add tests for Aut(Σ₄.₁)
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@ -22,7 +22,8 @@ julia = "1.3, 1.4, 1.5"
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[extras]
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BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
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PermutationGroups = "8bc5a954-2dfc-11e9-10e6-cd969bffa420"
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Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
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[targets]
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test = ["Test", "BenchmarkTools"]
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test = ["Test", "BenchmarkTools", "PermutationGroups"]
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@ -1,5 +1,3 @@
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include("symplectic_twists.jl")
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struct SurfaceGroup{T, S, R} <: AbstractFPGroup
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genus::Int
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boundaries::Int
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@ -8,6 +6,8 @@ struct SurfaceGroup{T, S, R} <: AbstractFPGroup
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rws::R
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end
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include("symplectic_twists.jl")
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genus(S::SurfaceGroup) = S.genus
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function Base.show(io::IO, S::SurfaceGroup)
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@ -0,0 +1,193 @@
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using PermutationGroups
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@testset "Wajnryb presentation for Σ₄₁" begin
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genus = 4
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G = New.SpecialAutomorphismGroup(New.FreeGroup(2genus))
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T = let G = G; (Tas, Tαs, Tes) = New.mcg_twists(genus)
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Ta = G.(Tas)
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Tα = G.(Tαs)
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Tes = G.(Tes)
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[Ta; Tα; Tes]
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end
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a1 = T[1]^-1 # Ta₁
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a2 = T[5]^-1 # Tα₁
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a3 = T[9]^-1 # Te₁₂
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a4 = T[6]^-1 # Tα₂
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a5 = T[12]^-1 # Te₂₃
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a6 = T[7]^-1 # Tα₃
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a7 = T[14]^-1 # Te₃₄
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a8 = T[8]^-1 # Tα₄
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b0 = T[2]^-1 # Ta₂
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a0 = (a1*a2*a3)^4*b0^-1 # from the 3-chain relation
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X = a4*a5*a3*a4 # auxillary, not present in the Primer
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b1 = X^-1*a0*X
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b2 = T[10]^-1 # Te₁₃
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As = T[[1,5,9,6,12,7,14,8]] # the inverses of the elements a
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@testset "commutation relations" begin
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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.
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for (j, aj) in enumerate(As)
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if abs(i-j) > 1
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@test ai*aj == aj*ai
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elseif i ≠ j
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@test ai*aj != aj*ai
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end
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end
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if i != 4
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@test a0*ai == ai*a0
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end
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end
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end
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@testset "braid relations" begin
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for (i, ai) in enumerate(As) #the element ai here is actually the inverse of ai before. It does not matter for braid relations.
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for (j, aj) in enumerate(As)
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if abs(i-j) == 1
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@test ai*aj*ai == aj*ai*aj
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end
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end
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end
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@test a0*a4*a0 == a4*a0*a4 # here, a0 and a4 are as before
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end
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@testset "Lantern relation" begin
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@testset "b2 definition" begin
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@test b2 == (a2*a3*a1*a2)^-1*b1*(a2*a3*a1*a2)
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# some additional tests, checking what explicitly happens to the generators of the π₁ under b2
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d = New.domain(b2)
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im = New.evaluate(b2)
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z = im[7]*d[7]^-1
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@test im[1] == d[1]
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@test im[2] == z*d[2]*z^-1
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@test im[3] == z*d[3]*z^-1
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@test im[4] == d[4]
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@test im[5] == d[5]*z^-1
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@test im[6] == z*d[6]*z^-1
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@test im[7] == z*d[7]
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@test im[8] == d[8]
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end
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@testset "b2: commutation relations" begin
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@test b2*a1 == a1*b2
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@test b2*a2 != a2*b2
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@test b2*a3 == a3*b2
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@test b2*a4 == a4*b2
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@test b2*a5 == a5*b2
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@test b2*a6 != a6*b2
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end
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@testset "b2: braid relations" begin
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@test a2*b2*a2 == b2*a2*b2
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@test a6*b2*a6 == b2*a6*b2
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end
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@testset "lantern" begin
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u = (a6*a5)^-1*b1*(a6*a5)
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x = (a6*a5*a4*a3*a2*u*a1^-1*a2^-1*a3^-1*a4^-1) # yet another auxillary
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# x = (a4^-1*a3^-1*a2^-1*a1^-1*u*a2*a3*a4*a5*a6)
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@time New.evaluate(x)
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b3 = x*a0*x^-1
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@time New.evaluate(b3)
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@test a0*b2*b1 == a1*a3*a5*b3
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end
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end
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@testset "Te₁₂ definition" begin
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G = parent(first(T))
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F₈ = New.object(G)
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(a, b, c, d, α, β, γ, δ) = gens(F₈)
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A = KnuthBendix.alphabet(G)
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λ = [i == j ? one(G) : G([A[New.λ(i,j)]]) for i in 1:8, j in 1:8]
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ϱ = [i == j ? one(G) : G([A[New.ϱ(i,j)]]) for i in 1:8, j in 1:8]
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g = one(G)
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# @show g
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# @show g(Groups.domain(G))
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# β ↦ α*β
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g *= λ[6,5]
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@test New.evaluate(g)[6] == α*β
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# α ↦ a*α*b^-1
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g *= λ[5,1]*inv(ϱ[5,2])
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@test New.evaluate(g)[5] == a*α*b^-1
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# β ↦ b*α^-1*a^-1*α*β
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g *= inv(λ[6,5])
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@test New.evaluate(g)[6] == b*α^-1*a^-1*α*β
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# b ↦ α
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g *= λ[2,5]*inv(λ[2,1]);
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@test New.evaluate(g)[2] == α
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# b ↦ b*α^-1*a^-1*α
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g *= inv(λ[2,5]);
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@test New.evaluate(g)[2] == b*α^-1*a^-1*α
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# b ↦ b*α^-1*a^-1*α*b*α^-1
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g *= inv(ϱ[2,5])*ϱ[2,1];
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@test New.evaluate(g)[2] == b*α^-1*a^-1*α*b*α^-1
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# b ↦ b*α^-1*a^-1*α*b*α^-1*a*α*b^-1
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g *= ϱ[2,5];
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@test New.evaluate(g)[2] == b*α^-1*a^-1*α*b*α^-1*a*α*b^-1
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x = b*α^-1*a^-1*α
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@test New.evaluate(g) == # (a, b, c, d, α, β, γ, δ)
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(a, x*b*x^-1, c, d, α*x^-1, x*β, γ, δ)
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@test g == T[9]
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end
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Base.conj(t::New.Transvection, p::Perm) =
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New.Transvection(t.id, t.i^p, t.j^p, t.inv)
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function Base.conj(elt::New.FPGroupElement, p::Perm)
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G = parent(elt)
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A = New.alphabet(elt)
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return G([A[conj(A[idx], p)] for idx in New.word(elt)])
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end
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@testset "Te₂₃ definition" begin
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Te₁₂, Te₂₃ = T[9], T[12]
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G = parent(Te₁₂)
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F₈ = New.object(G)
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(a, b, c, d, α, β, γ, δ) = gens(F₈)
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img_Te₂₃ = New.evaluate(Te₂₃)
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y = c*β^-1*b^-1*β
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@test img_Te₂₃ == (a, b, y*c*y^-1, d, α, β*y^-1, y*γ, δ)
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σ = perm"(1,2,3)(5,6,7)(8)"
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Te₂₃_σ = conj(Te₁₂, σ)
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# @test New.word(Te₂₃_σ) == New.word(Te₂₃)
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@test New.evaluate(Te₂₃_σ) == New.evaluate(Te₂₃)
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@test Te₂₃ == Te₂₃_σ
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end
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@testset "Te₃₄ definition" begin
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Te₁₂, Te₃₄ = T[9], T[14]
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G = parent(Te₁₂)
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F₈ = New.object(G)
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(a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)
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σ = perm"(1,3)(2,4)(5,7)(6,8)"
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Te₃₄_σ = conj(Te₁₂, σ)
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@test Te₃₄ == Te₃₄_σ
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end
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end
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@ -35,6 +35,8 @@ using LinearAlgebra
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include("fp_groups.jl")
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include("AutFn.jl")
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include("AutSigma_41.jl")
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include("benchmarks.jl")
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end
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end
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