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Algebra, topology, group theory
My main interests are algebra and topology. I especially enjoy using topological methods to study infinite discrete groups. In some of my early work, for instance, I studied Euler characteristics of groups. I obtained formulas relating the Euler characteristic (a topological concept) to purely algebraic properties of groups. When applied in special cases, these formulas unexpectedly led to new results in algebraic number theory. Later, I found topological methods for studying two interesting families of groups: infinite simple groups, and groups that can be presented by means of a complete rewriting system. More recently, my work has had unexpected applications to probability theory: I have used methods of algebra and topology to analyze an interesting family of random walks.
My main current interest is the theory of buildings.
Euler Characteristics of Discrete Groups and G-Spaces, Invent. Math. 27 (1974), 229–264.
Cohomology of Groups, Graduate Texts in Mathematics 87, Springer-Verlag, New York, 1982.
Semigroups, Rings and Markov Chains, J. Theoret. Probab. 13 (2000), 871–938.
Forest diagrams for elements of Thompson’s group F (with James Belk), Internat. J. Algebra Comput. 15 (2005), 815–850.
Buildings: Theory and Applications (with Peter Abramenko), Graduate Texts in Mathematics 248, Springer, New York, 2008.