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/1deca6dc74ed9a6c730682a13a25995a4dfe51c7bb33fc546ee15761b1139dbd.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/38b3291fcf05c26872282c037fcb91c3da8ddd2545c81e803481734a6d9fb10d.png

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

sss = ss.delaunay_decomposition()
sss
Delaunay cell decomposition of Translation Surface in H_2(2) built from 2 pentagons
sss.plot()
../_images/c620c61db96233384756ec08f6c9ac13b98de75a413408591190d9cae29dc524.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/511bd0e914bf19535df6ac67f64704777c75767378a2756cc8305fcfcc732077.png
ss = s.delaunay_decomposition()
gs = ss.graphical_surface(edge_labels=False, polygon_labels=False)
gs.make_all_visible(limit=20)
gs.plot()
../_images/bde9dd8b1f083f63ede22ce455d93bb2cb0e83dfc1df99cfe48c058fea7d712f.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/306c96c17a6a1be1007bc0909514e0f0c19e5f66cd0b286fe29946b9bca1c5e4.png
mss = ms.delaunay_decomposition()
gs = mss.graphical_surface(edge_labels=False, polygon_labels=False)
gs.make_all_visible(limit=20)
gs.plot()
../_images/87b8c0c89d5825841c43a67f837a98e0077aafa86827c3f50260cf479eeebc87.png

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