Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T20:22:43.780Z Has data issue: false hasContentIssue false
  • English
  • Français

Higher generating subgroups and Cohen–Macaulay complexes

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  17 December 2019

Benjamin Brück
Benjamin Brück*
Affiliation:
Bielefeld University, PO Box 10 01 31, D-33501Bielefeld, Germany ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We show how to find higher generating families of subgroups, in the sense of Abels and Holz, for groups acting on Cohen–Macaulay complexes. We apply this to groups with a BN-pair to prove higher generation by parabolic and Levi subgroups and describe higher generating families of parabolic subgroups in Aut(Fn).

Keywords

higher generating subgroupsCohen–Macaulay complexesgroups with BN-pairTits buildingsautomorphism groups of free groups

MSC classification

Primary: 20F65: Geometric group theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 1 , February 2020 , pp. 275 - 285
Copyright
Copyright © Edinburgh Mathematical Society 2019

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

1.Abels, H. and Holz, S., Higher generation by subgroups, J. Algebra 160 (2) (1993), 310–341.CrossRefGoogle Scholar
2.Abramenko, P. and Brown, K. S., Buildings, Graduate Texts in Mathematics, Volume 248 (Springer, New York, 2008).CrossRefGoogle Scholar
3.Björner, A., Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, Adv. Math. 52(3) (1984), 173–212.CrossRefGoogle Scholar
4.Björner, A., Topological methods, in Handbook of combinatorics, Volumes 1, 2, pp. 1819–1872 (Elsevier Science. B. V., Amsterdam, 1995).Google Scholar
5.Björner, A., Wachs, M. and Welker, V., On sequentially Cohen-Macaulay complexes and posets, Israel J. Math. 169 (2009), 295–316.CrossRefGoogle Scholar
6.Bredon, G. E., Introduction to compact transformation groups, Pure and Applied Mathematics, Volume 46 (Academic Press, New York–London, 1972).Google Scholar
7.Brown, K. S., The coset poset and probabilistic zeta function of a finite group, J. Algebra 225(2) (2000), 989–1012.CrossRefGoogle Scholar
8.Bux, K.-U., Fluch, M. G., Marschler, M., Witzel, S. and Zaremsky, M. C. B., The braided Thompson's groups are of type F∞, J. Reine Angew. Math. 718 (2016), 59–101. With an appendix by Zaremsky.Google Scholar
9.Charney, R. and Davis, M. W., The K(π, 1)-problem for hyperplane complements associated to infinite reflection groups, J. Amer. Math. Soc. 8(3) (1995), 597–627.Google Scholar
10.Hatcher, A. and Vogtmann, K., The complex of free factors of a free group, Q. J. Math. 49(196) (1998), 459–468.CrossRefGoogle Scholar
11.Holz, S., Endliche Identifizierbarkeit von Gruppen. PhD thesis, Universität Bielefeld, 1985.Google Scholar
12.Lutz, F. H., Triangulated manifolds with few vertices and vertex-transitive group actions. PhD thesis, Technische Universität Berlin, 1999.Google Scholar
13.Meier, J., Meinert, H. and VanWyk, L., Higher generation subgroup sets and the Σ-invariants of graph groups, Comment. Math. Helv. 73(1) (1998), 22–44.10.1007/s000140050044CrossRefGoogle Scholar
14.Quillen, D., Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. Math. 28(2) (1978), 101–128.CrossRefGoogle Scholar
15.Rego, Y. S., On the finiteness length of some soluble linear groups (arxiv.org/abs/1901. 06704, 2019).Google Scholar
16.Segev, Y., Group actions on finite acyclic simplicial complexes, Israel J. Math. 82(1–3) (1993), 381–394.CrossRefGoogle Scholar
17.Shareshian, J. and Woodroofe, R., Order complexes of coset posets of finite groups are not contractible, Adv. Math. 291 (2016), 758–773.CrossRefGoogle Scholar
18.Solomon, L., The Steinberg character of a finite group with BN-pair, in Theory of finite groups, Symposium, Harvard University, Cambridge, MA, 1968, pp. 213–221 (Benjamin, New York, 1969).Google Scholar
19.Stanley, R. P., Combinatorics and commutative algebra, Progress in Mathematics, Volume 41 (Springer, 1996).Google Scholar
20.von Heydebreck, A., Homotopy properties of certain complexes associated to spherical buildings, Israel J. Math. 133 (2003), 369–379.CrossRefGoogle Scholar