On the finite presentation of subdirect products and the nature of residually free groups

Abstract

We establish virtual surjection to pairs (VSP) as a general criterion for the finite presentability of subdirect products of groups: if Γ1,.,Γn are finitely presented andS<Γ1×· · ·×Γn projects to a subgroup of finite index in each Γi×Γj, then S is finitely presentable, indeed there is an algorithm that will construct a finite presentation for S. We use the VSP criterion to characterize the finitely presented residually free groups. We prove that the class of such groups is recursively enumerable. We describe an algorithm that, given a finite presentation of a residually free group, constructs a canonical embedding into a direct product of finitely many limit groups. We solve the (multiple) c..