Polynomial growth in holonomy groups of foliations

Abstract

A notion of growth in finitely generated groups has been introduced by Milnor [8]. One type of growth considered there is called polynomial growth and it has been conjectured that a finitely generated group has polynomial growth if, and only if, it has a nilpotent subgroup of finite index. That groups which have a nilpotent subgroup of finite index have polynomial growth has been shown by Wolf [22] (see also Bass [1]). So far, the converse has been proved for solvable groups (Milnor-Wolf [9, 22]) and for linear groups (Tits [21]). In the present note we show that the conjecture is true for finitely generated groups of ditterentiable germs. There are also related results about groups of homeomorphisms which may be of independent interest. Our interest in groups of germs was motivated by the study of holonomy groups of foliations and we give some applications in that direction. It turns out, for example, that holonomy groups of codimension one, transversely oriented foliations of class C 2 which have polynomial growth, must actually be abelian. These results are applied in the last section to generalize a result of Haefliger concerning analytic foliations of codimension one.