Orderable groups with Engel-like conditions

TL;DR

This paper investigates orderable groups where certain subgroups generated by iterated commutators are polycyclic with bounded Hirsch length, showing such groups have a structured composition involving a finitely generated nilpotent subgroup and a nilpotent quotient.

Contribution

It establishes bounds on the structure of orderable groups with polycyclic subgroups generated by iterated commutators, extending understanding of Engel-like conditions.

Findings

Existence of bounded nilpotent subgroup N within G.

G/N is nilpotent of bounded class.

N is finitely generated and has bounded Hirsch length.

Abstract

Let x be an element of a group G. For a positive integer n let E_n(x) be the subgroup generated by all commutators [...[[y,x],x],...,x] over y in G, where x is repeated n times. There are several recent results showing that certain properties of groups with small subgroups E_n(x) are close to those of Engel groups. The present article deals with orderable groups in which, for some n, the subgroups E_n(x) are polycyclic. Let h,n be positive integers and G an orderable group in which E_n(x) is polycyclic with Hirsch length at most h for every x in G. It is proved that there are (h,n)-bounded numbers h* and c* such that G has a finitely generated normal nilpotent subgroup N with h(N)<h* and G/N nilpotent of class at most c*.