Victor Petrogradsky
Ulianovsk University, Russia
Types of growth for finitely generated algebras.
Abstract: We mainly discuss different types of growth for Lie algebras. There are natural examples of finitely generated Lie algebras that have growths between polynomial and exponential. We suggest an hierarchy of types of intermediate growth, which consists of a countable series of functions. In terms of these functions we describe the growth of finitely generated solvable Lie algebras that have a fixed solubility length q and which are free under this condition. We obtain an application of this result to free solvable groups. Namely, we describe an asymptotic behaviour of the lower central series for these groups. Respective generating functions play an important role. They are analytic in the unit circle and we obtain an hierarchy of types of growth for these functions, too. We also observe growths for finitely generated algebras of other types, such as associative algebras, free commuttaive and anticommutative algebras.