Discreteness Locus

This section is based on the joint work with Y. Komori, T. Sugawa and M. Wada. The following pictures are slices of the parameter space of once-punctured torus groups.

1. Bers slice

1-1. Base point = hexagonal torus.

1-2. Base point = square torus.

1-3. Other Bers slices

2. Earle slice

3. Maskit slice (Tr b = 2)

4. Linear slice (trace constant slice)

4-1. Tr b = 2 + 6i (see Figure 10.20 in Indra's Pearls)

4-2. Tr b = 1.91 + 0.05i (see Figure 10.19 in Indra's Pearls)

4-3. Other examples

• old version
• Sugawa's page
• OPTi by M. Wada

Convex hull boundary (pleated surface)

I am impressed by the nice picture of Figure 12.6 in "Indra's Pearls". (The picture was produced by Yair Minsky. See his limit set pictures.) I wanted to draw these nice pictures. Here are some of the pictures I produced.

Tr a = 2.7, Tr b = 2.7

2D picture [PDF(0.4M)] 2D picture [PDF(0.5M)] animation (7.2M)

Tr a = 2.5, Tr b = 2.5

2D picture [PDF(0.5M)]

Tr a = 2.2, Tr b = 2.2

2D picture [PDF(0.7M)] 2D picture [PDF(0.1M)]

Tr a = 2, Tr b = 2

2D picture [PDF(1.2M)]

Tr a = 3, Tr b = 2+0.5i

2D picture [PDF(0.3M)]

Tr a = 3, Tr b = 1.91+0.05i

2D picture [PDF(4.5M)]

Tr a = 3.5, Tr b = 2.578456017+0.64347683583i

animation (7.7M)

Tr a = 2.03125, Tr b = 1.750069784+0.9998778635i

2D picture [PDF(4.7M)]

Tr a = 4, Tr b = 2.071677342+0.761020644i

2D picture [PDF(3.3M)] animation (6.8M)

Tr a = 10, Tr b = 2.683021785+0.558833926i

2D picture [PDF(1.2M)]

Tr a = 64, Tr b = 4.862001682+0.6774259456i

animation (6.6M)