Lifting Semistability in Finitely Generated Ascending HNN-Extensions
Annales de l'Institut Fourier, Volume 74 (2024) no. 1, pp. 349-365.

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),sS 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),sS, 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.

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),sS 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),sS, 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.

Received:
Revised:
Accepted:
Online First:
Published online:
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)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2024__74_1_349_0,
     author = {Lasheras, Francisco F. and Mihalik, Michael},
     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},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3599/}
}
TY  - JOUR
AU  - Lasheras, Francisco F.
AU  - Mihalik, Michael
TI  - Lifting Semistability in Finitely Generated Ascending HNN-Extensions
JO  - Annales de l'Institut Fourier
PY  - 2024
SP  - 349
EP  - 365
VL  - 74
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3599/
DO  - 10.5802/aif.3599
LA  - en
ID  - AIF_2024__74_1_349_0
ER  - 
%0 Journal Article
%A Lasheras, Francisco F.
%A Mihalik, Michael
%T Lifting Semistability in Finitely Generated Ascending HNN-Extensions
%J Annales de l'Institut Fourier
%D 2024
%P 349-365
%V 74
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3599/
%R 10.5802/aif.3599
%G en
%F AIF_2024__74_1_349_0
Lasheras, Francisco F.; Mihalik, Michael. Lifting Semistability in Finitely Generated Ascending HNN-Extensions. Annales de l'Institut Fourier, Volume 74 (2024) no. 1, pp. 349-365. doi : 10.5802/aif.3599. https://aif.centre-mersenne.org/articles/10.5802/aif.3599/

[1] Harpe, Pierre Topics in Geometric Group Theory, Chicago Lectures in Mathematics, University of Chicago Press, 2000

[2] Geoghegan, Ross Topological Methods in Group Theory, Graduate Texts in Mathematics, 243, Springer, 2008 | DOI

[3] Geoghegan, Ross; Mihalik, Michael L. Free abelian cohomology of groups and ends of universal covers, J. Pure Appl. Algebra, Volume 36 (1985), pp. 123-137 | DOI | MR | Zbl

[4] Lasheras, Francisco F. Ascending HNN-extensions and properly 3-realisable groups, Bull. Aust. Math. Soc., Volume 72 (2005) no. 2, pp. 187-196 | DOI | MR | Zbl

[5] Mardešić, Sibe; Segal, Jack Shape theory. The inverse system approach, North-Holland Mathematical Library, 26, North-Holland, 1982

[6] Mihalik, Michael L. Ends of groups with the integers as quotient, J. Pure Appl. Algebra, Volume 35 (1985), pp. 305-320 | DOI | MR | Zbl

[7] Mihalik, Michael L. Ends of Double Extension Groups, Topology, Volume 25 (1986), pp. 45-53 | DOI | MR | Zbl

[8] Mihalik, Michael L. Semistability at of finitely generated groups, and solvable groups, Topology Appl., Volume 24 (1986), pp. 259-269 | DOI | MR | Zbl

[9] Mihalik, Michael L. Semistability at , -ended groups and group cohomology, Trans. Am. Math. Soc., Volume 303 (1987), pp. 479-485 | MR | Zbl

[10] Mihalik, Michael L. Semistability and simple connectivity at of finitely generated groups with a finite series of commensurated subgroups, Algebr. Geom. Topol., Volume 16 (2016) no. 6, pp. 3615-3640 | DOI | MR | Zbl

[11] Mihalik, Michael L. Bounded depth ascending HNN-extensions and π 1 -semistability at infinity, Can. J. Math., Volume 72 (2020) no. 6, pp. 1529-1550 | DOI | MR | Zbl

[12] Mihalik, Michael L. Near ascending HNN-extensions and a combination result for semistability at infinity (2022) (https://arxiv.org/abs/2206.04152)

[13] Touikan, Nicholas On the one-endedness of graphs of groups, Pac. J. Math., Volume 278 (2015) no. 2, pp. 463-478 | DOI | MR | Zbl

Cited by Sources: