On the finite presentation of subdirect products and the nature of residually free groups
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2012
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| _version_ | 1866912483063627776 |
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| author | Bridson, Martin R. Howie, James Miller III, Charles F. Short, Hamish |
| author_facet | Bridson, Martin R. Howie, James Miller III, Charles F. Short, Hamish |
| contents | We establish {\em{virtual surjection to pairs}} (VSP) as a general criterion for the finite presentability of subdirect products of groups: if $Γ_1,...,Γ_n$ are finitely presented and $S<Γ_1\times...\timesΓ_n$ projects to a subgroup of finite index in each $Γ_i\timesΓ_j$, 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1205_6175 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | On the finite presentation of subdirect products and the nature of residually free groups Bridson, Martin R. Howie, James Miller III, Charles F. Short, Hamish Group Theory 20F65, 20E08, 20F67 We establish {\em{virtual surjection to pairs}} (VSP) as a general criterion for the finite presentability of subdirect products of groups: if $Γ_1,...,Γ_n$ are finitely presented and $S<Γ_1\times...\timesΓ_n$ projects to a subgroup of finite index in each $Γ_i\timesΓ_j$, 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 |
| title | On the finite presentation of subdirect products and the nature of residually free groups |
| topic | Group Theory 20F65, 20E08, 20F67 |
| url | https://arxiv.org/abs/1205.6175 |