Examples of uniform exponential growth in algebras

Abstract

In this paper, we discuss the concept and examples of algebras of uniform exponential growth. We prove that Golod–Shafarevich algebras and group algebras of Golod–Shafarevich groups are of uniform exponential growth. We prove that uniform exponential growth of the universal enveloping algebra of a Lie algebra [Formula: see text] implies uniform exponential growth of [Formula: see text], and conversely should [Formula: see text] be graded by the natural numbers. We prove that a restricted Lie algebra is of uniform exponential growth if and only if its universal enveloping algebra is. We proceed to give several conditions equivalent to the uniform exponential growth of the graded algebra associated to a group algebra filtered by powers of its fundamental ideal.