The principal investigator proposes proposes research in two main areas which lie at the intersection of geometric group theory and topology. Ever since Gromov's celebrated theorem on polynomial growth which related a purely algebraic property with the large scale geometry of a group, researchers have been interested in what one can learn from studying finitely generated infinite groups via the geometry of their associated Cayley graphs. One line of proposed research seeks to continue this program for certain families of self-similar groups and some generalizations of these groups. The second area of proposed research arises from topological fixed point theory and concerns groups for which every automorphism has infinite Reidemeister number, a property which has other topological consequences. The principal investigator and collaborator P. Wong are interested in a geometric interpretation of this number for the families of groups studied, building on their previous work. The self-similar groups and generalizations studied by the principal investigator include the family of groups whose Cayley graphs (with respect to an appropriate generating set) are Diestel-Leader graphs, or horocyclic products of trees, generalizing geometrically the classical lamplighter groups.
The principal investigator proposes several projects in the area of geometric group theory. This field studies fundamental mathematical objects called groups from a geometric point of view. Symmetry groups are a typical example of a mathematical group. The study of symmetry groups was used, for example, to discern that the shape of DNA was a double helix. The proposed work of the principal investigator is related to self-similar groups; the appearance of self-similarity in mathematics and nature is prevalent, from colorful fractals to a study of the Maine coastline. In mathematics, self similar groups are defined relative to their action on an infinite tree; trees, or graphs without circuits, are fundamental objects in the study of combinatorics and in computer science. In computer science, for example, trees are used to implement efficient search algorithms. A self similar group has instructions for a "rearrangement" of a tree in a particular way. The research questions above all relate to the interaction between the geometric and algebraic structure attached to such a group, and what one can learn about algebraic properties by studying geometry, and vice versa.
