Mathematical Sciences: Growth Functions and Geometry of Groups

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Plotnick plans to investigate relationships between combinatorial and geometric properties of certain groups, as reflected in their growth functions. The main objects of study will be discrete groups such as hyperbolic groups, especially in dimensions 2 and 3, and Coxeter groups. Methods used will be combinatorial (attempting to generalize work of Floyd/Plotnick in dimension 2), geometric (following Cannon), and homological (applying the theory of Kac-Moody Lie algebras, including Bruhat decompositions and flag manifolds, to certain topologically interesting classes of Coxeter groups). In particular, he seeks to understand certain symmetry properties enjoyed by these growth functions, and also the connection between growth functions and Euler characteristics of certain discrete groups.