Algorithms for finitely \(L\)-presented groups and their applications to some self-similar groups. (Q392149)
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scientific article; zbMATH DE number 6244690
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Algorithms for finitely \(L\)-presented groups and their applications to some self-similar groups. |
scientific article; zbMATH DE number 6244690 |
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Algorithms for finitely \(L\)-presented groups and their applications to some self-similar groups. (English)
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13 January 2014
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algorithms
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Reidemeister-Schreier theorem
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infinite presentations
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recursive presentations
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self-similar groups
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finite index subgroups
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finitely presented groups
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finite \(L\)-presentations
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nilpotent quotient algorithm
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Grigorchuk group
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Fabrykowski-Gupta groups
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lower central series
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In the article under review, the author gives an overview of algorithms for finitely \(L\)-presented groups. As applications, using these algorithms, the author investigates the class of self-similar groups \(\Gamma_p\) for \(3\leq p\leq 11\). After recalling the notion of a self-similar group, the author demonstrates the application of his algorithms to do the following:NEWLINENEWLINE \(\bullet\) to compute the isomorphism type of the lower central series sections \(\gamma_c\Gamma_p/\gamma_{c+1}\Gamma_p\).NEWLINENEWLINE \(\bullet\) to compute the isomorphism type of the Dwyer quotients \(M_c(\Gamma_p)\) of their Schur multiplier.NEWLINENEWLINE \(\bullet\) to determine the number of low-index subgroups of the groups \(\Gamma_p\).NEWLINENEWLINE \(\bullet\) to compute the isomorphism type of the sections \(\Gamma_p^{(c)}/\Gamma_p^{(c+1)}\) of the derived series.
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