Robert continued his project on configurations on hyperbolic surfaces (a pair of pants) that are realizable as geodesics. He attempted to show that by adding one more round in the configuration to one of the legs, similar results exist regarding realizable configurations.
However, the sums of exterior and interior angles are insufficient to produce a contradiction on the configuration suspected to be unrealizable. By taking into account the areas of various shapes as well, and using the formula
-Area+Exterior Angles=2pi*Euler Characteristic
It may be possible to find a contradiction.
Jingyu formalized the relevant definitions and suggested an approach to her project about self-intersection and mutual-intersection of multiwords. She defined cyclic reduced words (abbreviated as c.r.w.), intersection of c.r.w. as the number of minimal self-intersection of a representative of the homotopy class described by c.r.w., multi-words as a list of c.r.w., as well as self- and mutual-intersection of multiwords similar to the definition of intersection of c.r.w. She was to find a representative of a multiword such that the sum of its self- and mutual intersection is minimum. She realized that it is attained when both the self- and mutual-intersection are minimum, according to some existing theorems, and proposed an algorithm to find the representative.
She defined bigon as a shape of two vertices and two arcs connected to the vertices. A proper bigon is a bigon that cannot be eliminated through surgery, and a bigon that is not proper is called an improper bigon. The related theorem then states that the total intersection is minimal when all improper bigons are eliminated through surgery. This theorem implies that minimal total intersection is attained when self- and mutual-intersection are both minimum. It then follows that for a give configuration, we can reduce bigons from the outmost to the inmost, and end up with the desired configuration.
Keren's project is about finding the distribution of geometric length of a geodesic for a certain combinatorial length in a given hyperbolic surface, and the range of the geometric length to combinatorial length ratio. A hyperbolic surface can be projected to a Poincare disk model or an upper half-plane. In the Ppincare model, a surface is represented by a surface word, and the combinatorial length of a geodesic is the number of letters in the word of the curve. In the upper half-plane model, the actual geometric length can be computed from the trace of the transform matrices.
In the upper half-plane model, the x-coordinates of an axis of transformation of a geodesic can be determined by the slopes of eigenvectors of that transform matrix, while the length of the axis be related to the trace of the matrix. For a certain word of a geodesic, its length has the following relationship with the multiplication of all transform matices (denoted by M):
cosh(l/2)=(1/2)Tr(M)
Therefore we can calculate the geometric length of any given geodesic and study its distribution.
Also, Dr. Sullivan has talked about universal covering space.
He started with the fact that two simply connected geodesically complete surfaces of constant curvature -1 are isometric. This is also true for zero and curvature +1. Simply connected means the surface can shrink to a point, and geodesically complete means a curve never leaves the surface. This fact can be proved by the theory of covering spaces.
A corollary of this fact suggests that for any connected surface of -1 curvature and "complete", then its universal covering space is isometric to the noneuclidian plane, and its covering symmetry group becomes a group of noneuclidian motions. This group is called the Poincare group or fundamental group.
For any topological space X (with tiny assumption that X is path connected with locally "unique" paths up to deformation), choose a point x in X as the base point, then all the ways of going to a point from the base point can be drawn as a stack of points above the space. If you slide a pint locally, there is a cononical bijection between the ways. Moving the base point along a closed path produces a permutation of the stack of of the base point. This is the original definition of the fundamental group by Poincare.
Group of transformations of universal covering space is given as follows:
Given (X,x), we construct (X',x') with a map Pi: (X',x')->(X,x). The group of continuous bijections to fixed points Gamma= Pi_1(X,x) such that X'/Gamma=X and X' is simply connected.
(BTW X' and x' should actually be X- and x-tilde, I cannot type it)
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