## Convex hulls of limits sets of hyperbolic and AdS manifolds

**Goal:**
The ultimate goal of the project is to visualize limit sets of "quasifuchsian
hyperbolic manifolds" and "globally hyperbolic AdS manifolds". Each
of those objects is determined by a homeomorphism from a discrete
group *G*, defined by *2g* generators and one relation only,
to the group *SO(3,1)*, respectively *SO(2,2)*, of *4x4*
matrices preserving a symmetric bilinear form of signature *(3,1)*,
respectively *(2,2)*. Examples of such homeomorphisms can be
computed in a number of ways.

Once those homeomorphism are computed, one should be able to
compute and visualize their "limit sets", which are Jordan curves
(typically non-regular, and fractal) either in the sphere or in
a one-sheeted hyperboloid in *R*^{3}. One way to
visualize those limit sets is by computing their convex hull, which
have remarkable geometric properties.

Visualization could be achieved either by producing "movies" of
how limit sets vary when one varies some parameters of the homeomorphims
defining them, or by printing 3d models of their convex hulls for different
values of some parameters. It should be interesting to compare the results
in the "hyperbolic" and the "AdS" situations.

**Schedule:** to be determined.

**Persons involved:**
Advised by Jean-Marc Schlenker, NN.

**Difficulty level:**
medium/advanced.

**Tools:**
programs in python/sage, use of computational tools to compute
the first eigenvalues of the Laplace operator.

**Expected results:**
either disprove the conjecture, or give some evidence that
it might be correct by showing that, up to some approximation,
the first few eigenvalues of the Laplace operator remain
constant in an isometric deformation in a few simple examples.