n-Engel groups for large n

An n-Engel group is a group that satisfies a group law [x, y, ... y] = 1, where y is repeated n times. The following natural question was asked in the 1950s: is such a group locally nilpotent?  For n at most 4, the answer is affirmative. For larger n this question seems to be quite delicate. Namely, if one adds some extra restrictions together with the n-Engel identity, the resulting group is indeed locally nilpotent. For example, finitely generated n-Engel groups that are residually finite, or solvable, are nilpotent. However, we expect that finitely generated free n-Engel groups are not nilpotent for sufficiently large n.

The Engel problem has a connection with the Burnside problem, which asks whether a finitely generated group with a group law x^n = 1 is necessarily finite. The new proof of the Burnside problem that was recently obtained by E. Rips, K. Tent and myself helps to make progress in the Engel problem. I will talk about my current progress and difficulties along this way.