 
Summary: SOLVABLE GROUPS OF EXPONENTIAL
GROWTH AND HNN EXTENSIONS
Roger C. Alperin
An extraordinary theorem of Gromov, [Gv], characterizes the finitely gen
erated groups of polynomial growth; a group has polynomial growth iff it
is nilpotent by finite. This theorem went a long way from its roots in the
class of discrete subgroups of solvable Lie groups. Wolf, [W], proved that a
polycyclic group of polynomial growth is nilpotent by finite. This theorem
is primarily about linear groups and another proof by Tits appears as an
appendix to Gromov's paper. In fact if G is torsion free polycyclic and not
nilpotent then Rosenblatt, [R], constructs a free abelian by cyclic group in
G, in which the automorphism is expanding and thereby constructs a free
semigroup. The converse of this, that a finitely generated nilpotent by finite
group is of polynomial growth is relatively easy; but in fact one can also use
the nilpotent length to estimate the degree of polynomial growth as shown
by Guivarc'h, [Gh], Bass, [Bs], and Wolf, [W]. The theorem of Milnor, [M],
on the other hand shows that a finitely generated solvable group, not of ex
ponential growth, is polycyclic. Rosenblatt's version of this, [Rt], is that a
finitely generated solvable group without a two generator free subsemigroup
is polycyclic. We give another version of Milnor's theorem using the HNN
