Almost Engel compact groups

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• Evgeny Khukhro (Charlotte Scott Research Centre for Algebra, University of Lincoln, UK)
• Thursday 28 March 2019, 15:00-16:00
• Nuffield G13.

If you have a question about this talk, please contact Chris Parker.

We say that a group $G$ is almost Engel if for every $g\in G$ there is a finite set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$, that is, for every $x\in G$ there is a positive integer $n(x,g)$ such that $[...[[x,g],g],\dots ,g]\in {\mathscr E}(g)$ if $g$ is repeated at least $n(x,g)$ times. (Thus, Engel groups are precisely the almost Engel groups for which we can choose ${\mathscr E}(g)=\{ 1\}$ for all $g\in G$.)

We prove that if a compact (Hausdorff) group $G$ is almost Engel, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. If in addition there is a uniform bound $|{\mathscr E}(g)|\leq m$ for the orders of the corresponding sets, then the subgroup $N$ can be chosen of order bounded in terms of $m$. The proofs use the Wilson—Zelmanov theorem saying that Engel profinite groups are locally nilpotent.

This is joint work with Pavel Shumyatsky.

This talk is part of the Algebra seminar series.

This talk is included in these lists:

• Algebra seminar
• Nuffield G13
• School of Mathematics events

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