How do algebras grow?

Gromov proved in 1981 that finitely generated groups of polynomial growth are virtually nilpotent. The Grigorchuk group and far reaching generalizations constructed upon it provide examples of groups with intermediate (namely, super-polynomial but subexponential) growth.

For finitely generated associative algebras, a much wider class of growth functions is possible. In particular, many functions, including intermediate growth functions and oscillating functions, are realizable as growth functions of algebras (Belov, Bartholdi-Smoktunowicz, others); nil algebras (which are analogs of Burnside groups) of polynomial growth exist (Lenagan-Smoktunowicz); using matrix wreath products, many intermediate growth functions can be realized in important classes of algebras, including nil rings (Zelmanov); any `sufficiently regular' growth function which is more rapid than $n^{\log n}$ is the growth function of a simple algebra (simple groups with intermediate growth were only recently constructed), and many intermediate growth function more rapid than $\exp(\sqrt(n))$ are the growth functions of finitely generated domains, yielding quotient division rings with interesting properties (in particular, amenable but of infinite GK-transcendence degree).

We present new results on possible (and impossible) growth functions of important classes of algebras, answer several open questions posed by experts in the area and survey the main open problems in the field.