Evgeny Khukhro’s talk at the Algebra Seminar of the University of Warwick

Evgeny Khukhro gave an invited talk at the Algebra Seminar of the University of Warwick on 31 October 2024. He reported on joint results with Pavel Shumyatsky (Univ. of Brasilia). The latest of these results can be found in the preprint, which also contains references to a few earlier results of this international collaboration of Lincoln and Brazilian algebraists.

Abstract: Using Zelmanov’s deep results on Engel Lie algebras, Wilson and Zelmanov proved that any profinite Engel group is locally nilpotent, and Medvedev extended this result to Engel compact groups. We state generalizations of the Engel condition as restrictions on the so-called Engel sinks of group elements. For example, a group can be considered to be `almost Engel’ if all Engel sinks are finite. We proved that almost Engel compact groups are almost locally nilpotent (in certain precise terms). Similar results for finite groups have quantitative nature, with almost nilpotency expressed as a function of sizes of Engel sinks. Our most recent results concern imposing restrictions on Engel sinks of commutators (rather than all elements).