Residual finiteness and other properties of (relative) one-relator groups

This is a fairly general lecture. I will begin with some history of combinatorial group theory, starting with Max Dehn. The study of one-relator groups was begun by Wilhelm Magnus in the late 1920’s and early 1930’s (this study was suggested to Magnus by Dehn, who was Magnus’ PhD supervisor). The class of one-relator groups has been intensely studied since Magnus’ pioneering work. Nevertheless, there are still basic open questions concerning these groups. In particular, the problem of when one-relator groups are residually finite is still open. By definition, a one-relator group is defined by a presentation P = [x;R] where x is the generating set and R is a word on x which is the single defining relator. A more general situation is to consider relative one-relator groups. These are given by a relative one-relator presentation P = [x;H;R]. Here H is a group (to be thought of as a “coefficient group”), and R is a word on the alphabet x and the elements of H. The group G(P)defined by P is the quotient of the free product F(x) * H (where F(x) is the free group on x), by the normal closure of R. One would hope that the properties of G(P) should be governed by the “shape” of the “x skeleton” (the word on x obtained from R by deleting all terms from H), and the algebraic properties of the coefficient group H. Here I will introduce the “unique max-min property” for the shape. It turns out then that G(P) is residually finite if and only if H is residually finite. This theorem can then be used to get some corollaries concerning ordinary one-relator groups. Further results (including results concerning the conjugacy problem) are also obtained. Time permitting, I will also mention some other recent work by Anton Klyachko.