Some Residually Solvable One-Relator Groups

Abstract

This communication records some observations made in the course of studying one-relator groups from the point of view of residual solvability. As a contribution to clas- sication eorts we single out some relator types that render the corresponding one-relator groups residually solvable. a collection of facts and examples gathered while attempting to char- acterize the residually solvable one-relator groups in terms (of the form) of the (single) dening relator. In what follows we prove suf- ciency results for certain cases when the relator is a commutator, and then raise some questions. The class of one-relator groups shows a varied pattern of behavior with respect to residual properties. We begin with reviewing some of the literature that motivated our interest in the topic. G. Baum- slag in (3) showed that positive one-relator groups, which is to say that the relator has only positive exponents, are residually solvable. In the same paper he provided a specic example to demonstrate that not all one-relator groups are residually solvable. A free-by- cyclic group is necessarily residually solvable. As well are the free- by-solvable Baumslag{Solitar groups Bm;n (the groups with presen- tation ha;b;a 1 b m a = b n i for pairs of non-zero integers m;n), by a result of Peter Kropholler (15) who showed that in these groups the second derived subgroup is free. The Baumslag{Solitar groups