fix inverses and symplectic twists to finally arrive at SAut(π₁Σ)

src/groups/mcg.jl

@@ -20,10 +20,18 @@ end
 function SurfaceGroup(genus::Integer, boundaries::Integer)
     @assert genus > 1
 
+    # The (confluent) rewriting systems comes from
+    # S. Hermiller, Rewriting systems for Coxeter groups
+    # Journal of Pure and Applied Algebra
+    # Volume 92, Issue 2, 7 March 1994, Pages 137-148
+    # https://doi.org/10.1016/0022-4049(94)90019-1
+    # Note: the notation is "inverted":
+    # a_g of the article becomes A_g here.
+
     ltrs = String[]
     for i in 1:genus
         subscript = join('₀'+d for d in reverse(digits(i)))
-        append!(ltrs, ["a" * subscript, "A" * subscript, "b" * subscript, "B" * subscript])
+        append!(ltrs, ["A" * subscript, "a" * subscript, "B" * subscript, "b" * subscript])
     end
     Al = Alphabet(reverse!(ltrs))
 
@@ -66,17 +74,17 @@ relations(S::SurfaceGroup) = S.relations
 function symplectic_twists(π₁Σ::SurfaceGroup)
     g = genus(π₁Σ)
 
-    saut = SpecialAutomorphismGroup(FreeGroup(2g))
+    saut = SpecialAutomorphismGroup(FreeGroup(2g), maxrules=100)
 
-    Aij  = [SymplecticMappingClass(π₁Σ, saut, :A, i, j) for i in 1:g for j in 1:g if i≠j]
+    Aij  = [SymplecticMappingClass(saut, :A, i, j) for i in 1:g for j in 1:g if i≠j]
 
-    Bij  = [SymplecticMappingClass(π₁Σ, saut, :B, i, j) for i in 1:g for j in i+1:g]
+    Bij  = [SymplecticMappingClass(saut, :B, i, j) for i in 1:g for j in 1:g if i≠j]
 
-    mBij = [SymplecticMappingClass(π₁Σ, saut, :B, i, j, minus=true) for i in 1:g for j in i+1:g]
+    mBij = [SymplecticMappingClass(saut, :B, i, j, minus=true) for i in 1:g for j in 1:g if i≠j]
 
-    Bii  = [SymplecticMappingClass(π₁Σ, saut, :B, i, i) for i in 1:g]
+    Bii  = [SymplecticMappingClass(saut, :B, i, i) for i in 1:g]
 
-    mBii = [SymplecticMappingClass(π₁Σ, saut, :B, i, i, minus=true) for i in 1:g]
+    mBii = [SymplecticMappingClass(saut, :B, i, i, minus=true) for i in 1:g]
 
     return [Aij; Bij; mBij; Bii; mBii]
 end
@@ -86,5 +94,10 @@ KnuthBendix.alphabet(G::AutomorphismGroup{<:SurfaceGroup}) = rewriting(G)
 function AutomorphismGroup(π₁Σ::SurfaceGroup; kwargs...)
     S = vcat(symplectic_twists(π₁Σ)...)
     A = Alphabet(S)
-    return AutomorphismGroup(π₁Σ, S, A, ntuple(i->gens(π₁Σ, i), 2genus(π₁Σ)))
+
+    # this is to fix the definitions of symplectic twists:
+    # with i->gens(π₁Σ, i) the corresponding automorphisms return
+    # reversed words
+    domain = ntuple(i->inv(gens(π₁Σ, i)), 2genus(π₁Σ))
+    return AutomorphismGroup(π₁Σ, S, A, domain)
 end

src/groups/symplectic_twists.jl

@@ -175,8 +175,6 @@ struct SymplecticMappingClass{T, F} <: GSymbol
     minus::Bool
     inv::Bool
     autFn_word::T
-    perm::Vector{Int}
-    invperm::Vector{Int}
     f::F
 end
 
@@ -185,8 +183,7 @@ Base.:(==)(a::SymplecticMappingClass, b::SymplecticMappingClass) = a.autFn_word
 Base.hash(a::SymplecticMappingClass, h::UInt) = hash(a.autFn_word, h)
 
 function SymplecticMappingClass(
-    Σ::SurfaceGroup,
-    sautFn,
+    sautFn::AutomorphismGroup{<:FreeGroup},
     id::Symbol,
     i::Integer,
     j::Integer;
@@ -195,11 +192,12 @@ function SymplecticMappingClass(
 )
     @assert i > 0 && j > 0
     id === :A && @assert i ≠ j
-    @assert 2genus(Σ) == ngens(object(sautFn))
+    @assert iseven(ngens(object(sautFn)))
+    genus = ngens(object(sautFn))÷2
 
-    A = KnuthBendix.alphabet(sautFn)
-    λ = ΡΛ(:λ, A, 2genus(Σ))
-    ϱ = ΡΛ(:ϱ, A, 2genus(Σ))
+    A = alphabet(sautFn)
+    λ = ΡΛ(:λ, A, 2genus)
+    ϱ = ΡΛ(:ϱ, A, 2genus)
 
     w = if id === :A
         Te(λ, ϱ, i, j) *
@@ -229,14 +227,14 @@ function SymplecticMappingClass(
         throw("Type not recognized: $id")
     end
 
+    # w is a word defined in the context of A (= alphabet(sautFn))
+    # so this "coercion" is correct
     a = sautFn(w)
-    g = genus(Σ)
-    perm = [2g:-2:1; (2g-1):-2:1]
 
     f = compiled(a)
     # f = t -> evaluate!(t, a)
 
-    res = SymplecticMappingClass(id, UInt(i), UInt(j), minus, inverse, a, perm, invperm(perm), f)
+    res = SymplecticMappingClass(id, UInt(i), UInt(j), minus, inverse, a, f)
 
     return res
 end
@@ -255,7 +253,7 @@ function Base.inv(m::SymplecticMappingClass)
     inv_w = inv(m.autFn_word)
     # f(t) = evaluate!(t, inv_w)
     f = compiled(inv_w)
-    return SymplecticMappingClass(m.id, m.i, m.j, m.minus, !m.inv, inv_w, m.perm, m.invperm, f)
+    return SymplecticMappingClass(m.id, m.i, m.j, m.minus, !m.inv, inv_w, f)
 end
 
 function evaluate!(
@@ -263,9 +261,9 @@ function evaluate!(
     smc::SymplecticMappingClass,
     tmp=nothing,
 ) where {N,T}
-
-    t = smc.f(t[smc.perm])[smc.invperm]
+    t = smc.f(t)
+    for i in 1:N
+        normalform!(t[i])
+    end
     return t
-end
-    end
 end