Korean J. Math.  Vol 29, No 3 (2021)  pp.603-612
DOI: https://doi.org/10.11568/kjm.2021.29.3.603

Residual finiteness and Abelian subgroup separability of some high dimensional graph manifolds

Raeyong Kim

Abstract


We generalize $3$-manifolds supporting non-positively curved metric to construct manifolds which have the following properties : (1) They are not locally $\mathrm{CAT}(0)$. (2) Their fundamental groups are residually finite. (3) They have subgroup separability for some abelian subgroups.


Keywords


residually finite group, abelian subgroup separability, high dimensional graph manifold

Subject classification

20E26, 57M05, 27F65

Sponsor(s)

This research was supported by the Daegu University Research Grant, 2017.

Full Text:

PDF

References


Ian Agol, Daniel Groves, and Jason Manning. The virtual haken conjecture. Doc. Math, 18 (1) (2013), 1045–1087. (Google Scholar)

Hyman Bass. Some remarks on group actions on trees. Communications in algebra, 4 (12) (1976), 1091–1126. (Google Scholar)

Martin R Bridson and Andr ́e Haefliger. Metric spaces of non-positive curvature, Volume 319. Springer Science & Business Media, 2013. (Google Scholar)

Roberto Frigerio, Jean-Fran ̧cois Lafont, and Alessandro Sisto. Rigidity of high dimensional graph manifolds., Asterisque Series, Soci ́et ́e math ́ematique de France, 2015. (Google Scholar)

Marshall Hall. Coset representations in free groups. Transactions of the American Mathematical Society, 67 (2) (1949), 421–432. (Google Scholar)

Emily Hamilton. Abelian subgroup separability of haken 3-manifolds and closed hyperbolic n- orbifolds. Proceedings of the London Mathematical Society, 83(3) (2001), 626–646. (Google Scholar)

John Hempel. Residual finiteness for 3-manifolds. In Combinatorial Group Theory and Topology.(AM-111), Volume 111, 379–396. Princeton University Press, 2016. (Google Scholar)

Wilhelm Magnus. Residually finite groups. Bulletin of the American Mathematical Society, 75 (2) (1969), 305–316. (Google Scholar)

Anatolii Malcev. On isomorphic matrix representations of infinite groups. Matematicheskii Sbornik, 50 (3) (1940), 405–422. (Google Scholar)

Isabelle Pays and Alain Valette. Sous-groupes libres dans les groupes d’automorphismes d’arbres. Enseignement Math ́ematique, 37 (1991), 151–174. (Google Scholar)

John G Ratcliffe. Foundations of hyperbolic manifolds, 2nd ́ed. Grad. Texts Math, 149. (Google Scholar)

Peter Scott. Subgroups of surface groups are almost geometric. Journal of the London Mathematical Society, 2 (3) (1978), 555–565. (Google Scholar)

Hongbin Sun. Non-lerfness of arithmetic hyperbolic manifold groups and mixed 3-manifold groups. Duke Mathematical Journal, 168 (4) (2019), 655–696. (Google Scholar)

Jean-Pierre Serre, Trees. Translated from the french original by John Stillwell. corrected 2nd printing of the 1980 english translation. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. (Google Scholar)

Daniel T Wise. From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry: 3-manifolds, Right-angled Artin Groups, and Cubical Geometry, Volume 117. American Mathematical Soc., 2012. (Google Scholar)


Refbacks

  • There are currently no refbacks.


ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr