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/cbd1ccf86cceae119014cb5b385e37fea9d49663a746407f41f2582095904734.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/84a774530ffc683943552ca1e8ecadc5e270249730cdf0529aa95393b4073051.png

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

sss = ss.delaunay_decomposition()
sss
/usr/share/miniconda3/envs/test/lib/python3.11/site-packages/flatsurf/geometry/categories/similarity_surfaces.py:2877: UserWarning: delaunay_decomposition() has been deprecated and will be removed in a future version of sage-flatsurf; use delaunay_decompose().codomain() instead
  warnings.warn(
Delaunay cell decomposition of Translation Surface in H_2(2) built from 2 pentagons
sss.plot()
../_images/55aa9a46a13e58f6276492f01474f83a3302b452252118515788dc9ccf9bc5eb.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/41c6554c919f9eaac7beb61a5abc03663a6b2798e9345127064948cd0fc1e71d.png
ss = s.delaunay_decomposition()
gs = ss.graphical_surface(edge_labels=False, polygon_labels=False)
gs.make_all_visible(limit=20)
gs.plot()
../_images/4f46df70f7276ab81410a3dc6759909f0e8a785e7c3a87a4ae56df8b1cd7a4fe.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/18a6aca4333ecf34d8ff06c36a068f84006adba5395058559b3a8b82f119dda9.png
mss = ms.delaunay_decomposition()
gs = mss.graphical_surface(edge_labels=False, polygon_labels=False)
gs.make_all_visible(limit=20)
gs.plot()
../_images/f826181fc2010714868fc15e1f536fe46a2f0d3de1358d32281185b093d4eeb0.png

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