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/e18fcc9eea1e1e4457039ea9253c2bbdf7aad32247b44af0dc59ce6effe6719b.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/5fc4e9efdc81d4f946b34d43ad30372120a35abc4b2812f64aaf609eee8c7ddd.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/2851dd88603542a38c606553ea83551e2821cc3e7895bf5269a8e6739a48c103.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/39c7366a396f32150b678909891fe590ffa67d19ae681ca1c9c10d9ffc10aa05.png
ss = s.delaunay_decomposition()
gs = ss.graphical_surface(edge_labels=False, polygon_labels=False)
gs.make_all_visible(limit=20)
gs.plot()
../_images/0f0db1b261e20e61f38fabb0939c9363775903855d0cbf7b096c748f5880e4dc.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/2807e3eb82435462f86b71d56773d2b3f361a5bfa2da11bcbf653f61a171a735.png
mss = ms.delaunay_decomposition()
gs = mss.graphical_surface(edge_labels=False, polygon_labels=False)
gs.make_all_visible(limit=20)
gs.plot()
../_images/fe20e7c94f4f9a60cef24ce034922b1324946f4a8d533753af7e358888f7a1e8.png

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