US-SPAIN Cooperative Research: Metric Properties of Thompson's Group

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The goal of this project is the understanding of the metric properties of Thompson's groups. Thompson defined a fascinating family of groups, which appear in a wide range of mathematical disciplines, including logic, homotopy theory and the measure theory of discrete groups. Thompson's groups are often the simplest or only known examples of a number of strange group-theoretic phenomena. Despite a great deal of study by a number of researchers, many properties of these groups are poorly understood. This project investigates the metric structure of Thompson's groups by developing a better understanding of the Cayley graphs of these groups. This research builds upon earlier results of the researchers, working together in some cases and separately in others, relating to various metric properties of Thompson's groups.This award supports US-Spanish cooperative research in mathematics. It provides travel support for Sean Cleary of The City College of New York and Jennifer Taback of the University at Albany to visit Jose Burillo of the Universitat Politecnia de Catalunya in Barcelona. This mathematical research lies in the area of group theory. A group is an abstract structure, which can often be visualized as a collection of symmetries of a geometrical object. Some common properties of a set of such symmetries can be made abstract in systematic ways. In this research, groups themselves are studied as geometric objects, and geometric insight can help to understand the algebraic structure of some groups more effectively. This research focuses on a family of groups known as Thompson's groups, after the researcher who first defined them. Thompson's groups can be understood geometrically using pairs of binary trees, an approach that relates this group to questions in theoretical computer science. The broader impacts of this project include these applications to algorithmic questions in computer science related to the shape of binary trees, international involvement and exchange between researchers interested in similar projects, and possible student involvement in research.