Apply veech group elements to homology

[saraedum]

We would like to make this interface work (bringing together #211, #163, flatsurf/flatsurf#308, #213 for the homology bits)

S = ...
M = matrix([[1, 2], [0, 1]])
H = SimplicialHomology(S)
automorphism = S.apply_automorphism(M)
# shortcut for:
# deformation = S.apply_matrix(M)
# undeformation = deformation.codomain().isomorphism(S) # what if this is not unique? Just pick one (with a keyword argument to return all or fail)
# return automorphism = undeformation * deformation

M = H.matrix(automorphism)
# shortcut for:
# for γ in H.gens():
#    γγ = automorphism(γ)
#    and decompose into a matrix

One application could be something like this:

MatrixGroup([SimplicialHomology(S).matrix(S.automorphism(g))
             for g in S.veech_group().gens()]).change_ring(GF(3)).order()

[saraedum]

cc @sfreedman67

[videlec]

Elements of the Veech group (aka matrices) are not maps between surfaces. As a user interface, I would rather propose something like

S = ...
A = S.affine_automorphism_group()
M = matrix([[1, 2], [0, 1]])
f = A.derivative().section()(M)
H = SimplicialHomology(S)
Mhom = H.hom(f).matrix()

Here

• A.derivative() is the morphism from the affine group to GL(2,R) (whose image is the Veech group and whose kernel is the group of translation automorphisms)
• A.derivative().section() is choosing for you an automorphism representative with given derivative (if it exists)

[saraedum]

@videlec that makes sense. I'll make sure to change that once #211 is in.