Bers cut

Bers section of the (2-dimensional) Teichmüller space of the dotted torus .

In mathematics are Bers cuts (Engl. Bers slices ) the images of certain embeddings of Teichmüller space into the space of quasi fox rule groups . They often have a fractal shape.

Bers cuts and the skinning map defined with their help play a role in many proofs of low-dimensional geometry, for example in Thurston's proof of the geometrization of hook manifolds .

construction

Let be a closed surface and the associated surface group . With is called the Teichmüller space of and with the space of all those homomorphisms whose image is a Quasi-Fuchsian group. ${\displaystyle S}$ ${\displaystyle \Gamma }$ ${\displaystyle {\mathcal {T}}(\Gamma )}$ ${\displaystyle S}$ ${\displaystyle QF(\Gamma )}$ ${\displaystyle \Gamma \to PSL(2,\mathbb {C} )}$

Simultaneous uniformization gives a bijection

${\displaystyle QF(\Gamma )={\mathcal {T}}(\Gamma )\times {\mathcal {T}}(\Gamma )}$.

For a fixed one is called that ${\displaystyle X\in {\mathcal {T}}(\Gamma )}$

${\displaystyle {\mathcal {T}}(\Gamma )\times \left\{X\right\}}$

corresponding subset of the ( belonging) Bers cut. ${\displaystyle QF(\Gamma )}$ ${\displaystyle X}$

Bers compactification

By embedding the marked hyperbolic manifolds in the module space , homotopy-equivalent to , one can embed the Bers cut in this module space. His picture is relatively compact , his compactification is called the Bers compactification of the Teichmüller area. ${\displaystyle QF(\Gamma )}$ ${\displaystyle AH(\Gamma )}$ ${\displaystyle S\times \left[0,1\right]}$

Kerckhoff and Thurston have shown that the effect of the mapping class group on the Bers compactification of the Teichmüller space is not continuous. In particular, the Bers compactification does not agree with Thurston's compactification of the Teichmüller area.

Skinning map

For a geometrically finite hyperbolic 3-manifold its conformal boundary gives a point in the Teichmüller space . On the other hand, the image is from a quasi-fox group and thus gives a point in . The figure so defined ${\displaystyle M}$ ${\displaystyle {\mathcal {T}}(\partial M)}$ ${\displaystyle \Gamma }$ ${\displaystyle \pi _{1}\partial M\to \pi _{1}M}$ ${\displaystyle QF(\Gamma )={\mathcal {T}}(\Gamma )\times {\mathcal {T}}(\Gamma )}$

${\displaystyle {\mathcal {T}}(\Gamma )\to {\mathcal {T}}(\Gamma )\times {\mathcal {T}}(\Gamma )}$

is the identity map on the first component , so it is of the form

${\displaystyle X\to (X,\sigma _{M}(X))}$.

The image

${\displaystyle \sigma _{M}\colon {\mathcal {T}}(\Gamma )\to {\mathcal {T}}(\Gamma )}$

is called skinning map .

Thurston's Bounded Image Theorem states that the image of the skinning map has a finite diameter. It is an essential step in the proof of the hyperbolization of hook manifolds.

literature

• Lipman Bers : Uniformization, moduli, and Kleinian groups. Bull. London Math. Soc. 4: 257-300 (1972). pdf
• Komori-Sugawa: Bers embedding of the Teichmüller space of a once-punctured torus. Conform. Geom. Dyn. 8 (2004), 115-142 pdf
• Komori-Sugawa-Wada-Yamashita: Drawing Bers embeddings of the Teichmüller space of once-punctured tori. Experiment. Math. 15 (2006), no. 1, 51-60. pdf