Groups with Exponential Subgroup Growth

Abstract

We saw in the last chapter that finitely generated free groups have super-exponential subgroup growth; so for a group to have subgroup growth of merely exponential type is certainly some kind of restriction. Can it be characterized algebraically? This question seems difficult to answer, because the groups with exponential subgroup growth encompass a huge variety of examples. This is not really surprising, because a very mild algebraic condition is in fact sufficient to ensure that the growth is at most exponential: Theorem 3.1 Let T be a finitely generated group. Suppose that there exists a finite group which is not isomorphic to any upper section of T. Then T has at most exponential subgroup growth type.