Slices of the Discreteness Locus in Projective Structures

The Bers and linear slice images were created using a program developed by Yohei Komori, Toshiyuki Sugawa, Masaaki Wada, and Yasushi Yamashita.

Bers Slices

1. Centered on the standard component
   1. t = 0.5, radius 0.22
   2. t = 0.5, radius 3
   3. t = (0.5 + 0.866i), radius 4 (hex torus)
   4. t = (0.5 + 0.966i), radius 4
   5. t = (0.6 + 0.866i), radius 4
   6. t = (0.7 + 0.866i), radius 4
2. Splintering of components
1. t = (0.5 + 0.868i) and a smaller version of the same
2. t = (0.6 + 0.866i)
3. t = (0.5 + 0.873i)
4. Zooming in on the region between two components, t = (0.5 + 0.866i) (hex torus)
   1. radius = 0.5
   2. radius = 0.1
   3. radius = 0.05
   4. radius = 0.01 (ok, just kidding...)
5. Moving islands
   1. t = 0.595 + 0.866i
   2. t = 0.600 + 0.866i
   3. t = 0.605 + 0.866i
6. Within the harbor
   1. Island
   2. Sandbar
   3. Inlet
3. Rectangular tori with short geodesics (grids and archipelagoes)
   1. t=0.01, radius=11
   2. t=0.002, radius=2
   3. t=0.002, radius=0.5
   4. t=0.0002, radius=0.5
   5. t=0.0002, radius=0.1
   6. Centered image, showing cardioid shape of standard component
   7. Larger centered image
4. Animations
   1. Pinching a geodesic (rectangular tori)
   2. Near the hex torus (0.4 + 0.866i to 0.6+0.866i)

Linear Slices of V(S)

The representation variety for a once-punctured torus group is essentially parameterized by the traces of standard generators (whose commutator is parabolic). The images below show slices where one trace is fixed, coloring those values of the other trace for which the group is free and discrete.

1. Animations
1. Imaginary part (slow)
2. Imaginary part (fast)
3. Imaginary part (faster)
4. Real part (slow)