J. Milnor [Am. Math. Mon. 75, 685-686 (1968)] posed the following problem: “Is it true that the growth function of a finitely generated group must be equivalent either to some power function \(n^k\) or to the exponential function \(2^n\)?” The author announces his main theorem: There exists a continuum of nonisomorphic periodic groups on 2 generators with growth functions that are not equivalent to a power function nor to the exponential function.
A discrete group is said to be amenable if the space of bounded functions on the group admits an invariant mean. The groups of nonexponential growth are amenable. On the other hand any finitely generated infinite periodic group cannot be constructed from the finite and abelian groups by taking subgroups, quotients, group extensions and inductive limits. This is the answer to a question by Day.