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) ## TODO: how to evaluate automorphisms properly??!!! for g in T w = one(word(g)) dg = Groups.domain(g) gens_idcs = first.(word.(dg)) img = evaluate(g) A = alphabet(first(dg)) ltrs_map = Vector{eltype(dg)}(undef, length(KnuthBendix.letters(A))) for i in 1:length(KnuthBendix.letters(A)) if i in gens_idcs ltrs_map[i] = img[findfirst(==(i), gens_idcs)] else ltrs_map[i] = inv(img[findfirst(==(inv(A, i)), gens_idcs)]) end end for l in word(R) append!(w, word(ltrs_map[l])) end @test F(w) == 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 "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 @test_broken all(a * τᵍ == τᵍ * a for a in Groups.gens(G)) end end