GroupsWithPropertyT/groups/mappingclassgroup.jl

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module MappingClassGroups
using Nemo
using Groups
###############################################################################
#
# Generating set
#
###############################################################################
Comm(x,y) = x*y*x^-1*y^-1
function generatingset(N::Int)
if N == 2
MCGroup = Groups.FPGroup(["a1","a2","a3","a4","a5"]);
S = Nemo.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: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
elseif N < 2
throw("Genus must be at least 2!")
end
MCGroup = Groups.FPGroup(["a$i" for i in 0:2N])
S = Nemo.gens(MCGroup)
a0 = S[1]
A = S[2:end]
k = length(A)
relations = [
[Comm(A[i], A[j]) for i in 1:k for j in 1:k if abs(i-j) > 1]...,
[Comm(a0, A[i]) for i in 1:k if i != 4]...,
[A[i]*A[(i+1)]*A[i]*inv(A[i+1]*A[i]*A[i+1]) for i in 1:k-1]...,
A[4]*a0*A[4]*inv(a0*A[4]*a0)
]
# 3-chain relation
c = prod(reverse(A[1:4]))*prod(A[1:4])
b0 = c*a0*inv(c)
push!(relations, (A[1]*A[2]*A[3])^4*inv(a0*b0))
# Lantern relation
b1 = inv(A[4]*A[5]*A[3]*A[4])*a0*(A[4]*A[5]*A[3]*A[4])
b2 = inv(A[2]*A[3]*A[1]*A[2])*b1*(A[2]*A[3]*A[1]*A[2])
u = inv(A[6]*A[5])*b1*(A[6]*A[5])
x = prod(reverse(A[2:6]))*u*prod(inv.(A[1:4]))
b3 = x*a0*inv(x)
push!(relations, a0*b2*b1*inv(A[1]*A[3]*A[5]*b3))
# Hyperelliptic relation
X = prod(reverse(A))*prod(A)
function n(i::Int, b=b0)
if i == 1
return A[1]
elseif i == 2
return b
else
return w(i-2)*n(i-2)*w(i-2)
end
end
function w(i::Int)
(A[2i+4]*A[2i+3]*A[2i+2]* n(i+1))*(A[2i+1]*A[2i] *A[2i+2]*A[2i+1])*
(A[2i+3]*A[2i+2]*A[2i+4]*A[2i+3])*( n(i+1)*A[2i+2]*A[2i+1]*A[2i] )
end
push!(relations, X*n(N)*inv(n(N)*X))
relations = [relations; [inv(rel) for rel in relations]]
Groups.add_rels!(MCGroup, Dict(rel => MCGroup() for rel in relations))
return MCGroup, gens(MCGroup)
end
function generatingset(parsed_args)
N = parsed_args["N"]
return generatingset(N)
end
###############################################################################
#
# Misc
#
###############################################################################
function groupname(parsed_args)
N = parsed_args["N"]
return groupname(N), N
end
groupname(N::Int) = "MCG$(N)"
end #of module MappingClassGroups