Lie Methods in Growth of Groups and Groups of Finite Width

TL;DR

This paper explores the connection between Lie algebras and group growth, establishing growth gaps, and provides explicit examples of groups with finite width that challenge existing conjectures.

Contribution

It introduces a method linking graded Lie algebras to group growth, constructs specific finite width groups, and offers counterexamples to conjectures on just-infinite groups.

Findings

Identifies a growth gap between polynomial and exponential types in residually-p groups.

Constructs explicit examples of finite width groups and describes their Lie algebras.

Provides counterexamples to a conjecture on the structure of just-infinite groups of finite width.

Abstract

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 $e^{\sqrt n}$ 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.