The principal investigator proposes to study a variety of problems from both a combinatorial and a geometric viewpoint. Many of these problems concern Thompson's group, a fascinating group which appears in varied branches of mathematics, from logic to algebra to homotopy theory and group theory. The principal investigator plans to explore questions ranging from amenability to quasi-isometries of this group. Additionally, the investigator proposes to extend prior work on quasi-isometries of lattices, working with colleagues to finish a quasi-isometry classification of s-arithmetic lattices. This would complete a project worked on by many geometric group theorists that has led to the introduction of interesting new techniques. Finally, the principal investigator plans several projects concerning convexity properties of groups. While almost convexity is well understood, there are few examples of groups which are not almost convex but satisfy weaker convexity conditions. She also seeks to expand the list of classes of groups which are known to be almost convex.
The principal investigator proposes several projects in the area of group theory. A group is a mathematical structure, often introduced in the context of symmetries of a particular object. Certain properties of a set of symmetries can be extrapolated to describe abstract mathematical sets. In her research, the investigator studies groups as geometric objects, using the geometry to give greater insight into abstract properties. The investigator studies closely one particular group, named Thompson's group after the researcher who first defined it. Thompson's group can be understood geometrically using pairs of binary trees, an approach which relates this group to questions in theoretical computer science. Thus it provides an interesting interdisciplinary application of abstract mathematics. Additionally, Thompson's group is a favorite example or counter-example to many questions in mathematics. There are many open questions about this group which interest an international group of mathematicians. In particular, the geometry of this group is not well understood, a question which interests the investigator greatly. Mathematicians have a standard way of describing a "picture" of a group, which is sometimes difficult to construct. For certain groups, it is possible to ask a computer to construct this picture. The principal investigator is interested in exploring an algorithmic property of some groups, called almost convexity, which allows computer construction of the group.
