Discreteness Locus in Projective Structures
Definition:
Let B(D,G) denote hyperbolically bounded holomorphic quadratic
forms on the unit disk D for
the Fuchsian group G uniformizing a once-punctured torus X.
This group G and the Fuchsian group G'
uniformizing a 4-times punctured sphere Y are commensurable
with the Fuchsian group H uniformizing a 4-times punctured torus Z so that |G:H|=4 and |G':H|=2.
We may assume Y=C-{0,1,a}, where C denotes the
complex plane.
Then Q:=p*(dz2/z(z-1)(z-a) )
forms a basis
of the vector space B(D,G), where p:D-->Y is the canonical
projection by the action of G' because B(D,G) is one-dimensional.
For a complex number t, we can construct the developing map
Ft from the unit disk to the Riemann sphere so that
-
Ft(0)=0,
Ft'(0)=1,
Ft''(0)=0 and
-
{Ft, z}= tQ(z),
where {F,z} denotes the Schwarzian derivative of F:
(F''/F')'-(F''/F')2/2.
Furthermore, there exists a unique homomorphism rt:
G --> PSL(2,C) such that
for each element g in G.
This is called the monodromy homomorphism or holonomy homomorphism.
Set
-
T(G)={t; Ft is univalent in D
and admits a quasiconformal extension to the Riemann sphere}, and
-
K(G)={t; rt(G) is discrete in PSL(2,C)}.
T(G) is called the Bers embedding of the Teichmueller space
of G and known to be equal to the connected component of
Int K(G) which contains the origin (due to H. Shiga).
We (Yohei Komori, Toshiyuki Sugawa, Masaaki Wada and Yasushi Yamashita)
found a way to visualize these spaces.
Here are some pictures drawn by Y. Yamashita.
The colors in these pictures are according to the combinatorial nature
of the Ford region
of the corresponding monodromy image rt(G).
Example (Square Torus)
In the case below, X is the square torus with one point removed,
in other words, the completion of
X is the quotient space of the complex plane by the lattice group
generated by 1 and i.
We display the range {t=u+iv; |u|< A and |v|< A}
for each picture.
We can see the Bers embedding of Teichmueller space as a component
of colored regions at the center of pictures.
 |
 |
 |
 |
| A=.5 |
A=1 |
A=2 |
A=4 |
 |
 |
 |
| A=8 |
A=16 |
A=32 |
Example 2 (Equilateral Triangle)
In the case below, X has the symmetry under the rotation of order 3,
in other words, the completion of
X is the quotient space of the complex plane by the lattice group
generated by 1 and the cubic root of -1.
We display the range {t=u+iv; |u|< A and |v|< A}
for each picture.
 |
 |
 |
| A=.5 |
A=1 |
A=2 |
 |
 |
 |
| A=4 |
A=8 |
A=16 |
For any question or comment,
please contact us
Related links:
McMullen's Gallery (many beautiful pictures)
Yamashita's Home Page (more pictures)
Wada's OPTi (cool program)
Bers Slice Project (in Japanese)
Back
