Fricke room

In mathematics, the Fricke space (after Robert Fricke ) describes a module space whose objects are hyperbolic metrics on a closed surface . These objects concern the curvature of Riemannian manifolds . Equivalent is a modular space of the discrete , true representations from the fundamental group of the surface to the isometric group of the hyperbolic plane .

The large Riemann mapping theorem (uniformization theorem) shows that there is a unique hyperbolic metric in every equivalence class of Riemann surfaces by gender . The comparable Teichmüller room actually deals with the modular room of Riemannian surfaces, the Fricke room stands for the module room of hyperbolic metrics. ${\displaystyle \geq 2}$

Area of ​​gender 2

Coordinates

The Fricke space of a surface of the gender is -dimensional and homeomorphic to the open unit sphere in . ${\displaystyle {\mathcal {F}}(S_{g})}$ ${\displaystyle S_{g}}$ ${\displaystyle g\geq 2}$ ${\displaystyle (6g-6)}$ ${\displaystyle \mathbb {R} ^{6g-6}}$

The Fennel-Nielsen coordinates provide a possible parameterization using real parameters . Other coordination results from the identification with the Teichmüller room. ${\displaystyle 6g-6}$

In more modern approaches, the Fricke room is often identified with a component of the character variety , namely the component that contains the characters of all discrete , faithful representations . (Each hyperbolic metric corresponds to its monodromic representation.) The following coordinates, which can be traced back to Fricke, are called Fricke coordinates. ${\displaystyle X(\pi _{1}S_{g},PSL(2,\mathbb {R} ))}$

Fricke coordinates . Be the canonical presentation of the surface group and a discrete, faithful representation. Then are for ${\displaystyle \pi _{1}S_{g}=\langle \alpha _{1},\beta _{1},\ldots ,\alpha _{g},\beta _{g}\mid \Pi _{i=1}^{g}\left[\alpha _{i},\beta _{i}\right]\rangle }$ ${\displaystyle \rho \colon \pi _{1}S_{g}\to PSL(2,\mathbb {R} )}$ ${\displaystyle i=1,\ldots ,g}$

${\displaystyle \rho (\alpha _{i})=\left({\begin{array}{cc}a_{i}&b_{i}\\c_{i}&d_{i}\end{array}}\right),\rho (\beta _{i})=\left({\begin{array}{cc}a_{i}^{\prime }&b_{i}^{\prime }\\c_{i}^{\prime }&d_{i}^{\prime }\end{array}}\right)}$

Equivalence classes of matrices, where we o. B. d. A. can accept. The parameters of the Fricke space are then ${\displaystyle c_{i}>0,c_{i}^{\prime }>0}$ ${\displaystyle 6g-6}$

${\displaystyle (a_{1},c_{1},d_{1},a_{1}^{\prime },c_{1}^{\prime },d_{1}^{\prime },\ldots ,a_{g-1},c_{g-1},d_{g-1},a_{g-1}^{\prime },c_{g-1}^{\prime },d_{g-1}^{\prime })}$.

The uniformization set identifies the Teichmüller space with the Fricke space and, in particular, the Fricke coordinates also provide coordinates on the Teichmüller space. However, in this way one does not get the complex structure in the Teichmüller space, which was first explicitly coordinated by Teichmüller.

Areas with a border

A pair of pants is bordered by 3 closed curves.

For surfaces with a border , the Fricke space is defined as the module space of the marked hyperbolic metrics with a geodetic border modulo border-retaining isotopes.

Examples

trousers

The Fricke space of the trousers is parameterized by the traces of the three boundary curves for the upscale monodrome display . (Let these be oriented so that .) With these coordinates is the quotient of ${\displaystyle H=S_{0,3}}$ ${\displaystyle x=Tr(\rho (A)),y=Tr(\rho (B)),z=Tr(\rho (C))}$ ${\displaystyle A,B,C}$ ${\displaystyle SL(2,\mathbb {R} )}$ ${\displaystyle \rho \colon \pi _{1}H\to PSL(2,\mathbb {R} )}$ ${\displaystyle ABC=1\in \pi _{1}H}$ ${\displaystyle {\mathcal {F}}(S_{0,3})}$

${\displaystyle (-\infty ,-2)^{3}\bigcup (-\infty ,-2)\times (2,\infty )^{2}\bigcup (2,\infty )\times (-\infty ,-2)\times (2,\infty )\bigcup (2,\infty )^{2}\times (-\infty ,-2)\subset \mathbb {R} ^{3}}$

(whereby the 4 related components come about through the different elevations of the monodrome representation from to ) under the effect of , it can therefore be identified with. ${\displaystyle PSL(2,\mathbb {R} )}$ ${\displaystyle SL(2,\mathbb {R} )}$ ${\displaystyle Hom(\pi _{1}H,\pm 1)}$ ${\displaystyle (-\infty ,-2)^{3}}$

Dotted torus

The Fricke space of the dotted torus is parameterized by the traces , where the longitude and the meridian of the torus are designated. With these coordinates is the quotient of ${\displaystyle T=S_{1,1}}$ ${\displaystyle x=Tr(\rho (L)),y=Tr(\rho (M)),z=Tr(\rho (LM))}$ ${\displaystyle L}$ ${\displaystyle M}$ ${\displaystyle {\mathcal {F}}(S_{1,1})}$

${\displaystyle \left\{(x,y,z)\in \mathbb {R} ^{3}\colon x^{2}+y^{2}+z^{2}-xyz\leq 0\right\}\subset \mathbb {R} ^{3}}$

under the effect of , so he can be identified with. ${\displaystyle Hom(\pi _{1}T,\pm 1)}$ ${\displaystyle \left\{(x,y,z)\in (2,\infty )^{3}\colon x^{2}+y^{2}+z^{2}-xyz\leq 0\right\}}$

literature

• Fricke-Klein: Lectures on the theory of automorphic functions . Volume I: The group theoretical basics . Teubner, Leipzig 1897; Volume II: The functional theory versions and the applications . 1912.
• Imayoshi-Taniguchi: An introduction to Teichmüller spaces. Translated and revised from the Japanese by the authors. Springer-Verlag, Tokyo 1992, ISBN 4-431-70088-9 , chapter 2.5
• Goldman: Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. In: Handbook of Teichmüller theory . Vol. II, pp. 611-684. In: IRMA Lect. Math. Theor. Phys. , 13, Eur. Math. Soc., Zurich 2009, arxiv : 0901.1404v1 .

Individual evidence

1. ↑ Chapter 4.3 in Goldman, op.cit.
2. ↑ Chapter 4.4 in Goldman, op.cit.