Limit Sets of Kleinian Groups

Dendrites (Nov 2007)

These images show limit sets of singly degenerate punctured torus groups in the stable manifold of a pseudo-anosov mapping class (acting on the SL₂(C) character variety). The ending lamination λ of the degenerate end is specified by its slope, an irrational real number given by the ratio of intersection numbers of λ with a pair loops generating the fundamental group.

The images were created using kleinian, a program by David Wright. The group generators were computed in Mathematica, and the image generation process was automated using a Python script.

Golden ratio (slope = φ)

Preview image links to xvid AVI (2.4MB). View/download as:

• AVI, xvid codec (easiest format for most users)
• Animated GIF
• MNG
• Individual frames: start, middle, end

Bers Slices of P_1,1 (Jan 2002)

The images below show limit sets of complex projective structures on punctured tori. Animations show how the limit set changes as the projective structure is deformed while leaving the underlying complex structure unchanged (i.e. within a Bers slice).

Implicit in these considerations is the discreteness of the holonomy representation of the projective structures. The set of all projective structures with discrete holonomy is a complex and interesting parameter space.

Animations

• Exotic structures approaching a cusp of small complexity

These limit sets were drawn using kleinian, a program by David Wright; the accompanying Bers slice images were created using a program developed by Yohei Komori, Toshiyuki Sugawa, Masaaki Wada, and Yasushi Yamashita.

Maskit's Embedding of T_1,1 (Jun 2001)

Starting with the thrice-punctured sphere, given its unique complete hyperbolic metric, one can construct punctured tori by cutting off horoball neighborhoods of two cusps and gluing their circular boundaries by some möbius transformation. The result has a natural complex projective (möbius) structure, whose underlying conformal structure can be uniformized to a hyperbolic structure.

Pictures below are of two types. Some show just the limit set of the holonomy group of the complex projective structure on the punctured torus. Others also include the lift to the universal cover of the horocycle gluing locus.

• typical example
  • limit set & horocycles - gif ps
  • limit set - gif ps
• big-horocycle limit
  • limit set & horocycles - gif ps
  • limit set - gif ps
• glued with a (small) twist
  • limit set & horocycles - gif ps
  • limit set - gif ps
• twisted limit
  • limit set & horocycles - gif ps
  • limit set - gif ps
• half-twist limit
  • limit set & horocycles - gif ps
  • limit set - gif ps
• twisting animation - warning: animated gifs may take a while to download!
  • limit set & horocycles - mpeg gif
  • limit set - mpeg gif
• twisting animation in the half-plane
  • limit set & horocycles - mpeg gif
  • limit set - mpeg gif
• growing horocycle animation
  • limit set & horocycles - mpeg gif
  • limit set - mpeg gif

Generalization: This construction extends to a method for constructing holomorphic disks in Teichmüller spaces from surfaces with distinguished pairs of cusps.

These pictures and animations were created using mathematica and lim, a program written by Curt McMullen.