The Engel elements in generalized FC-groups

We generalize to FC*, the class of generalized FC-groups introduced in [F. de Giovanni, A. Russo, G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J. 28 (2002), 241-254], a result of Baer on Engel elements. More precisely, we prove that the sets of left Engel elements and bounded left Engel elements of an FC*-group G coincide with the Fitting subgroup; whereas the sets of right Engel elements and bounded right Engel elements of G are subgroups and the former coincides with the hypercentre. We also give an example of an FC*-group for which the set of right Engel elements contains properly the set of bounded right Engel elements.Comment: to appear in "Illinois Journal of Mathematics

Characteristic Classes for the Degenerations of Two-Plane Fields in Four Dimensions

There is a remarkable type of field of two-planes special to four dimensions known as an Engel distributions. They are the only stable regular distributions besides the contact, quasi-contact and line fields. If an arbitrary two-plane field on a four-manifold is slightly perturbed then it will be Engel at generic points. On the other hand, if a manifold admits an oriented Engel structure then the manifold must be parallelizable and consequently the alleged Engel distribution must have a degeneration loci -- a point set where the Engel conditions fails. By a theorem of Zhitomirskii this locus is a finite union of surfaces. We prove that these surfaces represent Chern classes associated to the distribution.Comment: LaTeX, 15 page

Algorithmic decidability of Engel's property for automaton groups

We consider decidability problems associated with Engel's identity ($[\cdots[[x,y],y],\dots,y]=1$ for a long enough commutator sequence) in groups generated by an automaton. We give a partial algorithm that decides, given $x,y$, whether an Engel identity is satisfied. It succeeds, importantly, in proving that Grigorchuk's $2$-group is not Engel. We consider next the problem of recognizing Engel elements, namely elements $y$ such that the map $x\mapsto[x,y]$ attracts to $\{1\}$. Although this problem seems intractable in general, we prove that it is decidable for Grigorchuk's group: Engel elements are precisely those of order at most $2$. Our computations were implemented using the package FR within the computer algebra system GAP