Uniform Growth of Polycyclic Groups

Abstract

The Milnor-Wolf Theorem characterizes the finitely generated solvable groups which have exponential growth; a finitely generated solvable group has exponential growth iff it is not virtually nilpotent. Wolf showed that a finitely generated nilpotent by finite group has polynomial growth; then extended this by proving that polycyclic groups which are not virtually nilpotent have expontial growth, [8]. On the other hand, Milnor, [5], showed that finitely generated solvable groups which are not polycyclic have exponential growth. In both approaches exponential growth can be deduced from the existence of a free semigroup, [1, 6]. In this article we elaborate on these results by proving that the growth rate of a polycyclic group Γ of exponential growth is uniformly exponential. This means that base of the rate of exponential growth β(S,Γ) is bounded away from 1, independent of the set of generators, S; that is, there is a constant β(Γ) so that β(S,Γ) ≥ β(Γ) > 1 for any finite generating set. The growth rate is also related to the spectral radius μ(S,G) of the random walk on the Cayley graph, with the given set of generators, [3]. The exponential polycyclic groups are an important class of groups for resolving the question of whether or not exponential growth is the same as uniform exponential growth, since they are in a sense very close to groups of polynomial growth. The ideas used here for polycylic groups take advantage of their linear and arithmetic properties. These may be important tools for proving uniform growth for other classes of groups. Other methods for proving uniform growth take advantage of special properties of presentations; for example, an excess of the number of generators over relations by at least 2 ensures the existence of a subgroup of finite index which maps onto a non-abelian free group. Here, of course we can not map to free groups, but we are in a sense able to map to a non-abelian free semigroup.