Hostname: page-component-68945f75b7-zpsnj Total loading time: 0 Render date: 2024-08-06T09:11:26.284Z Has data issue: false hasContentIssue false
  • English
  • Français

Construction of an infinitely generated group that is not a free product of surface groups and abelian groups, but which acts freely on an ℝ-tree

Published online by Cambridge University Press:  14 November 2011

Andeas Zastrow
Andeas Zastrow
Affiliation:
Fakultät und Institut für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

The existence of a group H as described in the title shows that the statement of Rips's Theorem for finitely generated groups cannot be extended without further complications to infinitely generated groups. The construction as given in this paper uses a careful combinatorial description of the fundamental group of the Hawaiian Earrings and a length function that can be put on a special subgroup. Then the existence of H follows using a theorem of Chiswell, Alperin and Moss.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 2 , 1998 , pp. 433 - 445
Copyright
Copyright © Royal Society of Edinburgh 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alperin, R. C. and Moss, K. N.. Complete trees for groups with a real valued length function. J. London Math. Soc. (2) 31 (1985), 55–68.CrossRefGoogle Scholar
2Bestvina, M. and Feighn, M.. Stable actions of groups on real trees. Invent. Math. 121 (1995), 287–321.CrossRefGoogle Scholar
3Chiswell, I. M.. Abstract length functions in groups. Math. Proc. Cambridge Philos. Soc. 80 (1976), 451–63.CrossRefGoogle Scholar
4Culler, Marc and Morgan, John W.. Group actions on ℝ-trees. Proc. London Math. Soc. (3) 55 (1987), 571–604.CrossRefGoogle Scholar
5Dunwoody, M. J.. Groups acting on protrees (Southampton Preprint Series 244, April 1994).Google Scholar
6Gaboriau, D., Levitt, G. and Paulin, F.. Pseudogroups of isometrics of ℝ: Rips's Theorem on free actions on ℝ-trees. Israel J. Math. 87 (1994), 403–28.CrossRefGoogle Scholar
7Griffiths, H. B.. The fundamental group of two spaces with a common point. Quart. J. Math. Oxford (2) 5 (1954), 175–90; correction: ibid. 6 (1955), 154–5.CrossRefGoogle Scholar
8Griffiths, H. B.. Infinite products of semi-groups and local connectivity. Proc. London Math. Soc. (3) 6 (1956), 455–80.CrossRefGoogle Scholar
9Johansson, Ingebirgt. Topologische Untersuchungen über unverzweigte Überlagerungsflächen. Skr. norske Vidensk. Akad. Oslo (math.-naturv. Kl.) 1 (1931), 1–69.Google Scholar
10Lubkin, Saul. Theory of covering spaces. Trans. Amer. Math. Soc. (1) 104 (1962), 205–38.CrossRefGoogle Scholar
11Lyndon, R. C.. Length functions in groups. Math. Scand. 12 (1963), 209–34.CrossRefGoogle Scholar
12Lyndon, R. and Schupp, P.. Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 89 (Berlin: Springer, 1977).Google Scholar
13Morgan, John and Morrision, Ian. A van-Kampen-Theorem for weak joins. Proc. London Math. Soc. (3) 53 (1986), 562–76.CrossRefGoogle Scholar
14Promislow, David. Equivalence classes of length functions on groups. Proc. London Math. Soc. (3) 51 (1985), 449–77.CrossRefGoogle Scholar
15Shalen, Peter B.. Dendrology of groups: an introduction. In Essays in Group Theory, ed. Gersten, S. M., Proceedings of the Seminar held at the MSRI in Berkeley (California) in late June 1985, MSRI-Publications, pp. 265–321 (New York: Springer, 1987).CrossRefGoogle Scholar
16Shalen, Peter B.. Dendrology and its applications. In Group Theory from a Geometrical Viewpoint, ed. Ghys, E., Haefliger, A. and Verjovski, A., Proceedings of the conference held at the ICTP in Trieste (Italy) 26 March-6 April 1990, pp. 543–616 (River Edge, NY: World Scientific, 1991).Google Scholar
17Smit, Bart de. The fundamental group of the Hawaiian Earring is not free. Internal. J. Algebra Cornput. 3 (1992), 33–7.CrossRefGoogle Scholar