Anna Wienhard - A beautiful world beyond hyperbolic geometry Anosov representations and higher Teichmüller spaces

Abstract

I will describe the beautiful world of discrete subgroups of Lie groups of higher rank, taking inspiration from and making connection to hyperbolic geometry.

Talk

Hyperbolic geometry

We look at the Poincaré Scheibe. The isometry group is given by $PSL(2,\mathbb{Z})$. Similarly we look at the Poincare ball model. An other model is the upper half plane model.

Instead of $SL(2,\mathbb{R})$ we think of the Simplectic group $Sp(2,\mathbb{R})$.

We look at hyperbolic surfaces. The Uniformization theorem is mentioned. The Teichmüller-space ist the space of hyperbolic structure. This is homeomorphic to a ball.

Markov numbers

Markov equation: $x^2+y^2+z^2-3xyz=0$ Morkov triple ar einteger solutions of this equation. From them we can construct the Markov tree.

Then they can be used to parametrize hyperbolic structure. First we choose a Triangulation. The Markov triples describe lengths of a triangle as they satisfy the Markov equation.

Hyperbolic 3-manifold

They can be constructed using knot complements and mapping tori. We have the following keywords

• Mostow rigidity
• Kleinsche Gruppe
• Grenzmenge
• Quasi-Fuchsian subgroup

Apollonian circle packing

• Apollonian circle packing

Keywords

Feedback

• Im a big fan of the work by Series, Thurston, Schleimer Maret