The GL(2,R) action, the Veech group, Delaunay decomposition

Initial version by Pat Hooper whooper@ccny.cuny.edu, Dec 16, 2017.



In [1]:

    
from flatsurf import *

Acting on surfaces by matrices.



In [2]:

    
s = translation_surfaces.veech_double_n_gon(5)



In [3]:

    
s.plot()









    Out[3]:



In [4]:

    
m=matrix([[2,1],[1,1]])

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



In [5]:

    
ss = m*s
ss









    Out[5]:





TranslationSurface built from 2 polygons



In [6]:

    
ss.plot()









    Out[6]:

To "renormalize" you can improve the presentation using the Delaunay decomposition.



In [7]:

    
sss = ss.delaunay_decomposition().copy(relabel=True)



In [8]:

    
sss.plot()









    Out[8]:

Set $s$ to be the double pentagon again.



In [9]:

    
s = translation_surfaces.veech_double_n_gon(5)

It is best to work in the field in which the surfact is defined.



In [10]:

    
p=s.polygon(0)
p









    Out[10]:





Polygon: (0, 0), (1, 0), (1/2*a^2 - 1/2, 1/2*a), (1/2, 1/2*a^3 - a), (-1/2*a^2 + 3/2, 1/2*a)

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



In [11]:

    
modulus = (p.vertex(3)[1]-p.vertex(2)[1])/(p.vertex(2)[0]-p.vertex(4)[0])
AA(modulus)









    Out[11]:





0.3632712640026804?



In [12]:

    
m = matrix(s.base_ring(),[[1, 1/modulus],[0,1]])
show(m)









    




 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 1 & \frac{2}{5} a^{3} \\ 0 & 1 \end{array}\right)$



In [13]:

    
show(matrix(AA,m))









    




 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 1 & 2.752763840942347? \\ 0 & 1 \end{array}\right)$

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



In [14]:

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









    Out[14]:





True

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



In [15]:

    
s=translation_surfaces.chamanara(1/2)



In [16]:

    
s.plot(edge_labels=False,polygon_labels=False)









    Out[16]:



In [17]:

    
ss=s.delaunay_decomposition()



In [18]:

    
ss.graphical_surface().make_all_visible(limit=20)



In [19]:

    
ss.plot(edge_labels=False,polygon_labels=False)









    Out[19]:



In [20]:

    
m = matrix([[2,0],[0,1/2]])



In [21]:

    
ms = m*s



In [22]:

    
ms.graphical_surface().make_all_visible(limit=20)
ms.plot(edge_labels=False,polygon_labels=False)









    Out[22]:



In [23]:

    
mss = ms.delaunay_decomposition()



In [24]:

    
mss.graphical_surface().make_all_visible(limit=20)
mss.plot(edge_labels=False,polygon_labels=False)









    Out[24]:

You can tell from the above picture that $m$ is in the Veech group.