Projective Structures on a Genus Two Surface

A genus 2 surface can be obtained from an octagon by making the identifications above. To make this a geometric manifold, the vertex must have a total angle of 2p , so we need an octagon with all 45 degree angles.

An octagon in the upper half plane model of the Hyperbolic plane. This octagon has a 45 degree angle at all of its vertices.

The same hyperbolic octagon in the poincare unit disk model. This shows the identification maps used to obtain the hyperbolic structure. These maps are all hyperbolic.
Animated View
By iterating the maps above, we obtain this picture for the developing map of a hyperbolic structure on the genus 2 surface.

Deform the hyperbolic map A to a loxodromic map in SL(2,C). This causes a stretching to the fundamental domain.

Small Deformation
Animation View
This is a deformation of the above structure by making a small deformation to A. This is for q Î (0 , .4 p )
Large Deformation
Animated View
This is for q Î (.7p , p )
Animation over range where image is discrete

Real projective deformation
A Twist Deformation

A prettier version of these pictures appeared in Notices with an article by William Goldman titled What is...a Projective Structure?