# 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 ${\rm{FP}}_\infty$ nor of Stallings-Bieri type
Type de document :
Article dans une revue
American Journal of Mathematics, Johns Hopkins University Press, 2013, 135 (4), pp.891-933. 〈10.1353/ajm.2013.0036〉

https://hal.archives-ouvertes.fr/hal-00821482
Contributeur : Hamish Short <>
Soumis le : vendredi 10 mai 2013 - 13:29:04
Dernière modification le : lundi 4 mars 2019 - 14:04:24

### Citation

Martin R. Bridson, James Howie, Charles F. Miller Iii, Hamish Short. On the finite presentation of subdirect products and the nature of residually free groups. American Journal of Mathematics, Johns Hopkins University Press, 2013, 135 (4), pp.891-933. 〈10.1353/ajm.2013.0036〉. 〈hal-00821482〉

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