turkmath.org

Let A be an (not necessarily associative) algebra over a field k generated by a finite set X. The growth function of A with respect to X is defined as the dimension of the subspace of A spanned by all monomials on X of length at most n. The asymptotic behavior of this function does not depend on the generating set X and it is called the growth of A. The growth rate is a widely studied invariant for finitely generated algebraic structures such as algebras, groups, semigroups. This talk will be a survey of history, open problems and some new results around this notion. The main focus will be on finitely presented algebras whose growth is intermediate between polynomial and exponential.