Clover nil restricted Lie algebras of quasi-linear growth

Abstract: The Grigorchuk and Gupta-Sidki groups play 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, Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic. Also, the author constructed a family of 2-generated restricted Lie algebras of slow polynomial growth with a nil $p$-mapping. Now, we construct a family of so called clover 3-generated restricted Lie algebras $T(\Xi)$, where a field of positive characteristic is arbitrary and $\Xi$ an infinite tuple of positive integers. We prove that $1\le \mathrm{GKdim}T(\Xi)\le3$, moreover, the set of Gelfand-Kirillov dimensions of clover Lie algebras with constant tuples is dense on $[1,3]$. We construct a subfamily of non-isomorphic nil restricted Lie algebras $T(\Xi_{q,\kappa})$, where $q\in\mathbb N$, $\kappa\in\mathbb R^+$, with extremely slow quasi-linear growth of type: $\gamma_{T(\Xi_{q,\kappa})}(m)=m\big(\ln^{(q)}!m\big)^{\kappa+o(1)}$, as $m\to\infty$. The present research is motivated by a construction by Kassabov and Pak of groups of oscillating growth. As an analogue, we construct nil restricted Lie algebras of intermediate oscillating growth in another paper. 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 $n$ the growth function behaves like $\exp(n/(\ln n)^{\lambda})$, for such periods the algebra is "resuscitating". The present construction of 3-generated nil restricted Lie algebras of quasi-linear growth is an important part of that result, responsible for the lower quasi-linear growth in that construction.