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

Abstract

We establish {\em{virtual surjection to pairs}} (VSP) as a general criterion for the finite presentability of subdirect products of groups: if ${\Gamma }_1,...,{\Gamma }_n$ are finitely presented and $S<{\Gamma }_1\times ...\times {\Gamma }_n$ projects to a subgroup of finite index in each ${\Gamma }_i\times {\Gamma }_j$, then $S$ is finitely presentable, indeed there is an algorithm that will construct a finite presentation for $S$. We use the VSP criterion to characterise 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) conjugacy problem and membership problem for finitely presentable subgroups of residually free groups. We also prove that there is an algorithm that, given a finite generating set for such a subgroup, will construct a finite presentation. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither ${\mathrm{FP}}_{\infty }$ nor of Stallings-Bieri type

44 pages. To appear in American Journal of Mathematics. This is a substantial rewrite of our previous Arxiv article 0809.3704, taking into account subsequent developments, advice of colleagues and referee's comments

References in corpus (4)

• Limit groups are CAT(0)
• Non-finiteness properties of fundamental groups of smooth projective varieties
• Finitely presented residually free groups
• Isoperimetric functions for subdirect products and Bestvina-Brady groups

Cited by in corpus (22)

• Kähler groups and subdirect products of surface groups
• Weak commutativity and finiteness properties of groups
• Generating subdirect products
• Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces
• Finite presentations for Kähler groups with arbitrary finiteness properties
• On the Subgroups of Right Angled Artin Groups and Mapping Class Groups
• On the finiteness length of some soluble linear groups
• The weak commutativity construction for Lie algebras
• On conjugacy separability of fibre products
• The homology of groups, profinite completions, and echoes of Gilbert Baumslag
• Complex hypersurfaces in direct products of Riemann surfaces
• Profinite isomorphisms and fixed-point properties
• Branched covers of elliptic curves and Kähler groups with exotic finiteness properties
• Modular structures and extended-modular-group-structures after Hecke pairs
• Subgroups of direct products of limit groups over Droms RAAGs
• Virtual Extensions of Modules
• The isomorphism problem for profinite completions of residually finite groups
• Solvable quotients of subdirect products of perfect groups are nilpotent
• On subgroups of right angled Artin groups with few generators
• Dehn functions of coabelian subgroups of direct products of groups
• Homological finiteness properties of fibre products
• Eraser morphisms and membership problem in groups and monoids