On the residual finiteness and other properties of (relative) one-relator groups

TL;DR

This paper investigates the residual finiteness and other properties of (relative) one-relator groups, establishing conditions under which these groups are residually finite and exploring related algebraic and geometric properties.

Contribution

It introduces the unique max-min property for group words, linking residual finiteness of relative one-relator groups to that of the subgroup H, and applies this to classical one-relator groups.

Findings

Residual finiteness of relative one-relator groups depends on the subgroup H.

New criteria for residual finiteness of one-relator groups are established.

Results on conjugacy problem and relative asphericity are obtained.

Abstract

A relative one-relator presentation has the form P = < X,H ; R > where X is a set, H is a group, and R is a group word on X and H. We show that if the group word on X obtained from R by deleting all the terms from H has what we call the unique max-min property, then the group defined by P is residually finite if and only if H is residually finite (Theorem 1). We apply this to obtain new results concerning the residual finiteness of (ordinary) one-relator groups (Theorem 4). We also obtain results concerning the conjugacy problem for one-relator groups (Theorem 5), and results concerning the relative asphericity of presentations of the form P (Theorem 6).