Extending partial automorphisms and the profinite topology on free groups
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- by Bernhard Herwig and Daniel Lascar PDF
- Trans. Amer. Math. Soc. 352 (2000), 1985-2021 Request permission
Abstract:
A class of structures $\mathcal {C}$ is said to have the extension property for partial automorphisms (EPPA) if, whenever $C_1$ and $C_2$ are structures in $\mathcal {C}$, $C_1$ finite, $C_1\subseteq C_2$, and $p_1,p_2,\dotsc ,p_n$ are partial automorphisms of $C_1$ extending to automorphisms of $C_2$, then there exist a finite structure $C_3$ in $\mathcal {C}$ and automorphisms $\alpha _1, \alpha _2,\dotsc ,\alpha _n$ of $C_3$ extending the $p_i$. We will prove that some classes of structures have the EPPA and show the equivalence of these kinds of results with problems related with the profinite topology on free groups. In particular, we will give a generalisation of the theorem, due to Ribes and Zalesskiĭstating that a finite product of finitely generated subgroups is closed for this topology.References
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Additional Information
- Bernhard Herwig
- Affiliation: Institut für Mathematische Logik, Universität Freiburg, D-79104 Freiburg, Germany
- Email: herwig@sun2.ruf.uni.freiburg.de
- Daniel Lascar
- Affiliation: Université Paris 7, CNRS, UPRESA 7056, UFR de Mathématiques, 2 Place Jussieu, Case 7012, 75251, Paris CEDEX 05, France
- Email: lascar@logique.jussieu.fr
- Received by editor(s): October 30, 1997
- Published electronically: October 21, 1999
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 1985-2021
- MSC (2000): Primary 20E05, 05C25; Secondary 05C20, 08A35
- DOI: https://doi.org/10.1090/S0002-9947-99-02374-0
- MathSciNet review: 1621745