Teichmuller theory

==introduction

==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