## Almost Engel compact groupsAdd to your list(s) Download to your calendar using vCal - 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:Note that ex-directory lists are not shown. |
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