Growth functions

Summary

We give a survey of the use of growth functions in algebra. In particular, we define Gelfand–Kirillov dimension and give an overview of some of the main results about this dimension, including Bergman’s gap theorem, the solution of the Artin–Stafford conjecture by Smoktunowicz, and the characterization of groups of polynomially bounded growth by Gromov. In addition, we give a summary of the main ideas employed in the proof of Gromov’s theorem and discuss the work of Lenagan and Smoktunowicz, which gives a counterexample to Kurosh’s conjecture with polynomially bounded growth.

The notion of growth is a fundamental object of study in the theory of groups and algebras, due to its utility in answering many basic questions in these fields. The concept of growth was introduced by Gelfand and Kirillov [1966] for algebras and by Milnor [1968] for groups, who showed that there is a strong relation between the growth of the fundamental group of a Riemannian manifold and its curvature. After the seminal works of Gelfand and Kirillov and of Milnor, the study of growth continued and many important advances were made. In particular, Borho and Kraft [1976] further developed the theory of growth in algebras, giving a systematic study of the theory of Gelfand–Kirillov dimension. In addition to this, Milnor [1968] and Wolf [1968] gave a complete characterization of solvable groups with polynomially bounded growth (see Section 2 for relevant definitions).

The reason for the importance of Gelfand–Kirillov dimension, Gelfand– Kirillov transcendence degree, and corresponding notions in the theory of groups is that it serves as a natural noncommutative analogue of Krull dimension (resp. transcendence degree) and thus provides a suitable notion of dimension for noncommutative algebras. Indeed, the first application of Gelfand–Kirillov dimension was to show that the quotient division algebras of the m-th and nth Weyl algebras are isomorphic if and only if m = n, by showing that their transcendence degrees differed when n ≠ m. Since this initial application, the theory of growth has expanded considerably and this notion now plays a fundamental role in both geometric group theory and noncommutative projective geometry, where it serves as a natural notion of dimension.