Orderable groups with Engel-like conditions
Group Theory
2017-07-20 v1
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*.
Cite
@article{arxiv.1707.06153,
title = {Orderable groups with Engel-like conditions},
author = {Pavel Shumyatsky},
journal= {arXiv preprint arXiv:1707.06153},
year = {2017}
}