Groups.jl/test/AutSigma_41.jl

using PermutationGroups
using Groups.KnuthBendix

@testset "Wajnryb presentation for Σ₄₁" begin

    genus = 4

    G = SpecialAutomorphismGroup(FreeGroup(2genus))

    # symplectic pairing in the free Group goes like this:
    # f1 ↔ f5
    # f2 ↔ f6
    # f3 ↔ f7
    # f4 ↔ f8

    T = let G = G
        (Tas, Tαs, Tes) = Groups.mcg_twists(G)
        Ta = G.(Tas)
        Tα = G.(Tαs)
        Tes = G.(Tes)

        [Ta; Tα; Tes]
    end

    a1 = T[1]^-1 # Ta₁
    a2 = T[5]^-1 # Tα₁
    a3 = T[9]^-1 # Te₁₂
    a4 = T[6]^-1 # Tα₂
    a5 = T[12]^-1 # Te₂₃
    a6 = T[7]^-1 # Tα₃
    a7 = T[14]^-1 # Te₃₄
    a8 = T[8]^-1 # Tα₄

    b0 = T[2]^-1 # Ta₂
    a0 = (a1 * a2 * a3)^4 * b0^-1 # from the 3-chain relation
    X = a4 * a5 * a3 * a4 # auxillary, not present in the Primer
    b1 = X^-1 * a0 * X
    b2 = T[10]^-1 # Te₁₃

    As = T[[1, 5, 9, 6, 12, 7, 14, 8]] # the inverses of the elements a


    @testset "preserving relator" begin
        F = Groups.object(G)

        R = prod(commutator(gens(F, i), gens(F, i+genus)) for i in 1:genus)
        
        for g in T
            @test g(R) == R
        end
    end

    @testset "commutation relations" begin
        for (i, ai) in enumerate(As) #the element ai here is actually the inverse of ai before. It does not matter for commutativity. Also, a0 is as defined before.
            for (j, aj) in enumerate(As)
                if abs(i - j) > 1
                    @test ai * aj == aj * ai
                elseif i ≠ j
                    @test ai * aj != aj * ai
                end
            end
            if i != 4
                @test a0 * ai == ai * a0
            end
        end
    end

    @testset "braid relations" begin
        for (i, ai) in enumerate(As) #the element ai here is actually the inverse of ai before. It does not matter for braid relations.
            for (j, aj) in enumerate(As)
                if abs(i - j) == 1
                    @test ai * aj * ai == aj * ai * aj
                end
            end
        end
        @test a0 * a4 * a0 == a4 * a0 * a4 # here, a0 and a4 are as before
    end

    @testset "3-chain relation" begin
        x = a4*a3*a2*a1*a1*a2*a3*a4 # auxillary; does not have a name in the Primer
        @test b0 == x*a0*x^-1
    end

    @testset "Lantern relation" begin

        @testset "b2 definition" begin
            @test b2 == (a2 * a3 * a1 * a2)^-1 * b1 * (a2 * a3 * a1 * a2)

            # some additional tests, checking what explicitly happens to the generators of the π₁ under b2
            d = Groups.domain(b2)
            im = evaluate(b2)
            z = im[7] * d[7]^-1

            @test im[1] == d[1]
            @test im[2] == z * d[2] * z^-1
            @test im[3] == z * d[3] * z^-1
            @test im[4] == d[4]

            @test im[5] == d[5] * z^-1
            @test im[6] == z * d[6] * z^-1
            @test im[7] == z * d[7]
            @test im[8] == d[8]

        end

        @testset "b2: commutation relations" begin
            @test b2 * a1 == a1 * b2
            @test b2 * a2 != a2 * b2
            @test b2 * a3 == a3 * b2
            @test b2 * a4 == a4 * b2
            @test b2 * a5 == a5 * b2
            @test b2 * a6 != a6 * b2
        end

        @testset "b2: braid relations" begin
            @test a2 * b2 * a2 == b2 * a2 * b2
            @test a6 * b2 * a6 == b2 * a6 * b2
        end

        @testset "lantern" begin
            u = (a6 * a5)^-1 * b1 * (a6 * a5)
            x = (a6 * a5 * a4 * a3 * a2 * u * a1^-1 * a2^-1 * a3^-1 * a4^-1) # yet another auxillary
            # x = (a4^-1*a3^-1*a2^-1*a1^-1*u*a2*a3*a4*a5*a6)
            @time evaluate(x)
            b3 = x * a0 * x^-1
            @time evaluate(b3)
            @test a0 * b2 * b1 == a1 * a3 * a5 * b3
        end
    end


    @testset "Te₁₂ definition" begin
        G = parent(first(T))
        F₈ = Groups.object(G)
        (a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)

        A = alphabet(G)

        λ = [i == j ? one(G) : G([A[Groups.λ(i, j)]]) for i in 1:8, j in 1:8]
        ϱ = [i == j ? one(G) : G([A[Groups.ϱ(i, j)]]) for i in 1:8, j in 1:8]

        g = one(G)

        # β ↦ α*β
        g *= λ[6, 5]
        @test evaluate(g)[6] == α * β

        # α ↦ a*α*b^-1
        g *= λ[5, 1] * inv(ϱ[5, 2])
        @test evaluate(g)[5] == a * α * b^-1

        # β ↦ b*α^-1*a^-1*α*β
        g *= inv(λ[6, 5])
        @test evaluate(g)[6] == b * α^-1 * a^-1 * α * β

        # b ↦ α
        g *= λ[2, 5] * inv(λ[2, 1])
        @test evaluate(g)[2] == α

        # b ↦ b*α^-1*a^-1*α
        g *= inv(λ[2, 5])
        @test evaluate(g)[2] == b * α^-1 * a^-1 * α

        # b ↦ b*α^-1*a^-1*α*b*α^-1
        g *= inv(ϱ[2, 5]) * ϱ[2, 1]
        @test evaluate(g)[2] == b * α^-1 * a^-1 * α * b * α^-1

        # b ↦ b*α^-1*a^-1*α*b*α^-1*a*α*b^-1
        g *= ϱ[2, 5]
        @test evaluate(g)[2] == b * α^-1 * a^-1 * α * b * α^-1 * a * α * b^-1

        x = b * α^-1 * a^-1 * α
        @test evaluate(g) ==
              (a, x * b * x^-1, c, d, α * x^-1, x * β, γ, δ)
            # (a, b,            c, d, α,        β,     γ, δ)
        @test g == T[9]
    end

    Base.conj(t::Groups.Transvection, p::Perm) =
        Groups.Transvection(t.id, t.i^p, t.j^p, t.inv)

    function Base.conj(elt::FPGroupElement, p::Perm)
        G = parent(elt)
        A = alphabet(elt)
        return G([A[conj(A[idx], p)] for idx in word(elt)])
    end

    @testset "Te₂₃ definition" begin
        Te₁₂, Te₂₃ = T[9], T[12]
        G = parent(Te₁₂)
        F₈ = Groups.object(G)
        (a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)

        img_Te₂₃ = evaluate(Te₂₃)
        y = c * β^-1 * b^-1 * β
        @test img_Te₂₃ == (a, b, y * c * y^-1, d, α, β * y^-1, y * γ, δ)

        σ = perm"(1,2,3)(5,6,7)(8)"
        Te₂₃_σ = conj(Te₁₂, σ)
        # @test word(Te₂₃_σ) == word(Te₂₃)

        @test evaluate(Te₂₃_σ) == evaluate(Te₂₃)
        @test Te₂₃ == Te₂₃_σ
    end

    @testset "Te₃₄ definition" begin
        Te₁₂, Te₃₄ = T[9], T[14]
        G = parent(Te₁₂)
        F₈ = Groups.object(G)
        (a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)

        σ = perm"(1,3)(2,4)(5,7)(6,8)"
        Te₃₄_σ = conj(Te₁₂, σ)
        @test Te₃₄ == Te₃₄_σ
    end

    @testset "hyperelliptic τ is central" begin

        A = alphabet(G)
        λ = Groups.ΡΛ(:λ, A, 2genus)
        ϱ = Groups.ΡΛ(:ϱ, A, 2genus)

        import Groups: Ta, Tα, Te

        halftwists = map(1:genus-1) do i
            j = i + 1
            x = Ta(λ, j) * inv(A, Ta(λ, i)) * Tα(λ, j) * Te(λ, ϱ, i, j)
            δ = x * Tα(λ, i) * inv(A, x)
            c =
                inv(A, Ta(λ, j)) *
                Te(λ, ϱ, i, j) *
                Tα(λ, i)^2 *
                inv(A, δ) *
                inv(A, Ta(λ, j)) *
                Ta(λ, i) *
                δ
            z =
                Te(λ, ϱ, j, i) *
                inv(A, Ta(λ, i)) *
                Tα(λ, i) *
                Ta(λ, i) *
                inv(A, Te(λ, ϱ, j, i))

            G(Ta(λ, i) * inv(A, Ta(λ, j) * Tα(λ, j))^6 * (Ta(λ, j) * Tα(λ, j) * z)^4 * c)
        end

        τ = (G(Ta(λ, 1) * Tα(λ, 1))^6) * prod(halftwists)

        # τ^genus is trivial but only in autπ₁Σ₄
        # here we check its centrality

        τᵍ = τ^genus

        symplectic_gens = let genus = genus, G = G
            π₁Σ = Groups.SurfaceGroup(genus, 0)
            autπ₁Σ = AutomorphismGroup(π₁Σ)
            letters = alphabet(autπ₁Σ).letters
            G.(word(l.autFn_word) for l in letters)
        end

        @test all(sg * τᵍ == τᵍ * sg for sg in symplectic_gens)
    end
end