Orderable groups with Engel-like conditions

Let x be an element of a group G. For a positive integer n let En(x) be the subgroup generated by all commutators [...[[y,x],x],…,x] over y∈G, where x is repeated n times. There are several recent results showing that certain properties of groups with small subgroups En(x) are close to those of Engel groups. The present article deals with orderable groups in which, for some n≥1, the subgroups En(x) are polycyclic. Let h≥0, n>0 be integers and G an orderable group in which En(x) is polycyclic with Hirsch length at most h for every x∈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⁎.