On finitely generated Engel branch groups

TL;DR

This paper constructs finitely generated Engel branch groups that are not nilpotent, providing new examples and advancing understanding of Engel groups with complex algebraic structures.

Contribution

It introduces the first known finitely generated non-nilpotent Engel groups, answering an open question and expanding the class of known examples.

Findings

Constructed finitely generated Engel branch groups that are not nilpotent

Groups act on rooted trees with rapidly contracting word lengths

Methods applicable to broader class of iterated identities

Abstract

We construct finitely generated Engel branch groups, answering a question of Fern\'andez-Alcober, Noce and Tracey on the existence of such objects. In particular, the groups constructed are not nilpotent, yielding the second known class of examples of finitely generated non-nilpotent Engel groups following a construction by Golod from 1969. To do so, we exhibit groups acting on rooted trees with growing valency on which word lengths of elements are contracting very quickly under section maps. Our methods apply in principle to a wider class of iterated identities, of which the Engel words are only a special case.