000
+```
+which finds $\lambda \geq 0.5857$ and decomposes $\Delta^2-\lambda\Delta$ into sum of $47$ hermitian squares in less than 20 seconds (including certification).
+
+If You see in the output (or in `full.log`) that the upper end of the interval where $\lVert\Delta^2 - \lambda\Delta - \sum{\xi_i}^*\xi_i\rVert_1$ belongs to is too large (resulting in positive `Floating point distance`, but negative `The Augmentation-projected actual distance`), decrease the `--tol` parameter, e.g.
+```
+julia SL.jl -N 2 -p 7 --radius 3 --iterations 100000 --tol 1e-9
+```
+to achieve a better estimate (the residuals $\ell_1$-norm should be around $\|B_d(e))\|\cdot tol$)
+
+## Symmetrization enhanced implementation
+
+A newer version of the software uses orbit and Wedderburn decomposition to effecitively find a (much) smaller optimisation problem to compute the spectral gap $\lambda$. In particular the solution to the original (naive) optimisation problem can be reconstructed from the solution of the symmetrised one.
+
+E.g. Run
+```shell
+julia SL_orbit.jl -N 4 --radius 2 --upper-bound 1.3
+```
+to find (and certify) the spectral gap for $SL(4, \mathbb{Z})$ is at least `1.2999...` in just under $2$ minutes time (for comparison this result requires over `5` hours in the old implementation on the same hardware).
+
+To replicate the results of _$\operatorname{Aut}(\textbf{F}_5)$ has property (T)_ You neet to run (on a `4`-core CPU)
+```shell
+julia ../AutFN_orbit.jl -N 5 --upper-bound 1.2 --iterations 24000000 --cpus 4
+```
+
+Note that this computation took more than `12` days and required at least `32`GB of ram (and possible more).
+
+# Help
+
+```shell
+julia SL.jl --help
+usage: SL.jl [--tol TOL] [--iterations ITERATIONS]
+             [--upper-bound UPPER-BOUND] [--cpus CPUS] [-N N] [-p P]
+             [--radius RADIUS] [-h]
+
+optional arguments:
+  --tol TOL             set numerical tolerance for the SDP solver
+                        (type: Float64, default: 1.0e-6)
+  --iterations ITERATIONS
+                        set maximal number of iterations for the SDP
+                        solver (default: 20000) (type: Int64, default:
+                        50000)
+  --upper-bound UPPER-BOUND
+                        Set an upper bound for the spectral gap (type:
+                        Float64, default: Inf)
+  --cpus CPUS           Set number of cpus used by solver (type:
+                        Int64)
+  -N N                  Consider elementary matrices EL(N) (type:
+                        Int64, default: 2)
+  -p P                  Matrices over field of p-elements (p=0 => over
+                        ZZ) (type: Int64, default: 0)
+  --radius RADIUS       Radius of ball B_r(e,S) to find solution over
+                        (type: Int64, default: 2)
+  -h, --help            show this help message and exit
+```
+
+# Specific version of [1703.09680](https://arxiv.org/abs/1703.09680)
+
+To checkout the specific versions of packages used for [Certifying Numerical Estimates of Spectral Gaps](https://arxiv.org/abs/1703.09680) run (inside the cloned `GroupswithPropertyT`)
+```shell
+git checkout 1703.09680v1
+```
+
+Unfortunately: You need to link `~/.julia/v0.5/GroupRings` to `~/.julia/v0.5/GroupAlgebras` due to change in the name of the package. Then run in `julia`
+```julia
+Pkg.checkout("GroupRings", "1703.09680v1")
+Pkg.checkout("PropertyT", "1703.09680v1")
+Pkg.resolve()
+```
+
+# Specific version of [1712.07167](https://arxiv.org/abs/1712.07167)
+
+You need to run `julia-0.6`.
+
+Clone `https://git.wmi.amu.edu.pl/kalmar/GroupsWithPropertyT` and checkout the `1712.07167` branch:
+```
+git clone https://git.wmi.amu.edu.pl/kalmar/GroupsWithPropertyT.git
+cd ./GroupsWithPropertyT
+git checkout 1712.07167
+```
+
+In `julia`s REPL execute
+```julia
+Pkg.add("ArgParse")
+Pkg.add("Nemo")
+Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/Groups.jl.git")
+Pkg.checkout("Groups", "1712.07167")
+Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/GroupRings.jl.git")
+Pkg.checkout("GroupRings", "1712.07167")
+Pkg.clone("https://git.wmi.amu.edu.pl/kalmar/PropertyT.jl.git")
+Pkg.checkout("PropertyT", "1712.07167")
+Pkg.checkout("SCS")
+Pkg.build("SCS")
+```
+
+This should resolve all the dependencies. Quit `julia` and place the `oSAutF5_r2` folder downloaded from [here](https://cloud.impan.pl/s/fGIpxvxdTYYkUxK) inside `GroupsWithPropertyT` folder. To verify the decomposition of $\Delta^2 - \lambda \Delta$ for the group run (if You have a `4`-core CPU at Your disposal)
+```julia
+julia AutFN_orbit.jl -N 5 --upper-bound=1.2 --cpus 4
+```
+If You want to generate `pm` and other files on Your own delete everything from the `oSAutF5_r2` folder but `1.2` folder and its contents and run the same command again.
+
+Note: You need at least `32`GB of RAM and spare `24`h of Your CPU.

SL.jl

@@ -1,173 +0,0 @@
-using ArgParse
-using GroupAlgebras
-using PropertyT
-
-using Nemo
-
-import SCS.SCSSolver
-
-
-function E(i::Int, j::Int, M::Nemo.MatSpace)
-    @assert i≠j
-    m = one(M)
-    m[i,j] = m[1,1]
-    return m
-end
-
-function SL_generatingset(n::Int)
-    indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
-    G = Nemo.MatrixSpace(Nemo.ZZ, n,n)
-    S = [E(i,j,G) for (i,j) in indexing];
-    S = vcat(S, [transpose(x) for x in S]);
-    S = vcat(S, [inv(x) for x in S]);
-    return unique(S), one(G)
-end
-
-function SLsize(n,p)
-    result = 1
-    for k in 0:n-1
-        result *= p^n - p^k
-    end
-    return div(result, p-1)
-end
-
-function SL_generatingset(n::Int, p::Int)
-    p == 0 && return SL_generatingset(n)
-    (p > 1 && n > 0) || throw(ArgumentError("Both n and p should be positive integers!"))
-    println("Size(SL(n,p)) = $(SLsize(n,p))")
-    F = Nemo.ResidueRing(Nemo.ZZ, p)
-    G = Nemo.MatrixSpace(F, n,n)
-    indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
-    S = [E(i, j, G) for (i,j) in indexing]
-    S = vcat(S, [transpose(x) for x in S])
-    S = vcat(S, [inv(s) for s in S])
-    return unique(S), one(G)
-end
-
-function products{T}(U::AbstractVector{T}, V::AbstractVector{T})
-    result = Vector{T}()
-    for u in U
-        for v in V
-            push!(result, u*v)
-        end
-    end
-    return unique(result)
-end
-
-function ΔandSDPconstraints(Id, S, radius)
-    radius *=2
-
-    sizes = Vector{Int}()
-    S = vcat([Id], S)
-    B = S
-    push!(sizes,length(B))
-    for i in 2:radius
-        B = products(S, B);
-        push!(sizes, length(B))
-    end
-    println("Generated balls of sizes $sizes")
-    k = div(radius,2)
-    basis = B[1:sizes[k]]
-
-    product_matrix = PropertyT.create_product_matrix(B, sizes[k]);
-    sdp_constraints = PropertyT.constraints_from_pm(product_matrix, length(B))
-    L_coeff = PropertyT.splaplacian_coeff(S, basis, length(B));
-    Δ = GroupAlgebraElement(L_coeff, product_matrix)
-
-    return Δ, sdp_constraints
-end
-
-#=
-To use file property(T).jl (specifically: check_property_T function)
-You need to define:
-
-function ΔandSDPconstraints(identity, S):: (Δ, sdp_constraints)
-
-=#
-
-function cpuinfo_physicalcores()
-    maxcore = -1
-    for line in eachline("/proc/cpuinfo")
-        if startswith(line, "core id")
-            maxcore = max(maxcore, parse(Int, split(line, ':')[2]))
-        end
-    end
-    maxcore < 0 && error("failure to read core ids from /proc/cpuinfo")
-    return maxcore + 1
-end
-
-function parse_commandline()
-    s = ArgParseSettings()
-
-    @add_arg_table s 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 matrices of size N (default: N=3)"
-            arg_type = Int
-            default = 3
-        "-p"
-            help = "Matrices over filed of p-elements (default: p=0 => over ZZ)"
-            arg_type = Int
-            default = 0
-        "--radius"
-            help = "Find the decomposition over B_r(e,S)"
-            arg_type = Int
-            default = 0
-    end
-
-    return parse_args(s)
-end
-
-function main()
-
-
-    parsed_args = parse_commandline()
-
-    tol = parsed_args["tol"]
-    iterations = parsed_args["iterations"]
-
-    solver = SCSSolver(eps=tol, max_iters=iterations, verbose=true)
-
-    N = parsed_args["N"]
-    upper_bound = parsed_args["upper-bound"]
-    p = parsed_args["p"]
-    if p == 0
-        name = "SL$(N)Z"
-    else
-        name = "SL$(N)_$p"
-    end
-    radius = parsed_args["radius"]
-    if radius == 0
-        name*"-$(string(upper_bound))"
-        radius = 2
-    else
-        name = name*"-$(string(upper_bound))-r=$radius"
-    end
-    S() = SL_generatingset(N, p)
-
-    if parsed_args["cpus"] ≠ nothing
-        if parsed_args["cpus"] > cpuinfo_physicalcores()
-            warn("Number of specified cores exceeds the physical core cound. Performance will suffer.")
-        end
-        Blas.set_num_threads(parsed_args["cpus"])
-    end
-    @time PropertyT.check_property_T(name, S, solver, upper_bound, tol, radius)
-    return 0
-end
-
-main()

SemiDirectProduct.jl

@@ -1,88 +0,0 @@
-module SemiDirectProduct
-
-import Base: convert, show, isequal, ==, size, inv
-import Base: +, -, *, //
-
-export SemiDirectProductElement, matrix_repr
-
-"""
-Implements elements of a semidirect product of groups H and N, where N is normal in the product. Usually written as H ⋉ N.
-The multiplication inside semidirect product is defined as
-    (h₁, n₁) ⋅ (h₂, n₂) = (h₁h₂, n₁φ(h₁)(n₂)),
-where φ:H → Aut(N) is a homomorphism.
-
-In the case below we implement H = GL(n,K) and N = Kⁿ, the Affine Group (i.e. GL(n,K) ⋉ Kⁿ) where elements of GL(n,K) act on vectors in Kⁿ via matrix multiplication.
-# Arguments:
-* `h::Array{T,2}` : square invertible matrix (element of GL(n,K))
-* `n::Vector{T,1}` : vector in Kⁿ
-* `φ = φ(h,n) = φ(h)(n)` :2-argument function which defines the action of GL(n,K) on Kⁿ; matrix-vector multiplication by default.
-"""
-immutable SemiDirectProductElement{T<:Number}
-    h::Array{T,2}
-    n::Vector{T}
-    φ::Function
-
-    function SemiDirectProductElement(h::Array{T,2},n::Vector{T},φ::Function)
-        # size(h,1) == size(h,2)|| throw(ArgumentError("h has to be square matrix"))
-        det(h) ≠ 0 || throw(ArgumentError("h has to be invertible!"))
-        new(h,n,φ)
-    end
-end
-
-SemiDirectProductElement{T}(h::Array{T,2}, n::Vector{T}, φ) =
-    SemiDirectProductElement{T}(h,n,φ)
-
-SemiDirectProductElement{T}(h::Array{T,2}, n::Vector{T}) =
-    SemiDirectProductElement(h,n,*)
-
-SemiDirectProductElement{T}(h::Array{T,2}) =
-    SemiDirectProductElement(h,zeros(h[:,1]))
-
-SemiDirectProductElement{T}(n::Vector{T}) =
-        SemiDirectProductElement(eye(eltype(n), n))
-
-convert{T<:Number}(::Type{T}, X::SemiDirectProductElement) =
-    SemiDirectProductElement(convert(Array{T,2},X.h),
-                             convert(Vector{T},X.n),
-                             X.φ)
-
-size(X::SemiDirectProductElement) = (size(X.h), size(X.n))
-
-matrix_repr{T}(X::SemiDirectProductElement{T}) =
-    [X.h X.n; zeros(T, 1, size(X.h,2)) [1]]
-
-show{T}(io::IO, X::SemiDirectProductElement{T}) = print(io,
-    "Element of SemiDirectProduct over $T of size $(size(X)):\n",
-    matrix_repr(X))
-
-function isequal{T}(X::SemiDirectProductElement{T}, Y::SemiDirectProductElement{T})
-    X.h == Y.h || return false
-    X.n == Y.n || return false
-    X.φ == Y.φ || return false
-    return true
-end
-
-function isequal{T,S}(X::SemiDirectProductElement{T}, Y::SemiDirectProductElement{S})
-    W = promote_type(T,S)
-    warn("Comparing elements with different coefficients! trying to promoting to $W")
-    X = convert(W, X)
-    Y = convert(W, Y)
-    return isequal(X,Y)
-end
-
-(==)(X::SemiDirectProductElement, Y::SemiDirectProductElement) = isequal(X, Y)
-
-function semidirect_multiplication{T}(X::SemiDirectProductElement{T},
-                                      Y::SemiDirectProductElement{T})
-    size(X) == size(Y) || throw(ArgumentError("trying to multiply elements from different groups!"))
-    return SemiDirectProductElement(X.h*Y.h, X.n + X.φ(X.h, Y.n))
-end
-
-(*){T}(X::SemiDirectProductElement{T}, Y::SemiDirectProductElement{T}) =
-    semidirect_multiplication(X,Y)
-
-inv{T}(X::SemiDirectProductElement{T}) =
-    SemiDirectProductElement(inv(X.h), X.φ(inv(X.h), -X.n))
-
-
-end

groups/Allgroups.jl

@@ -0,0 +1,56 @@
+module PropertyTGroups
+
+using PropertyT
+using AbstractAlgebra
+using Nemo
+using Groups
+using GroupRings
+
+export PropertyTGroup, SymmetrizedGroup, GAPGroup,
+    SpecialLinearGroup,
+    SpecialAutomorphismGroup,
+    HigmanGroup,
+    CapraceGroup,
+    MappingClassGroup
+
+export PropertyTGroup
+
+abstract type PropertyTGroup end
+
+abstract type SymmetrizedGroup <: PropertyTGroup end
+
+abstract type GAPGroup <: PropertyTGroup end
+
+function PropertyTGroup(args)
+    if haskey(args, "SL")
+        G = PropertyTGroups.SpecialLinearGroup(args)
+    elseif haskey(args, "SAut")
+        G = PropertyTGroups.SpecialAutomorphismGroup(args)
+    elseif haskey(args, "MCG")
+        G = PropertyTGroups.MappingClassGroup(args)
+    elseif haskey(args, "Higman")
+        G = PropertyTGroups.HigmanGroup(args)
+    elseif haskey(args, "Caprace")
+        G = PropertyTGroups.CapraceGroup(args)
+    else
+        throw("You must provide one of --SL, --SAut, --MCG, --Higman, --Caprace")
+    end
+    return G
+end
+
+include("autfreegroup.jl")
+include("speciallinear.jl")
+
+Comm(x,y) = x*y*x^-1*y^-1
+
+function generatingset(G::GAPGroup)
+    S = gens(group(G))
+    return unique([S; inv.(S)])
+end
+
+include("mappingclassgroup.jl")
+include("higman.jl")
+include("caprace.jl")
+include("actions.jl")
+
+end # of module PropertyTGroups

groups/actions.jl

@@ -0,0 +1,92 @@
+function (p::perm)(A::GroupRingElem)
+    RG = parent(A)
+    result = zero(RG, eltype(A.coeffs))
+    
+    for (idx, c) in enumerate(A.coeffs)
+        if c!= zero(eltype(A.coeffs))
+            result[p(RG.basis[idx])] = c
+        end
+    end
+    return result
+end
+
+###############################################################################
+#
+#  Action of WreathProductElems on Nemo.MatElem
+#
+###############################################################################
+
+function matrix_emb(n::DirectProductGroupElem, p::perm)
+    Id = parent(n.elts[1])()
+    elt = diagm([(-1)^(el == Id ? 0 : 1) for el in n.elts])
+    return elt[:, p.d]
+end
+
+function (g::WreathProductElem)(A::MatElem)
+    g_inv = inv(g)
+    G = matrix_emb(g.n, g_inv.p)
+    G_inv = matrix_emb(g_inv.n, g.p)
+    M = parent(A)
+    return M(G)*A*M(G_inv)
+end
+
+import Base.*
+
+doc"""
+    *(x::AbstractAlgebra.MatElem, P::Generic.perm)
+> Apply the pemutation $P$ to the rows of the matrix $x$ and return the result.
+"""
+function *(x::AbstractAlgebra.MatElem, P::Generic.perm)
+   z = similar(x)
+   m = rows(x)
+   n = cols(x)
+   for i = 1:m
+      for j = 1:n
+         z[i, j] = x[i,P[j]]
+      end
+   end
+   return z
+end
+
+function (p::perm)(A::MatElem)
+    length(p.d) == A.r == A.c || throw("Can't act via $p on matrix of size ($(A.r), $(A.c))")
+    return p*A*inv(p)
+end
+
+###############################################################################
+#
+#  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")
+    elt = A()
+    Id = parent(g.n.elts[1])()
+    flips = Groups.AutSymbol[Groups.flip_autsymbol(i) for i in 1:length(g.p.d) if g.n.elts[i] != Id]
+    Groups.r_multiply!(elt, flips, reduced=false)
+    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(p)) into $A")
+    return A(Groups.perm_autsymbol(p))
+end
+
+function (g::WreathProductElem)(a::Groups.Automorphism)
+    A = parent(a)
+    g = AutFG_emb(A,g)
+    res = A()
+    Groups.r_multiply!(res, g.symbols, reduced=false)
+    Groups.r_multiply!(res, a.symbols, reduced=false)
+    Groups.r_multiply!(res, [inv(s) for s in reverse!(g.symbols)])
+    return res
+end
+
+function (p::perm)(a::Groups.Automorphism)
+    g = AutFG_emb(parent(a),p)
+    return g*a*inv(g)
+end

groups/autfreegroup.jl

@@ -0,0 +1,21 @@
+struct SpecialAutomorphismGroup{N} <: SymmetrizedGroup
+    group::AutGroup
+end
+
+function SpecialAutomorphismGroup(args::Dict)
+    N = args["SAut"]
+    return SpecialAutomorphismGroup{N}(AutGroup(FreeGroup(N), special=true))
+end
+
+name(G::SpecialAutomorphismGroup{N}) where N = "SAutF$(N)"
+
+group(G::SpecialAutomorphismGroup) = G.group
+
+function generatingset(G::SpecialAutomorphismGroup)
+    S = gens(group(G));
+    return unique([S; inv.(S)])
+end
+
+function autS(G::SpecialAutomorphismGroup{N}) where N
+    return WreathProduct(PermutationGroup(2), PermutationGroup(N))
+end

groups/caprace.jl

@@ -0,0 +1,35 @@
+struct CapraceGroup <: GAPGroup end
+
+name(G::CapraceGroup) = "CapraceGroup"
+
+function group(G::CapraceGroup)
+
+   caprace_group = Groups.FPGroup(["x","y","z","t","r"])
+
+   x,y,z,t,r = gens(caprace_group)
+
+   relations = [
+   x^7,
+   y^7,
+   t^2,
+   r^73,
+   t*r*t*r,
+   Comm(x,y)*z^-1,
+   Comm(x,z),
+   Comm(y,z),
+   Comm(x^2*y*z^-1, t),
+   Comm(x*y*z^3, t*r),
+   Comm(x^3*y*z^2, t*r^17),
+   Comm(x, t*r^-34),
+   Comm(y, t*r^-32),
+   Comm(z, t*r^-29),
+   Comm(x^-2*y*z, t*r^-25),
+   Comm(x^-1*y*z^-3, t*r^-19),
+   Comm(x^-3*y*z^-2, t*r^-11)
+   ];
+
+   relations = [relations; [inv(rel) for rel in relations]]
+
+   Groups.add_rels!(caprace_group, Dict(rel => caprace_group() for rel in relations))
+   return caprace_group
+end

groups/higman.jl

@@ -0,0 +1,22 @@
+struct HigmanGroup <: GAPGroup end
+
+name(G::HigmanGroup) = "HigmanGroup"
+
+function group(G::HigmanGroup)
+
+   higman_group = Groups.FPGroup(["a","b","c","d"]);
+
+   a,b,c,d = gens(higman_group)
+
+   relations = [
+   b*Comm(b,a),
+   c*Comm(c,b),
+   d*Comm(d,c),
+   a*Comm(a,d)
+   ];
+
+   relations = [relations; [inv(rel) for rel in relations]]
+
+   Groups.add_rels!(higman_group, Dict(rel => higman_group() for rel in relations))
+   return higman_group
+end

groups/mappingclassgroup.jl

@@ -0,0 +1,83 @@
+struct MappingClassGroup{N} <: GAPGroup end
+
+MappingClassGroup(args::Dict) = MappingClassGroup{args["MCG"]}()
+
+name(G::MappingClassGroup{N}) where N = "MCG(N)"
+
+function group(G::MappingClassGroup{N}) where N
+
+    if N < 2
+        throw("Genus must be at least 2!")
+    elseif N == 2
+        MCGroup = Groups.FPGroup(["a1","a2","a3","a4","a5"]);
+        S = gens(MCGroup)
+
+        n = length(S)
+        A = prod(reverse(S))*prod(S)
+
+        relations = [
+           [Comm(S[i], S[j]) for i in 1:n for j in 1:n if abs(i-j) > 1]...,
+           [S[i]*S[i+1]*S[i]*inv(S[i+1]*S[i]*S[i+1]) for i in 1:G.n-1]...,
+           (S[1]*S[2]*S[3])^4*inv(S[5])^2,
+           Comm(A, S[1]),
+           A^2
+           ]
+
+       relations = [relations; [inv(rel) for rel in relations]]
+       Groups.add_rels!(MCGroup, Dict(rel => MCGroup() for rel in relations))
+       return MCGroup
+
+    else
+        MCGroup = Groups.FPGroup(["a$i" for i in 0:2N])
+        S = gens(MCGroup)
+
+        a0 = S[1]
+        A = S[2:end]
+        k = length(A)
+
+        relations = [
+            [Comm(A[i], A[j]) for i in 1:k for j in 1:k if abs(i-j) > 1]...,
+            [Comm(a0, A[i]) for i in 1:k if i != 4]...,
+            [A[i]*A[(i+1)]*A[i]*inv(A[i+1]*A[i]*A[i+1]) for i in 1:k-1]...,
+            A[4]*a0*A[4]*inv(a0*A[4]*a0)
+            ]
+
+        # 3-chain relation
+        c = prod(reverse(A[1:4]))*prod(A[1:4])
+        b0 = c*a0*inv(c)
+        push!(relations, (A[1]*A[2]*A[3])^4*inv(a0*b0))
+
+        # Lantern relation
+        b1 = inv(A[4]*A[5]*A[3]*A[4])*a0*(A[4]*A[5]*A[3]*A[4])
+        b2 = inv(A[2]*A[3]*A[1]*A[2])*b1*(A[2]*A[3]*A[1]*A[2])
+        u  = inv(A[6]*A[5])*b1*(A[6]*A[5])
+        x  = prod(reverse(A[2:6]))*u*prod(inv.(A[1:4]))
+        b3 = x*a0*inv(x)
+        push!(relations, a0*b2*b1*inv(A[1]*A[3]*A[5]*b3))
+
+        # Hyperelliptic relation
+        X = prod(reverse(A))*prod(A)
+
+        function n(i::Int, b=b0)
+            if i == 1
+                return A[1]
+            elseif i == 2
+                return b
+            else
+                return w(i-2)*n(i-2)*w(i-2)
+            end
+        end
+
+        function w(i::Int)
+           (A[2i+4]*A[2i+3]*A[2i+2]* n(i+1))*(A[2i+1]*A[2i]  *A[2i+2]*A[2i+1])*
+           (A[2i+3]*A[2i+2]*A[2i+4]*A[2i+3])*( n(i+1)*A[2i+2]*A[2i+1]*A[2i]  )
+        end
+
+        # push!(relations, X*n(N)*inv(n(N)*X))
+
+        relations = [relations; [inv(rel) for rel in relations]]
+        Groups.add_rels!(MCGroup, Dict(rel => MCGroup() for rel in relations))
+
+        return MCGroup
+    end
+end

groups/speciallinear.jl

@@ -0,0 +1,62 @@
+struct SpecialLinearGroup{N} <: SymmetrizedGroup
+    group::AbstractAlgebra.Group
+    p::Int
+    X::Bool
+end
+
+function SpecialLinearGroup(args::Dict)
+    N = args["SL"]
+    p = args["p"]
+    X = args["X"]
+
+    if p == 0
+        G = MatrixSpace(Nemo.ZZ, N, N)
+    else
+        R = Nemo.NmodRing(UInt(p))
+        G = MatrixSpace(R, N, N)
+    end
+    return SpecialLinearGroup{N}(G, p, X)
+end
+
+function name(G::SpecialLinearGroup{N}) where N
+    if G.p == 0
+       R = (G.X ? "Z[x]" : "Z")
+    else
+       R = "F$(G.p)"
+    end
+    return SL($(G.N),$R)
+end
+
+group(G::SpecialLinearGroup) = G.group
+
+function generatingset(G::SpecialLinearGroup{N}) where N
+    G.p > 0 && G.X && throw("SL(n, F_p[x]) not implemented")
+    SL = group(G)
+    return generatingset(SL, G.X)
+end
+
+# r is the injectivity radius of
+# SL(n, Z[X]) -> SL(n, Z) induced by X -> 100
+
+function generatingset(SL::MatSpace, X::Bool=false, r=5)
+    n = SL.cols
+    indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
+
+    if !X
+        S = [E(idx[1],idx[2],SL) for idx in indexing]
+    else
+        S = [E(i,j,SL,v) for (i,j) in indexing for v in [1, 100*r]]
+    end
+    return unique([S; inv.(S)])
+end
+
+function E(i::Int, j::Int, M::MatSpace, val=one(M.base_ring))
+    @assert i≠j
+    m = one(M)
+    m[i,j] = val
+    return m
+end
+
+function autS(G::SpecialLinearGroup{N}) where N
+    return WreathProduct(PermutationGroup(2), PermutationGroup(N))
+end

logging.jl

@@ -0,0 +1,58 @@
+using Memento
+
+function setup_logging(filename::String, handlername::Symbol=:log)
+    isdir(dirname(filename)) || mkdir(dirname(filename))
+    logger = Memento.config!("info", fmt="{date}| {msg}")
+    handler = DefaultHandler(filename, DefaultFormatter("{date}| {msg}"))
+    logger.handlers[String(handlername)] = handler
+    return logger
+end
+
+macro logtime(logger, ex)
+    quote
+        local stats = Base.gc_num()
+        local elapsedtime = Base.time_ns()
+        local val = $(esc(ex))
+        elapsedtime = Base.time_ns() - elapsedtime
+        local diff = Base.GC_Diff(Base.gc_num(), stats)
+        local ts = time_string(elapsedtime,
+            diff.allocd,
+            diff.total_time,
+            Base.gc_alloc_count(diff)
+            )
+        $(esc(info))($(esc(logger)), ts)
+        val
+    end
+end
+
+function time_string(elapsedtime, bytes, gctime, allocs)
+    str = @sprintf("%10.6f seconds", elapsedtime/1e9)
+    if bytes != 0 || allocs != 0
+        bytes, mb = Base.prettyprint_getunits(bytes, length(Base._mem_units), Int64(1024))
+        allocs, ma = Base.prettyprint_getunits(allocs, length(Base._cnt_units), Int64(1000))
+        if ma == 1
+            str*= @sprintf(" (%d%s allocation%s: ", allocs, Base._cnt_units[ma], allocs==1 ? "" : "s")
+        else
+            str*= @sprintf(" (%.2f%s allocations: ", allocs, Base._cnt_units[ma])
+        end
+        if mb == 1
+            str*= @sprintf("%d %s%s", bytes, Base._mem_units[mb], bytes==1 ? "" : "s")
+        else
+            str*= @sprintf("%.3f %s", bytes, Base._mem_units[mb])
+        end
+        if gctime > 0
+            str*= @sprintf(", %.2f%% gc time", 100*gctime/elapsedtime)
+        end
+        str*=")"
+    elseif gctime > 0
+        str*= @sprintf(", %.2f%% gc time", 100*gctime/elapsedtime)
+    end
+    return str
+end
+
+import Base: info, @time
+
+Base.info(x) = info(getlogger(), x)
+macro time(x)
+    return :(@logtime(getlogger(Main), $(esc(x))))
+end

main.jl

@@ -0,0 +1,61 @@
+using PropertyT
+
+include("FPGroups_GAP.jl")
+
+include("groups/Allgroups.jl")
+using PropertyTGroups
+
+import PropertyT.Settings
+
+function summarize(sett::PropertyT.Settings)
+    info("Threads:     $(Threads.nthreads())")
+    info("Workers:     $(workers())")
+    info("GroupDir:    $(PropertyT.prepath(sett))")
+    info(string(sett.G))
+    info("with generating set of size $(length(sett.S))")
+
+    info("Radius:      $(sett.radius)")
+    info("Precision:   $(sett.tol)")
+    info("Upper bound: $(sett.upper_bound)")
+    info("Solver:      $(sett.solver)")
+end
+
+function Settings(Gr::PropertyTGroup, args, solver)
+    r = get(args, "radius", 2)
+    gr_name = PropertyTGroups.name(Gr)*"_r$r"
+    G = PropertyTGroups.group(Gr)
+    S = PropertyTGroups.generatingset(Gr)
+
+    sol = solver
+    ub = get(args,"upper-bound", Inf)
+    tol = get(args,"tol", 1e-10)
+    ws = get(args, "warmstart", false)
+
+    if get(args, "nosymmetry", false)
+        return PropertyT.Settings(gr_name, G, S, r, sol, ub, tol, ws)
+    else
+        autS = PropertyTGroups.autS(Gr)
+        return PropertyT.Settings(gr_name, G, S, r, sol, ub, tol, ws, autS)
+    end
+end
+
+function main(::PropertyTGroup, sett::PropertyT.Settings)
+    isdir(PropertyT.fullpath(sett)) || mkpath(PropertyT.fullpath(sett))
+
+    summarize(sett)
+
+    return PropertyT.check_property_T(sett)
+end
+
+function main(::GAPGroup, sett::PropertyT.Settings)
+    isdir(PropertyT.fullpath(sett)) || mkpath(PropertyT.fullpath(sett))
+
+    summarize(sett)
+
+    S = [s for s in sett.S if s.symbols[1].pow == 1]
+    relations = [k*inv(v) for (k,v) in sett.G.rels]
+
+    prepare_pm_delta(PropertyT.prepath(sett), GAP_groupcode(S, relations), sett.radius)
+
+    return PropertyT.check_property_T(sett)
+end

positivity/check_positivity.jl

@@ -0,0 +1,197 @@
+using AbstractAlgebra
+using Groups
+using GroupRings
+using PropertyT
+
+using SCS
+solver(tol, iterations) =
+    SCSSolver(linearsolver=SCS.Direct,
+        eps=tol, max_iters=iterations,
+        alpha=1.95, acceleration_lookback=1)
+
+include("../main.jl")
+
+using PropertyTGroups
+
+args = Dict("SAut" => 5, "upper-bound" => 50.0, "radius" => 2, "nosymmetry"=>false, "tol"=>1e-12, "iterations" =>200000, "warmstart" => true)
+
+Gr = PropertyTGroups.PropertyTGroup(args)
+sett = PropertyT.Settings(Gr, args,
+    solver(args["tol"], args["iterations"]))
+
+@show sett
+
+fullpath = PropertyT.fullpath(sett)
+isdir(fullpath) || mkpath(fullpath)
+# setup_logging(PropertyT.filename(fullpath, :fulllog), :fulllog)
+
+function small_generating_set(RG::GroupRing{AutGroup{N}}, n) where N
+    indexing = [(i,j) for i in 1:n for j in 1:n if i≠j]
+
+    rmuls = [Groups.rmul_autsymbol(i,j) for (i,j) in indexing]
+    lmuls = [Groups.lmul_autsymbol(i,j) for (i,j) in indexing]
+    gen_set = RG.group.([rmuls; lmuls])
+
+    return [gen_set; inv.(gen_set)]
+end
+
+function computeX(RG::GroupRing{AutGroup{N}}) where N
+    Tn = small_generating_set(RG, N-1)
+
+    ℤ = Int64
+    Δn = length(Tn)*one(RG, ℤ) - RG(Tn, ℤ);
+
+    Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
+
+    @time X = sum(σ(Δn)*sum(τ(Δn) for τ ∈ Alt_N if τ ≠ σ) for σ in Alt_N);
+    return X
+end
+
+function Sq(RG::GroupRing{AutGroup{N}}) where N
+    T2 = small_generating_set(RG, 2)
+    ℤ = Int64
+    Δ₂ = length(T2)*one(RG, ℤ) - RG(T2, ℤ);
+
+    Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
+    elt = sum(σ(Δ₂)^2 for σ in Alt_N)
+    return elt
+end
+
+function Adj(RG::GroupRing{AutGroup{N}}) where N
+    T2 = small_generating_set(RG, 2)
+
+    ℤ = Int64
+    Δ₂ = length(T2)*one(RG, ℤ) - RG(T2, ℤ);
+
+    Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
+
+    adj(σ::perm, τ::perm, i=1, j=2) = Set([σ[i], σ[j]]) ∩ Set([τ[i], τ[j]])
+    adj(σ::perm) = [τ for τ in Alt_N if length(adj(σ, τ)) == 1]
+
+    @time elt = sum(σ(Δ₂)*sum(τ(Δ₂) for τ in adj(σ)) for σ in Alt_N);
+    # return RG(elt.coeffs÷factorial(N-2)^2)
+    return elt
+end
+
+function Op(RG::GroupRing{AutGroup{N}}) where N
+    T2 = small_generating_set(RG, 2)
+
+    ℤ = Int64
+    Δ₂ = length(T2)*one(RG, ℤ) - RG(T2, ℤ);
+
+    Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
+
+    adj(σ::perm, τ::perm, i=1, j=2) = Set([σ[i], σ[j]]) ∩ Set([τ[i], τ[j]])
+    adj(σ::perm) = [τ for τ in Alt_N if length(adj(σ, τ)) == 0]
+
+    @time elt = sum(σ(Δ₂)*sum(τ(Δ₂) for τ in adj(σ)) for σ in Alt_N);
+    # return RG(elt.coeffs÷factorial(N-2)^2)
+    return elt
+end
+
+const ELT_FILE = joinpath(dirname(PropertyT.filename(sett, :Δ)), "SqAdjOp_coeffs.jld")
+const WARMSTART_FILE = PropertyT.filename(sett, :warmstart)
+
+if isfile(PropertyT.filename(sett,:Δ)) && isfile(ELT_FILE) &&
+    isfile(PropertyT.filename(sett, :OrbitData))
+    # cached
+    Δ = PropertyT.loadGRElem(PropertyT.filename(sett,:Δ), sett.G)
+    RG = parent(Δ)
+    orbit_data = load(PropertyT.filename(sett, :OrbitData), "OrbitData")
+    sq_c, adj_c, op_c = load(ELT_FILE, "Sq", "Adj", "Op")
+    # elt = ELT_FILE, sett.G)
+    sq = GroupRingElem(sq_c, RG)
+    adj = GroupRingElem(adj_c, RG)
+    op = GroupRingElem(op_c, RG);
+else
+    info("Compute Laplacian")
+    Δ = PropertyT.Laplacian(sett.S, sett.radius)
+    RG = parent(Δ)
+
+    info("Compute Sq, Adj, Op")
+    @time sq, adj, op = Sq(RG), Adj(RG), Op(RG)
+
+    PropertyT.saveGRElem(PropertyT.filename(sett, :Δ), Δ)
+    save(ELT_FILE, "Sq", sq.coeffs, "Adj", adj.coeffs, "Op", op.coeffs)
+
+    info("Compute OrbitData")
+    if !isfile(PropertyT.filename(sett, :OrbitData))
+        orbit_data = PropertyT.OrbitData(parent(Y), sett.autS)
+        save(PropertyT.filename(sett, :OrbitData), "OrbitData", orbit_data)
+    else
+        orbit_data = load(PropertyT.filename(sett, :OrbitData), "OrbitData")
+    end
+end;
+
+orbit_data = PropertyT.decimate(orbit_data);
+
+elt = adj+2op;
+
+const SOLUTION_FILE = PropertyT.filename(sett, :solution)
+
+if !isfile(SOLUTION_FILE)
+    
+    SDP_problem, varλ, varP = PropertyT.SOS_problem(elt, Δ, orbit_data; upper_bound=sett.upper_bound)
+
+    begin
+        using SCS
+        scs_solver = SCS.SCSSolver(linearsolver=SCS.Direct,
+            eps=sett.tol,
+            max_iters=args["iterations"],
+            alpha=1.95,
+            acceleration_lookback=1)
+
+        JuMP.setsolver(SDP_problem, scs_solver)
+    end
+
+    λ = Ps  = nothing
+    ws = PropertyT.warmstart(sett)
+
+    # using ProgressMeter
+
+    # @showprogress "Running SDP optimization step... " for i in 1:args["repetitions"]
+    while true
+        λ, Ps, ws = PropertyT.solve(PropertyT.filename(sett, :solverlog),
+        SDP_problem, varλ, varP, ws);
+
+        if all((!isnan).(ws[1]))
+            save(WARMSTART_FILE, "warmstart", ws, "λ", λ, "Ps", Ps)
+            save(WARMSTART_FILE[1:end-4]*"_$(now()).jld", "warmstart", ws, "λ", λ, "Ps", Ps)
+        else
+            warn("No valid solution was saved!")
+        end
+    end
+    
+    info("Reconstructing P...")
+    @time P = PropertyT.reconstruct(Ps, orbit_data);
+    save(SOLUTION_FILE, "λ", λ, "P", P)
+end
+
+P, λ = load(SOLUTION_FILE, "P", "λ")
+@show λ;
+
+@time const Q = real(sqrtm(P));
+
+function SOS_residual(eoi::GroupRingElem, Q::Matrix)
+    RG = parent(eoi)
+    @time sos = PropertyT.compute_SOS(RG, Q);
+    return eoi - sos
+end
+
+info("Floating Point arithmetic:")
+EOI = elt - λ*Δ
+b = SOS_residual(EOI, Q)
+@show norm(b, 1);
+
+info("Interval arithmetic:")
+using IntervalArithmetic
+Qint = PropertyT.augIdproj(Q);
+@assert all([zero(eltype(Q)) in sum(view(Qint, :, i)) for i in 1:size(Qint, 2)])
+
+EOI_int = elt - @interval(λ)*Δ;
+Q_int = PropertyT.augIdproj(Q);
+@assert all([zero(eltype(Q)) in sum(view(Q_int, :, i)) for i in 1:size(Q_int, 2)])
+b_int = SOS_residual(EOI_int, Q_int)
+@show norm(b_int, 1);
+
+info("λ is certified to be > ", (@interval(λ) - 2^2*norm(b_int,1)).lo)

run.jl

@@ -0,0 +1,108 @@
+using ArgParse
+
+###############################################################################
+#
+#  Parsing command line
+#
+###############################################################################
+
+function parse_commandline()
+    settings = ArgParseSettings()
+
+    @add_arg_table settings 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"
+            arg_type = Int
+            required = false
+        "--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
+        "--nosymmetry"
+            help = "Don't use symmetries of the Laplacian"
+            action = :store_true
+
+        "--SL "
+            help = "GROUP: the group generated by elementary matrices of size n by n"
+            arg_type = Int
+            required = false
+        "-p"
+            help = "Matrices over field of p-elements (p=0 => over ZZ) [only with --SL]"
+            arg_type = Int
+            default = 0
+        "-X"
+            help = "Consider EL(N, ZZ⟨X⟩) [only with --SL]"
+            action = :store_true
+
+        "--SAut"
+            help = "GROUP: the automorphisms group of the free group on N generators"
+            arg_type = Int
+            required = false
+
+        "--MCG"
+            help = "GROUP: mapping class group of surface of genus N"
+            arg_type = Int
+            required = false
+
+        "--Higman"
+            help = "GROUP: the Higman Group"
+            action = :store_true
+
+        "--Caprace"
+            help = "GROUP: for Caprace Group"
+            action = :store_true
+    end
+    return parse_args(settings)
+end
+
+const PARSEDARGS = parse_commandline()
+
+set_parallel_mthread(PARSEDARGS, workers=false)
+
+include("CPUselect.jl")
+include("logging.jl")
+include("main.jl")
+
+using SCS.SCSSolver
+# using Mosek
+# using CSDP
+# using SDPA
+
+solver(tol, iterations) =
+    SCSSolver(linearsolver=SCS.Direct,
+        eps=tol, max_iters=iterations,
+        alpha=1.95, acceleration_lookback=1)
+
+# Mosek.MosekSolver(
+#    MSK_DPAR_INTPNT_CO_TOL_REL_GAP=tol,
+#    MSK_IPAR_INTPNT_MAX_ITERATIONS=iterations,
+#    QUIET=false)
+
+# CSDP.CSDPSolver(axtol=tol, atytol=tol, objtol=tol, minstepp=tol*10.0^-1, minstepd=tol*10.0^-1)
+
+# SDPA.SDPASolver(epsilonStar=tol, epsilonDash=tol)
+
+const Gr = PropertyTGroups.PropertyTGroup(PARSEDARGS)
+const sett = PropertyT.Settings(Gr, PARSEDARGS,
+    solver(PARSEDARGS["tol"], PARSEDARGS["iterations"]))
+
+fullpath = PropertyT.fullpath(sett)
+isdir(fullpath) || mkpath(fullpath)
+setup_logging(PropertyT.filename(fullpath, :fulllog), :fulllog)
+
+main(Gr, sett)

runtests.jl

@@ -0,0 +1,222 @@
+using Base.Test
+
+include("main.jl")
+
+testdir = "tests_"*string(now())
+mkdir(testdir)
+include("logging.jl")
+logger=setup_logging(joinpath(testdir, "tests.log"))
+info(testdir)
+
+cd(testdir)
+
+# groupname = name(G)
+# ub = PARSEDARGS["upper-bound"]
+#
+# fullpath = joinpath(groupname, string(ub))
+# isdir(fullpath) || mkpath(fullpath)
+
+separator(n=60) = info("\n"*("\n"*"="^n*"\n"^3)*"\n")
+
+
+function SL_tests(args)
+
+
+    args["SL"] = 2
+    args["p"] = 3
+    G = PropertyTGroup(args)
+    @test main(G) == true
+    separator()
+
+    let args = args
+        args["SL"] = 2
+        args["p"] = 5
+        G = PropertyTGroup(args)
+        @test main(G) == false
+        separator()
+
+        args["warmstart"] = true
+        G = PropertyTGroup(args)
+        @test main(G) == false
+        separator()
+
+        args["upper-bound"] = 0.1
+        G = PropertyTGroup(args)
+        @test main(G) == true
+        separator()
+    end
+
+    args["SL"] = 2
+    args["p"] = 7
+    G = PropertyTGroup(args)
+    @test main(G) == false
+    separator()
+
+    args["SL"] = 3
+    args["p"] = 7
+    G = PropertyTGroup(args)
+    @test main(G) == true
+    separator()
+
+    # begin
+    #     args["iterations"] = 25000
+    #     args["N"] = 3
+    #     args["p"] = 0
+    #     args["upper-bound"] = Inf
+    #
+    #     G = PropertyTGroups.SpecialLinearGroup(args)
+    #     @test main(G) == false
+    #     separator()
+    #
+	#     args["warmstart"] = false
+    #     args["upper-bound"] = 0.27
+    #     G = PropertyTGroups.SpecialLinearGroup(args)
+    #     @test main(G) == false
+    #     separator()
+    #
+    #     args["warmstart"] = true
+    #     G = PropertyTGroups.SpecialLinearGroup(args)
+    #     @test main(G) == true
+    #     separator()
+    # end
+
+    return 0
+end
+
+function SAut_tests(args)
+
+    G = PropertyTGroup(args)
+    @test main(G) == false
+    separator()
+
+    args["warmstart"] = true
+    G = PropertyTGroup(args)
+    @test main(G) == false
+    separator()
+
+    args["upper-bound"] = 0.1
+    G = PropertyTGroup(args)
+    @test main(G) == false
+    separator()
+
+    return 0
+end
+
+@testset "Groups with(out) (T)" begin
+
+    @testset "GAPGroups" begin
+        args = Dict(
+            "Higman" => true,
+            "iterations"=>5000,
+            "tol"=>1e-7,
+            "upper-bound"=>Inf,
+            "cpus"=>2,
+            "radius"=>2,
+            "warmstart"=>false,
+            "nosymmetry"=>true,
+        )
+
+        G = PropertyTGroup(args)
+        @test main(G) == false
+
+        args = Dict(
+            "Caprace" => true,
+            "iterations"=>5000,
+            "tol"=>1e-7,
+            "upper-bound"=>Inf,
+            "cpus"=>2,
+            "radius"=>2,
+            "warmstart"=>false,
+            "nosymmetry"=>true,
+        )
+
+        G = PropertyTGroup(args)
+        @test main(G) == false
+
+        args = Dict(
+            "MCG" => 3,
+            "iterations"=>5000,
+            "tol"=>1e-7,
+            "upper-bound"=>Inf,
+            "cpus"=>2,
+            "radius"=>2,
+            "warmstart"=>false,
+            "nosymmetry"=>true,
+        )
+
+        G = PropertyTGroup(args)
+        @test main(G) == false
+    end
+
+    @testset "SLn's" begin
+        @testset "Non-Symmetrized" begin
+
+            args = Dict(
+                "SL" => 2,
+                "p" => 3,
+                "X" => false,
+                "iterations"=>50000,
+                "tol"=>1e-7,
+                "upper-bound"=>Inf,
+                "cpus"=>2,
+                "radius"=>2,
+                "warmstart"=>false,
+                "nosymmetry"=>true,
+            )
+
+            SL_tests(args)
+        end
+
+        @testset "Symmetrized" begin
+
+            args = Dict(
+                "SL" => 2,
+                "p" => 3,
+                "X" => false,
+                "iterations"=>20000,
+                "tol"=>1e-7,
+                "upper-bound"=>Inf,
+                "cpus"=>2,
+                "radius"=>2,
+                "warmstart"=>false,
+                "nosymmetry"=>false,
+            )
+
+            SL_tests(args)
+        end
+    end
+
+    @testset "SAutF_n's" begin
+
+        @testset "Non-Symmetrized" begin
+
+            args = Dict(
+                "SAut" => 2,
+                "iterations"=>5000,
+                "tol"=>1e-7,
+                "upper-bound"=>Inf,
+                "cpus"=>2,
+                "radius"=>2,
+                "warmstart"=>false,
+                "nosymmetry"=>true,
+            )
+            SAut_tests(args)
+        end
+
+        @testset "Symmetrized" begin
+            args = Dict(
+                "SAut" => 3,
+                "iterations"=>500,
+                "tol"=>1e-7,
+                "upper-bound"=>Inf,
+                "cpus"=>2,
+                "radius"=>2,
+                "warmstart"=>false,
+                "nosymmetry"=>false,
+            )
+            SAut_tests(args)
+        end
+    end
+
+
+end