Mathematical Sciences: Growth Functions and Geometry of Groups

普洛尼克计划研究某些群的组合和几何性质之间的关系，这反映在它们的增长函数中。主要的研究对象将是离散群，如双曲群，特别是在2维和3维上，以及Coxeter群。所使用的方法将是组合的(试图将Floyd/Plotnick的工作推广到2维)、几何(遵循Cannon)和同调(将Kac-Moody李代数的理论，包括Bruhat分解和旗形，应用于某些拓扑上感兴趣的Coxeter群类)。特别是，他试图理解这些增长函数所享有的某些对称性，以及增长函数与某些离散群的欧拉特征之间的联系。

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.