Schreier's type formulae and two scales for growth of Lie algebras and groups

TL;DR

This paper surveys how Schreier's formulae and generating functions are used to analyze the growth and codimension growth of free Lie algebras and groups, providing explicit formulas and asymptotic results.

Contribution

It introduces two new scales for growth in Lie algebras and groups based on explicit generating function formulas and asymptotic analysis.

Findings

Explicit formulas for growth generating functions of free solvable Lie algebras.

Asymptotic growth estimates derived from these formulas.

Introduction of two growth scales for Lie algebra growth types.

Abstract

Let $G$ be a free group of rank $n$ and $H\subset G$ its subgroup of finite index. Then $H$ is also a free group and the rank $m$ of $H$ is determined by Schreier's formula $m-1=(n-1)\cdot|G:H|.$ Any subalgebra of a free Lie algebra is also free. But a straightforward analogue of Schreier's formula for free Lie algebras does not exist, because any subalgebra of finite codimension has an infinite number of generators. But the appropriate Schreier's formula for free Lie algebras exists in terms of formal power series. There exists also a version in terms of exponential generating functions. This is a survey on how these formulas are applied to study 1) growth of finitely generated Lie algebras and groups and 2) the codimension growth of varieties of Lie algebras. First, these formulae allow to specify explicit formulas for generating functions of respective types for free solvable (or more generally, polynilpotent) Lie algebras. Second, these explicit formulas for generating functions are used to derive asymptotic for these two types of the growth. These results can be viewed as analogues of the Witt formula for free Lie algebras and groups. In case of Lie algebras, we obtain two scales for respective types of growth. We also shortly mention the situation on growth for other types of linear algebras.