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 from the literature:
The blue curve corresponds to the locus where the image of a^{-1}b is
ellipto-parabolic. On its left, this element becomes elliptic and we have
not tried to picture the limit set. On its right, this element is loxodromic.
The cusp of the blue curve corresponds to a unipotent representation,
which is the holonomy representation of a uniformized CR-spherical
spherical CR structure on the 8-knot complement, as described by Deraux-Falbel.
Its limit set is pictured on load below the representation of the character variety.
By a result of Acosta, locally around this point,
the pictured representations are
holonomy representation of a spherical CR-structure on a non-hyperbolic
Dehn surgery on the 8-knot complement. But this representation is not proven
to be uniformizable.
By results of Parker, Wang, along the line of parameters u real, the
representation are discrete and uniformize this same Dehn surgery. In fact,
these representations correspond to (3,3,4)-complex hyperbolic triangular
groups.
By hovering above a green point, you get below 3 snapshots of
the limit set of the associated representation. Indeed, these limit
sets lie in a 3-dimensional sphere (the boundary at infinity of the
complex hyperbolic plane). They are projected to the space R3 by a
stereographic projection. The 3 snapshots are then taken along each
axis.
By clicking on such a point, you will open the access to a 3d
visualization.
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Geometry Viewer
infoadd_circleclose
u = 3
Cartan map
There is still a lot of work to do to build what I would like for this
landscape. This includes - but is not limited to - the following:
Compute approximations of useful invariants (Hausdorf measure...)
Other landscapes for other families of representations...
For most of these tasks, I am aware of a solution... but there are some
difficulties (time, sheer size of the files, my poor JS abilities...).