Victor Petrogradsky
UnB
Growth of finitely generated Lie algebras and groups.
Abstract: We discuss the growth of finitely generated Lie algebras. Free finitely generated Lie algebras have exponential growth. But there are some natural examples of Lie algebras that have growths between polynomial and exponent. For example, Lie algebras of vector fields have such an intermediate growth. We suggest an hierarchy of types of intermediate growth, which consists of a countable series of functions. In terms of this 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. These algebras belong to the level q of the hierarchy, where the level q=1 corresponds to finite dimensional Lie algebras. The level q=2 corresponds to the polynomial growth. We obtain an application of this result to free solvable groups, we describe an asymptotic of the factors of the lower central series for the free solvable groups of finite rank. We also consider results on growth of finitely generated solvable Lie superalgebras and growth of almost solvable Lie algebras. For free groups there is a well-known Schreier's formula. We find and use different analogues of this formula for free Lie algebras in terms of generating functions of two kinds above.