RUI: Twisted Conjugacy, Reidemeister Number and Thompson's Groups

Bowdoin College, Brunswick ME

Investigators

Abstract

Thompson's group F appears in varied branches of mathematics, from logic to algebra to homotopy theory and group theory. It can be understood either as a finitely or infinitely generated group, with a standard finite generating set. The principal investigator proposes to remove the dependence on this generating set by studying metric properties of the group with respect to arbitrary generating sets. Building on previous work, she will investigate whether this group is automatic, as well as a quasi-isometry classification for the standard generalizations F(p) of F. The remaining work proposed under this grant concerns twisted conjugacy classes and the Reidemeister number of a group automorphism. If f is a selfmap of a compact manifold, then the number of twisted conjugacy classes of the induced map on the fundamental group is called the Reidemeister number R(f). The Reidemeister number provides an upper bound on the Nielsen number N(f) for f, which is difficult to compu te. If the Reidemeister number is infinite for any selfmap of the manifold, this eliminates the possibility of a special type of theorem equating N(f) and R(f). The principal investigator proposes to extend the classes of (fundamental) groups with this property, and build on prior work to see when it is invariant under quasi-isometry. She will also investigate a geometric interpretation of twisted conjugacy classes, and the relationship of this property to subgroup separability. 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 mathematical groups. The study of symmetry groups was used, for example, to discern that the shape of DNA was a double helix. The principal investigator studies closely one particular group, named Thompson's group after the researcher who first defined it. Thompson's group can be understood algebraically, geometrically and analytically, allowing one to study it from differing viewpoints. The geometric perspective equates each group element with a pair of "tree-like" graphs, an approach which relates this group to questions in theoretical computer science. In this way it provides an interesting interdisciplinary application of abstract mathematics, with direct applications to the problem of transforming data to maximize the efficiency of some computer search algorithms. The principal investigator proposes sev eral problems related to understanding the geometry of this group. The remainder of her proposal concerns several problems at the intersection of geometric group theory and topology.