Growth of Groups
Eric M. Freden · Princeton University Press eBooks · 2017
This chapter considers the growth of a group. It begins with the case of a group with a given finite generating set, in which the ball B(1, r) of radius r centered at the identity 1 is the set of group elements whose word length is less than or equal to r. Given a group that is fixed and some r greater than or equal to 0, the chapter invokes Gromov's polynomial growth theorem to determine how many group elements are in B(1, r) and how many group elements are in S(1, r). It also explores the use of simple counting methods to compute several growth series before concluding with an overview of cone types, formal languages and context-free grammars, and the DSV method used to compute the growth of grammar productions. The discussion includes exercises and research projects.