Solvable Groups of Exponential Growth and Hnn Extensions

An extraordinary theorem of Gromov, [4], characterizes the finitely generated 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, [11], 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, [9], 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, [5], Bass, [1], and Wolf, [11]. The theorem of Milnor, [8], on the other hand shows that a finitely generated solvable group, not of exponential growth, is polycyclic. Rosenblatt’s version of this, [9], 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 construction. A consequence is that a finitely generated solvable group G has the ERF (extended residually finite) property iff G is polycyclic. We briefly review the HNN construction. Generally, the HNN has a given base group B and two subgroups, H1, H2 together with an (external) element t of infinite order which conjugates H1 to H2, Γ =< B, t | tH1t −1 = H2 > . For solvable groups, a good example, is the group Γ1 =< a, t | tat −1 = t > . Many one relator groups have HNN decompositions; for example, consider Γ2 =< a, t | a = [tat , tat] > . This is, in fact, the HNN extension with base H =< a0, a1, a2 | a0 = [a1, a2] > and free subgroups F1 =< a0 = a, a1 = tat −1 >, F2 =< tat, a2 = t at > amalgamated, so that Γ2 =< H, t | tF1t −1 = F2 >. These HNN constructions are ascending in the sense that a conjugation of the subgroup ascends or gets strictly larger in Γ. We say it is ascending with base B if Γ is generated by B and t, so that