| Abstrakt: |
The Grigorchuk and Gupta–Sidki groups play a fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2 [V. M. Petrogradsky, Examples of self-iterating Lie algebras, J. Algebra302(2) (2006) 881–886], Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [I. P. Shestakov and E. Zelmanov, Some examples of nil Lie algebras, J. Eur. Math. Soc. (JEMS)10(2) (2008) 391–398]. Now, we construct a family of so called clover 3-generated restricted Lie algebras T (Ξ) , where a field of positive characteristic is arbitrary and Ξ an infinite tuple of positive integers. All these algebras have a nil p -mapping. We prove that 1 ≤ GKdim T (Ξ) ≤ 3. We compute Gelfand–Kirillov dimensions of clover restricted Lie algebras with periodic tuples and show that these dimensions for constant tuples are dense on [ 1 , 3 ]. We construct a subfamily of nil restricted Lie algebras T (Ξ q , κ) , with parameters q ∈ ℕ , κ ∈ ℝ + , having extremely slow quasi-linear growth of type: γ T (Ξ q , κ) (m) = m ln ⋯ ln ︸ q m κ + o (1) , as m → ∞. The present research is motivated by construction by Kassabov and Pak of groups of oscillating growth [M. Kassabov and I. Pak, Groups of oscillating intermediate growth. Ann. Math. (2)177(3) (2013) 1113–1145]. As an analogue, we construct nil restricted Lie algebras of intermediate oscillating growth in [V. Petrogradsky, Nil restricted Lie algebras of oscillating intermediate growth, preprint (2020), arXiv:2004.05157]. We call them Phoenix algebras because, for infinitely many periods of time, the algebra is "almost dying" by having a "quasi-linear" growth as above, for infinitely many n it has a rather fast intermediate growth of type exp (n / (ln n) λ) , for such periods the algebra is "resuscitating". The present construction of three-generated nil restricted Lie algebras of quasi-linear growth is an important part of that result, responsible for the lower quasi-linear bound in that construction. [ABSTRACT FROM AUTHOR] |
