| The first part of our work is a statistical and geometric study of properties of Thompson's Group F. We enumerate the number of elements of F which are represented by a reduced pair of n-caret trees, and give asymptotic estimates. We also discuss the effects on word length and number of carets of right multiplication by a standard generator x0 or x1. We enumerate the average number of carets along the left edge of an n-caret tree, and use an Euler transformation to snake some conjectures relating to right multiplication by a generator. We describe a computer algorithm which produces Fordham's Table, and discuss using the computer algorithm to find a corresponding Fordham's Table for different generating sets for F. We expound upon the work of Cleary and Taback by completely classifying dead end elements of Thompson's group, and use the classification to discuss the spread of dead end elements and describe interesting elements we call deep roots. We discuss how deep roots may aid in answering the amenability problem for Thompson's group.; The second part of our work deals with random facet pairings of simplexes. We show that a random endpoint pairings of segments most often results in a disconnected one-manifold, and relate this to a game called "The Human Knot." When the dimension of the simplexes is greater than 1, however, a random facet pairing most often results in a connected pseudo manifold. This result can be stated in terms of graph theory as follows. Most regular multi graphs are connected, long as the common valence is at least three. |
