Introduction

Group theoretical investigation and classification efforts take advantage of the various group theoretical constructions: the attempt is to recapture information about groups of interest from their various subgroups, homomorphic images and extensions. The particularly successful theory of one-relator groups - which greatly benefitted from the pioneering work of W. Magnus - provided motivation for investigating one-relator quotients of free products as well as extensions and automorphisms of free groups. Questions of SQ-universality and recognizability, stemming from decision problems, also added interest in one-relator groups possessing free quotients, and in various groups with free subgroups.

Recall that if P is a group property, then a group G is virtuallyP if it has a subgroup of finite index satisfying P. Alternatively we also call G a virtual P-group. G is P-by-finite if G has a normal subgroup of finite index satisfying P. If P is a subgroup inherited property, such as torsion-freeness, freeness, or solvability then virtually P and P-by-finite are equivalent.

The structure of virtually-free groups {free-by-finite groups} is rather well understood {see Section 3} and generalizes the structure of free groups in expected ways. Virtually free groups also have connections with automatic groups and hyperbolic groups {see Section 3}.

The present paper represents the start of a general program to extend knowledge about virtually one-relator groups. The aim is to line up known results and give some extensions of relevance to the proposed project. Since a good number of properties of free groups appear in one-relator groups it is hoped that a similar parallelism can be detected for virtually one-relator groups.