Definition:

Let *B(D,G)* denote the set of hyperbolically bounded holomorphic
quadratic differentials on the unit disk *D* for the Fuchsian group
*G* uniformizing a once-punctured torus *X*. This group *G*
is commensurable with the Fuchsian group *G'* uniformizing a 4-times
punctured sphere *Y. *We may assume that *Y=C-{0,1,a}*, where
*C* denotes the complex plane. Then *Q:=p ^{*}*(

For a complex number

*F*(0)=0,_{t}*F*'(0)=1,_{t}*F*''(0)=0 and_{t}- {
*F*}=_{t}, z*tQ*(*z*),

Furthermore, there exists a unique homomorphism

Set

*T(G)*={*t*;*F*is univalent in_{t}*D*and admits a quasiconformal extension to the Riemann sphere}, and*K(G)*={*t*;*r*(_{t}*G*) is discrete in PSL(2,*C*)}.

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)

Bers Slice Project (part 2) (in Japanese)