Direct factors of profinite completions and decidability
We consider finitely presented,residually finite groups $G$ and finitely generated normal subgroups $A$ such that the inclusion $A\hookrightarrow G$ induces an isomorphism from the profinite completion of $A$ to a direct factor of the profinite completion of $G$. We explain why $A$ need not be a dir...
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| Format: | Journal article |
| Language: | English |
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2008
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| _version_ | 1797090221577207808 |
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| author | Bridson, M |
| author_facet | Bridson, M |
| author_sort | Bridson, M |
| collection | OXFORD |
| description | We consider finitely presented,residually finite groups $G$ and finitely generated normal subgroups $A$ such that the inclusion $A\hookrightarrow G$ induces an isomorphism from the profinite completion of $A$ to a direct factor of the profinite completion of $G$. We explain why $A$ need not be a direct factor of a subgroup of finite index in $G$; indeed $G$ need not have a subgroup of finite index that splits as a non-trivial direct product. We prove that there is no algorithm that can determine whether $A$ is a direct factor of a subgroup of finite index in $G$. |
| first_indexed | 2024-03-07T03:15:24Z |
| format | Journal article |
| id | oxford-uuid:b59b1eca-dd6e-4d26-84eb-c6855360b139 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T03:15:24Z |
| publishDate | 2008 |
| record_format | dspace |
| spelling | oxford-uuid:b59b1eca-dd6e-4d26-84eb-c6855360b1392022-03-27T04:34:43ZDirect factors of profinite completions and decidabilityJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:b59b1eca-dd6e-4d26-84eb-c6855360b139EnglishSymplectic Elements at Oxford2008Bridson, MWe consider finitely presented,residually finite groups $G$ and finitely generated normal subgroups $A$ such that the inclusion $A\hookrightarrow G$ induces an isomorphism from the profinite completion of $A$ to a direct factor of the profinite completion of $G$. We explain why $A$ need not be a direct factor of a subgroup of finite index in $G$; indeed $G$ need not have a subgroup of finite index that splits as a non-trivial direct product. We prove that there is no algorithm that can determine whether $A$ is a direct factor of a subgroup of finite index in $G$. |
| spellingShingle | Bridson, M Direct factors of profinite completions and decidability |
| title | Direct factors of profinite completions and decidability |
| title_full | Direct factors of profinite completions and decidability |
| title_fullStr | Direct factors of profinite completions and decidability |
| title_full_unstemmed | Direct factors of profinite completions and decidability |
| title_short | Direct factors of profinite completions and decidability |
| title_sort | direct factors of profinite completions and decidability |
| work_keys_str_mv | AT bridsonm directfactorsofprofinitecompletionsanddecidability |