A Landscape of Limit Sets Deformations

Description and context

The visualisation on this page gives an idea of how Limit sets deform by changing a representation inside the character variety of the 8-knot complement with target group PU(2,1). Here, each point correspond to a (computed) representation of the fundamental group of the 8-knot complement in SU(2,1), as parametrized in Guilloux-Will. It is a group generated by two elements, a and b, of order 3, with the additional property that the product ab has order 4. The parameter u below defines the four parameter of Guilloux-Will. Namely, it sets: z1=z3=1, z2 = u and z4 is the conjugate of z2.

Some information and context from the literature:

R. Alexandre has used similar computations for his paper on redundancy of hyperbolic triangle groups.

The visualization

u = 3
tile snapshot
Cartan map

Contributors, sources and to-do

The source code for the computation has been developed by Raphael Alexandre and Antonin Guilloux; the code for this visualization by Antonin Guilloux and Thi Thu Quyen Nguyen, with the help of V. Cornet and J. Tierny.

The code for the computations and visualization can be found in two git depots: CR-limit-sets, and HypSurfCR. Feel free to ask for access if you want to replicate/contribute.

There is still a lot of work to do to build what we would like for this landscape. This includes - but is not limited to - the following: