Lifting Semistability in Finitely Generated Ascending HNN-Extensions
[Semistabilité des extensions HNN ascendantes]
Lasheras, Francisco F.1 ; Mihalik, Michael2
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, Online first, 17 p.

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 st=ϕ(s),s∈S〉 qu’on suppose être semistable à l’infini. Soit R ¯ le noyau du morphisme tautologique du groupe libre F({t}∪S) sur H. Alors il existe un sous-ensemble fini R 0 ⊆R ¯ tel que toute extension HNN de type fini H 1 =〈S,t;R 1 ,t -1 st=ϕ(s),s∈S〉, ayant R 0 ⊆R 1 ⊂R ¯, n’a qu’un seul bout et est semistable à l’infini. De plus H 1 admet une telle présentation avec R 1 ⊂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 st=ϕ(s),s∈S〉 is semistable at infinity. Let R ¯ be the kernel of the obvious homomorphism from the free group F({t}∪S) onto H, then there is a finite subset R 0 ⊆R ¯ such that those finitely generated ascending HNN-extensions H 1 =〈S,t;R 1 ,t -1 st=ϕ(s),s∈S〉, with R 0 ⊆R 1 ⊂R ¯, are all 1-ended and semistable at infinity as well. Furthermore H 1 has such a presentation with R 1 ⊂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.

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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
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) 
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Lasheras, Francisco F.; Mihalik, Michael. Lifting Semistability in Finitely Generated Ascending HNN-Extensions. Annales de l'Institut Fourier, Online first, 17 p.

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