Abstract
We extend the definition of the bounded reduction property to endomorphisms of automatic group and find conditions for it to hold. We study endomorphisms with L-quasiconvex image and prove that those with finite kernel satisfy a synchronous version of the bounded reduction property. Finally, we use these techniques to prove L-quasiconvexity of the equalizer of two endomorphisms under certain (strict) conditions.
Résumé
Nous étendons la définition de la propriété de réduction bornée aux endomorphismes de groupes automatiques et trouvons des conditions pour qu’elle soit vérifiée. Nous étudions les endomorphismes avec une image L-quasiconvexe et montrons que ceux avec un noyau fini satisfont une version synchrone de la propriété de réduction bornée. Enfin, nous utilisons ces techniques pour prouver la L-quasiconvexité de l’égaliseur de deux endomorphismes sous certaines conditions (strictes).
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Acknowledgements
The author is grateful to Pedro V. Silva for fruitful discussions of these topics, which greatly improved the paper, and to the anonymous referee for many comments and suggestions that greatly improved the exposition and the contents of the paper. The author was supported by the grant SFRH/BD/145313/2019 funded by Fundação para a Ciência e a Tecnologia (FCT).
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Carvalho, A. On endomorphisms of automatic groups. Ann. Math. Québec 48, 21–44 (2024). https://doi.org/10.1007/s40316-022-00196-8
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DOI: https://doi.org/10.1007/s40316-022-00196-8