Towards a general approach to Burnside-type problems

The Burnside problem asks whether a finitely generated group satisfying the group law $x^n = 1$ for a fixed $n$ is finite. For large enough exponents $n$ the answer is negative. It turns out that the case of odd and even exponents requires different consideration. Now we restrict ourselves to odd exponents. The case of odd $n$ is solved in the celebrated works of P. Novikov and S. Adian (1968, combinatorial approach) and A. Olshanskii (1982, geometric approach). One can ask these kinds of questions for other group laws. One of such questions is the Engel problem (stated around the 1920s) that asks whether a finitely generated group satisfying the group law $[x, y, \ldots, y] = 1$ (where $[,]$ is a left normalized commutator repeated $n$ times) is nilpotent. For $n \leqslant 4$ the answer is affirmative; the case $n > 4$ remains open.

In our recent work on the Burnside problem (joint with E. Rips and K. Tent), we significantly improved the existing approaches to the Burnside problem. As a result, we decreased the known lower bound for odd exponents of infinite Burnside groups. The developed framework is relatively technically simple. This gives an opportunity to use it for the Engel problem and even to transform it to ``meta-method'' applicable to other Burnside-type problems.