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UNIFORM GROWTH OF POLYCYCLIC GROUPS ROGER C. ALPERIN
 

Summary: UNIFORM GROWTH OF POLYCYCLIC GROUPS
ROGER C. ALPERIN
1. Introduction
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 nilpo-
tent 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 of exponential growth is uniformly exponential. This means that
base of the rate of exponential growth (S, ) is bounded away from 1, independent
of the set of generators, S; that is, there is a constant () so that (S, ) () >
1 for any finite generating set. The growth rate is also related to the spectral radius
(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 exponen-

  

Source: Alperin, Roger C. - Department of Mathematics, San Jose State University

 

Collections: Mathematics