In the first, mostly expository, part of this paper, a graded Lie algebra is
associated to every group G given with an N-series of subgroups. The
asymptotics of the Poincare series of this algebra give estimates on the growth
of the group G. This establishes the existence of a gap between polynomial
growth and growth of type en​ in the class of residually-p groups,
and gives examples of finitely generated p-groups of uniformly exponential
growth. In the second part, we produce two examples of groups of finite width
and describe their Lie algebras, introducing a notion of Cayley graph for
graded Lie algebras. We compute explicitly their lower central and dimensional
series, and outline a general method applicable to some other groups from the
class of branch groups. These examples produce counterexamples to a conjecture
on the structure of just-infinite groups of finite width.Comment: to appear in volume 275 of the London Mathematical Society Lecture
Notes serie