School of Mathematics & Physics, University of Lincoln
In a new joint work, Evgeny Khukhro (Lincoln) and Pavel Shumyatsky (University of Brasilia) succeeded in applying two different Lie ring methods in the study of profinite groups with automorphisms whose fixed points satisfy generalized Engel conditions. The results have just been submitted as a paper “On profinite groups with automorphisms whose fixed points have countable Engel sinks“. In one of the main cases of the proof, pro-p groups are considered, where the Lie algebra constructed from the Zassenhaus p-filtration is analysed using Zelmanov’s theorem on Lie algebras satisfying a polynomial identity and generated by elements all of whose products are ad-nilpotent, in conjunction with the Bahturin–Zaitsev theorem on polynomial identities of Lie algebras with automorphisms. This analysis provides a reduction to uniformly powerful pro-p groups, for which a different Lie algebra over p-adic integers is used canonically connected with the group via the Baker–Campbell–Hausdorff formula.