Teichmuller theory

 

 

review of hyperbolic geometry

• horocycle
  
  • http://en.wikipedia.org/wiki/Horocycle
  • http://web1.kcn.jp/hp28ah77/us15_horo.htm
• exponentiated hyperbolic distances between horocycles drawn around vertices of a polygon with geodesic sides and cusps at the vertices
• lamination
• shear coordinates
• lambda length
• http://moniker.name/worldmaking/?p=744
• http://orion.math.iastate.edu/dept/thesisarchive/MSCC/OLearyMSCCSS06.pdf
• \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.

 

 

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

 

 

 

history

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=