This project is concerned with a variety of problems involving the action of groups on graphs related to the so-called amalgam method. The amalgam method refers to the family of results and techniques which have been developed to examine the structure of vertex stabilizers, inspired originally by Goldschmidt's theorem on trivalent graphs. The principal investigator intends to extend these methods, to apply them to the classification of sporadic simple groups, to examine the connections between amalgam-type geometric situations and group presentations and, in general, to investigate the extent to which the structure of vertex stabilizers determines the global structure of a graph. This research will focus on the area of mathematics which bridges graph theory and group theory. This blending of combinatorics and algebra has been a very fruitful endeavor enhancing both subjects.
