Mathematical Sciences: Geometric Group Theory and Topology

本程序不仅研究有限生成群的整类生长函数的不变量，而且研究该类内的特定生长函数的不变量。我们的目标是扩展已知生长函数总是有理的群的类别，并进一步研究几何群可能出现的生长函数。试图表明生长函数的合理性是独立于发电机组的，这将是坎农作品的精神所在。对发生的生长函数的研究将继续Floyd和Plotnick在平面群上的工作。将特别注意生长函数的互易性以及当值为1时欧拉特性的互易性。关于几何群的许多令人惊讶和容易陈述的事实已经被经验揭示，现在是时候给这个领域带来一些秩序了。

This program is mainly a study of invariants of not only the entire class of growth functions of a finitely generated group but also particular growth functions within that class. The goals are to extend the class of groups whose growth functions are known to be always rational, and to study further the growth functions that can occur for geometric groups. The attempt to show that rationality of the growth functions is independent of the generating set will be in the spirit of Cannon's work. The study of the growth functions that occur will continue the work of Floyd and Plotnick on planar groups. Special attention will be paid to reciprocity of the growth functions and to when the value at 1 is the reciprocal of the Euler characteristic. Many surprising and easily stated facts about geometric groups have been uncovered empirically, and it is time to bring some order into the area.