Subgroups of <i>HNN</i> Groups and Groups with one Defining Relation

TLDR

HNN groups have been studied in multiple works and involve tree‑product constructions. The paper derives a structure theorem for subgroups of HNN groups using prior results and presents several applications. The authors employ the framework of [6], defining HNN groups with base \(K\), associated subgroups, and a free part generated by stable letters \(t_i\) to establish the theorem.

Abstract

HNN groups have appeared in several papers, e.g., [ 3; 4; 5; 6; 8 ]. In this paper we use the results in [ 6 ] to obtain a structure theorem for the subgroups of an HNN group and give several applications. We shall use the terminology and notation of [ 6 ]. In particular, if K is a group and { φ i } is a collection of isomorphisms of subgroups { L i } into K, then we call the group 1 the HNN group with base K, associated subgroups { L i ,φ i ( L i )} and free part the group generated by t 1 , t 2 , …. (We usually denote φ i ( L i ) by M i or L –i . ) The notion of a tree product as defined in [ 6 ] will also be needed.