Lifting Semistability in Finitely Generated Ascending HNN-Extensions

Lasheras, Francisco F.; Mihalik, Michael

doi:10.5802/aif.3599

Lifting Semistability in Finitely Generated Ascending HNN-Extensions
[Semistabilité des extensions HNN ascendantes]

Lasheras, Francisco F.¹ ; Mihalik, Michael ²

¹ Departamento de Geometría y Topología Universidad de Sevilla, Fac. Matemáticas C/. Tarfia s/n, 41012 Sevilla (Spain)
² Department of Mathematics Vanderbilt University Nashville TN, 37240 (USA)

Annales de l'Institut Fourier, Tome 74 (2024) no. 1, pp. 349-365.

Résumé
Abstract

La question fondamentale de cet article est de savoir sous quelles conditions la semistabilité d’un groupe $H$ entraîne la semistabilité d’un groupe $G$ qui admet une surjection sur $H$ . Nous allons y répondre dans le cadre des extensions HNN ascendantes. Plus précisement, considérons une extension HNN de type fini ayant un seul bout $H = 〈 S, t; R, t^{- 1} s t = ϕ (s), s \in S 〉$ qu’on suppose être semistable à l’infini. Soit $\bar{R}$ le noyau du morphisme tautologique du groupe libre $F ({t} \cup S)$ sur $H$ . Alors il existe un sous-ensemble fini $R_{0} \subseteq \bar{R}$ tel que toute extension HNN de type fini $H_{1} = 〈 S, t; R_{1}, t^{- 1} s t = ϕ (s), s \in S 〉$ , ayant $R_{0} \subseteq R_{1} \subset \bar{R}$ , n’a qu’un seul bout et est semistable à l’infini. De plus $H_{1}$ admet une telle présentation avec $R_{1} \subset R$ . Notons qu’il y a un épimorphisme de $H_{1}$ dans $H$ . A l’heure actuelle, nous ne savons pas si toutes les extensions HNN ascendantes sont semistables à l’infini.

If a finitely generated group $G$ maps epimorphically onto a group $H$ , we are interested in the question: When does the semistability of $H$ imply $G$ is semistable? In this paper, we give an answer within the class of ascending HNN-extensions. More precisely, our main theorem states: Suppose that the $1$ -ended finitely generated ascending HNN-extension $H = 〈 S, t; R, t^{- 1} s t = ϕ (s), s \in S 〉$ is semistable at infinity. Let $\bar{R}$ be the kernel of the obvious homomorphism from the free group $F ({t} \cup S)$ onto $H$ , then there is a finite subset $R_{0} \subseteq \bar{R}$ such that those finitely generated ascending HNN-extensions $H_{1} = 〈 S, t; R_{1}, t^{- 1} s t = ϕ (s), s \in S 〉$ , with $R_{0} \subseteq R_{1} \subset \bar{R}$ , are all $1$ -ended and semistable at infinity as well. Furthermore $H_{1}$ has such a presentation with $R_{1} \subset R$ . Note that there is an obvious epimorphism from $H_{1}$ to $H$ . It is unknown whether all finitely presented ascending HNN-extensions are semistable at infinity.

Reçu le : 2021-09-27
Révisé le : 2022-07-14
Accepté le : 2022-09-23
Première publication : 2023-07-03
Publié le : 2024-05-17

DOI : 10.5802/aif.3599

Classification : 20F69, 20F65, 20E22
Keywords: Proper homotopy, semistability at infinity, ascending HNN-extension, group presentation
Mot clés : homotopie propre, semistabilité à l’infini, extension HNN ascendante, présentation de groupe

Affiliations des auteurs :

Lasheras, Francisco F. ¹ ; Mihalik, Michael ²

¹ Departamento de Geometría y Topología Universidad de Sevilla, Fac. Matemáticas C/. Tarfia s/n, 41012 Sevilla (Spain)
² Department of Mathematics Vanderbilt University Nashville TN, 37240 (USA)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Lifting {Semistability} in {Finitely} {Generated} {Ascending} {HNN-Extensions}},
     journal = {Annales de l'Institut Fourier},
     pages = {349--365},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {74},
     number = {1},
     year = {2024},
     doi = {10.5802/aif.3599},
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Lasheras, Francisco F.; Mihalik, Michael. Lifting Semistability in Finitely Generated Ascending HNN-Extensions. Annales de l'Institut Fourier, Tome 74 (2024) no. 1, pp. 349-365. doi : 10.5802/aif.3599. https://aif.centre-mersenne.org/articles/10.5802/aif.3599/

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