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Dynamics of implicit operations and tameness of pseudovarieties of groups


Author: Jorge Almeida
Journal: Trans. Amer. Math. Soc. 354 (2002), 387-411
MSC (1991): Primary 20E18, 20M05, 20M07; Secondary 20F10, 20E07, 20E05
DOI: https://doi.org/10.1090/S0002-9947-01-02857-4
Published electronically: August 20, 2001
MathSciNet review: 1859280
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Abstract: This work gives a new approach to the construction of implicit operations. By considering ``higher-dimensional'' spaces of implicit operations and implicit operators between them, the projection of idempotents back to one-dimensional spaces produces implicit operations with interesting properties. Besides providing a wealth of examples of implicit operations which can be obtained by these means, it is shown how they can be used to deduce from results of Ribes and Zalesski{\u{\i}}\kern.15em, Margolis, Sapir and Weil, and Steinberg that the pseudovariety of $p$-groups is tame. More generally, for a recursively enumerable extension closed pseudovariety of groups $\mathbf{V}$, if it can be decided whether a finitely generated subgroup of the free group with the pro- $\mathbf{V}$ topology is dense, then $\mathbf{V}$ is tame.


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Additional Information

Jorge Almeida
Affiliation: Centro de Matemática da Universidade do Porto, P. Gomes Teixeira, 4099-002 Porto, Portugal
Email: jalmeida@fc.up.pt

DOI: https://doi.org/10.1090/S0002-9947-01-02857-4
Keywords: Profinite topology, implicit operation, pseudovariety, free group, extension closed, finite semigroup, semidirect product
Received by editor(s): February 10, 2000
Received by editor(s) in revised form: March 28, 2001
Published electronically: August 20, 2001
Additional Notes: The author gratefully acknowledges support by FCT through the Centro de Matemática da Universidade do Porto, by the project Praxis/2/2.1/MAT/63/94 (Portugal), and by NSERC (United Kingdom)
Article copyright: © Copyright 2001 American Mathematical Society

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