The GL(2,R) Action, the Veech Group, Delaunay Decomposition#

Acting on surfaces by matrices.#

from flatsurf import translation_surfaces

s = translation_surfaces.veech_double_n_gon(5)
s.plot()
../_images/54b3cef6d141a97e0e385de4a48a1071cfeebc61002de800810024f9a43c9573.png
m = matrix([[2, 1], [1, 1]])

You can act on surfaces with the \(GL(2,R)\) action

ss = m * s
ss
Translation Surface in H_2(2) built from 2 pentagons
ss.plot()
../_images/54e5c8994e482e0c7fd749b3dc222c6be5f931d2d8d0bf1b32629ea3048b6c8f.png

To “renormalize” you can improve the presentation using the Delaunay decomposition.

sss = ss.delaunay_decomposition()
sss
Translation Surface in H_2(2) built from 6 triangles
sss.plot()
../_images/30b93b9c27d760b9c23f73b4103e555fc71b0511cecee155e56ffa9719667ca4.png

The Veech group#

Set \(s\) to be the double pentagon again.

s = translation_surfaces.veech_double_n_gon(5)

The surface has a horizontal cylinder decomposition all of whose moduli are given as below

p = s.polygon(0)
modulus = (p.vertex(3)[1] - p.vertex(2)[1]) / (p.vertex(2)[0] - p.vertex(4)[0])
AA(modulus)
0.3632712640026804?
m = matrix(s.base_ring(), [[1, 1 / modulus], [0, 1]])
show(m)
\(\displaystyle \left(\begin{array}{rr} 1 & \frac{2}{5} a^{3} \\ 0 & 1 \end{array}\right)\)
show(matrix(AA, m))
\(\displaystyle \left(\begin{array}{rr} 1 & 2.752763840942347? \\ 0 & 1 \end{array}\right)\)

The following can be used to check that \(m\) is in the Veech group of \(s\).

s.canonicalize() == (m * s).canonicalize()
True

Infinite surfaces#

Infinite surfaces support multiplication by matrices and computing the Delaunay decomposition. (Computation is done “lazily”)

s = translation_surfaces.chamanara(1 / 2)
s.plot(edge_labels=False, polygon_labels=False)
../_images/b8b1f5cc6b47483b392c85f81ac47faa83fca0e67dbe5880853de6c41b78d214.png
ss = s.delaunay_decomposition()
gs = ss.graphical_surface(edge_labels=False, polygon_labels=False)
gs.make_all_visible(limit=20)
gs.plot()
../_images/4b4c38c7bb0575d6d38277cabfa94b507a097353ee765244a2c472e1958bab4e.png
m = matrix([[2, 0], [0, 1 / 2]])
ms = m * s
gs = ms.graphical_surface(edge_labels=False, polygon_labels=False)
gs.make_all_visible(limit=20)
gs.plot()
../_images/492bdd8e91c26eb123705a4f3d57f396dc56103682ed4f4f174ae21e8f6042f1.png
mss = ms.delaunay_decomposition()
gs = mss.graphical_surface(edge_labels=False, polygon_labels=False)
gs.make_all_visible(limit=20)
gs.plot()
../_images/fcdc3ffced30799dec79c51b73eea60eaeda58ea981672500abf79d68f484665.png

You can tell from the above picture that \(m\) is in the Veech group.