Nil algebras, Lie algebras and wreath products with intermediate and oscillating growth

TL;DR

This paper constructs finitely generated nil algebras with customizable growth rates, including oscillating behaviors, and explores their properties, such as Gelfand-Kirillov dimension, answering open questions in algebra.

Contribution

It introduces methods to realize arbitrary growth functions in finitely generated nil algebras and provides examples with oscillating growth and sub-additive Gelfand-Kirillov dimension.

Findings

Any increasing submultiplicative function can be realized as a nil algebra's growth.

Constructed nil algebras with strongly oscillating growth functions.

Provided examples of primitive algebras with sub-additive Gelfand-Kirillov dimension.

Abstract

We construct finitely generated nil algebras with prescribed growth rate. In particular, any increasing submultiplicative function is realized as the growth function of a nil algebra up to a polynomial error term and an arbitrarily slow distortion. We then move on to examples of nil algebras and domains with strongly oscillating growth functions and construct primitive algebras for which the Gelfand-Kirillov dimension is strictly sub-additive with respect to tensor products, thus answering a question raised by Krempa-Okninski and Krause-Lenagan.