Teichmuller theory

 

 

Teichmuller space of a marked surface

Given marked surface (S,M) , the Teichmuller space T(S,M) is the space of metrics on (S,M) such that

• are hyperbolic  (constant curvature -1)
• have geodesic boundary at boundary of S
• local neighborhood of point on boundary S can be mapped isometrically to neighborhood of a point here on one side of geodesic
• have cusps at points in M

Considered up to diffeomorphism homotopic to identity.

Facts

(1) T(S,M) contractible

(2) T(S,M) is manifold of dimension 6g-6+2p+3b+c where g = genus, p=# of puncture, b = # boundary component, c=# of marked points on boundary

 

 

\def

An ideal triangle in (S,M) is a triangle with vertices at M, whose sides are geodesics.

 

\def

A horocycle at marked point p is a set of points "equidistant" to p. In lift to H^2, looks like circle tangent to boundary at p.

 

\def decorated Teichmuller space \tilde{T}(S,M) is

• a point in T(S,M)
• a choic of horocycle around each cusp from M

\def (Penner) For a arc A in (S,M) and \Sigma\in\tilde{T}(S,M),

the length of A with respect to \Sigma is

l_{\Sigma(A) = length on geodesic representative of A between intersections with horocycles around ends. (negative if 2 horocycles intersect) 

 

The \lambda - length is \lambda_{\Sigma}(A) : = e^{l_{\Sigma}(A)/2}

Note  :  \lambda_{\Sigma}(A) in \mathbb{R}_{> 0 }

 

\prop

In an ideal quadrilateral, the Ptolemy relation holds.

 

\thm (Penner)

For any triangulation (A_{i})

 

 

 

shear coordinate

 

 

 

• Introduction to Teichmüller theory, old and new, Athanase Papadopoulos