Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T07:03:19.605Z Has data issue: false hasContentIssue false
  • English
  • Français

INDICABLE GROUPS AND ENDOMORPHIC PRESENTATIONS

Published online by Cambridge University Press:  12 December 2011

MUSTAFA GÖKHAN BENLI
MUSTAFA GÖKHAN BENLI*
Affiliation:
Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 77843, USA e-mail: mbenli@math.tamu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we look at presentations of subgroups of finitely presented groups with infinite cyclic quotients. We prove that if H is a finitely generated normal subgroup of a finitely presented group G with G/H cyclic, then H has ascending finite endomorphic presentation. It follows that any finitely presented indicable group without free semigroups has the structure of a semidirect product H ⋊ ℤ, where H has finite ascending endomorphic presentation.

Keywords

20F05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 2 , May 2012 , pp. 335 - 344
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Bartholdi, L., Endomorphic presentations of branch groups, J. Algebra, 268 (2) (2003), 419–443. ISSN .Google Scholar
2.Bartholdi, L., Grigorchuk, R. I. and Šuniḱ, Z., Branch groups, in Handbook of algebra, vol. 3 (Hazewinkel, M., Editor) (North-Holland, Amsterdam, Netherlands, 2003), 989–1112.Google Scholar
3.Baumslag, G., Topics in combinatorial group theory, Lectures in Mathematics ETH Zürich (Birkhäuser Verlag, Basel, Switzerland, 1993). ISBN 3-7643-2921-1.CrossRefGoogle Scholar
4.Baumslag, G. and Roseblade, J. E., Subgroups of direct products of free groups, J. London Math. Soc. (2) 30 (1) (1984), 44–52. ISSN 0024-6107.Google Scholar
5.Grigorchuk, R. I., An example of a finitely presented amenable group that does not belong to the class EG, Mat. Sb. 189 (1) (1998), 79–100. ISSN .Google Scholar
6.Grigorchuk, R., Solved and unsolved problems around one group, in Infinite groups: Geometric, combinatorial and dynamical aspects, vol. 248 of Progress in Mathematics (Bartholdi, L., Checcherini-Silberstein, T., Smirnova-Nagnibeda, T. and Zuk, A., Editors) (Birkhäuser, Basel, 2005), 117–218.CrossRefGoogle Scholar
7.Grigorchuk, R., Savchuk, D. and Šunić, Z., The spectral problem, substitutions and iterated monodromy in Probability and mathematical physics, vol. 42 of CRM proceedings & lecture notes (Dawson, D. A., Jaksic, V. and Vainberg, B., Editors) (American Mathematical Society, Providence, RI, 2007), 225–248.CrossRefGoogle Scholar
8.Grigorchuk, R. I. and Zuk, A., Spectral properties of a torsion-free weakly branch group defined by a three state automaton, in Computational and statistical group theory, vol. 298 of Contemporary mathematics series (American Mathematical Society, Providence, RI, 2002), 57–82.CrossRefGoogle Scholar
9.Hartung, R., A Reidemeister–Schreier theorem for finitely l-presented groups. URL: http://arxiv.org/abs/1108.2403.Google Scholar
10.Higman, G., Subgroups of finitely presented groups, Proc. Roy. Soc. Ser. A 262 (1961), 455–475.Google Scholar
11.Howie, J., On locally indicable groups, Math. Z. 180 (4) (1982), 445–461. ISSN .CrossRefGoogle Scholar
12.Kropholler, P. H., Amenability and right orderable groups, Bull. Lond. Math. Soc. 25 (4) (1993), 347–352. ISSN 0024-6093.CrossRefGoogle Scholar
13.Longobardi, P., Maj, M. and Rhemtulla, A. H., Groups with no free subsemigroups, Trans. Am. Math. Soc. 347 (4) (1995), 1419–1427. ISSN 0002-9947.Google Scholar
14.Lysënok, I. G., A set of defining relations for the Grigorchuk group, Mat. Zametki. 38 (4) (1985), 503–516, 634. ISSN .Google Scholar
15.Morris, D. W., Amenable groups that act on the line, Algebr. Geom. Topol. 6 (2006), 2509–2518. ISSN .Google Scholar
16.Rosset, S., A property of groups of non-exponential growth, Proc. Am. Math. Soc. 54 (1976), 24–26. ISSN .Google Scholar