Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-11T02:02:36.170Z Has data issue: false hasContentIssue false
  • English
  • Français

Higher horospherical limit sets for G-modules over CAT(0)-spaces

Part of: Special aspects of infinite or finite groups Connections with homological algebra and category theory

Published online by Cambridge University Press:  12 March 2021

ROBERT BIERI  and
ROSS GEOGHEGAN
ROBERT BIERI
Affiliation:
Robert Bieri, Fachbereich Mathematik, Johann Wolfgang Goethe–Universität Frankfurt, D-60054 Frankfurt am Main, Germany, and Department of Mathematical Sciences, Binghamton University (SUNY), Binghamton, NY 13902-6000, U.S.A. e-mail: bieri@math.uni-frankfurt.de
ROSS GEOGHEGAN
Affiliation:
Department of Mathematical Sciences, Binghamton University (SUNY), Binghamton, NY 13902-6000, U.S.A. e-mail: ross@math.binghamton.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The “zeroth stage” of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the “nth stage” for any n.

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 20J05: Homological methods in group theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 1 , July 2021 , pp. 133 - 163
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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

REFERENCES

Bieri, R. and Groves, J. R. J.. Metabelian groups of type (FP)∞ are virtually of type (FP). Proc. London Math. Soc. (3) 45 (1982), no. 2, 365–384.Google Scholar
Bieri, R. and Geoghegan, R.. Connectivity properties of group actions on non-positively curved spaces. Mem. Amer. Math. Soc. 161 (2003), no. 765, xiv+83.Google Scholar
Bieri, R. and Geoghegan, R.. Sigma invariants of direct products of groups. Groups Geom. Dyn. 4 (2010), no. 2, 251–261.10.4171/GGD/82CrossRefGoogle Scholar
Bieri, R. and Geoghegan, R.. Limit sets for modules over groups on CAT(0) spaces: from the Euclidean to the hyperbolic. Proc. Lond. Math. Soc. (3) 112 (2016), no. 6, 1059–1102.10.1112/plms/pdw018CrossRefGoogle Scholar
Bridson, M. R. and Haefliger, A.. Metric spaces of non-positive curvature. Grundlehren Math. Wiss. [Fundamental Principles of Mathematical Sciences], vol. 319 (Springer–Verlag, Berlin, 1999).10.1007/978-3-662-12494-9_18CrossRefGoogle Scholar
Bieri, R.. Deficiency and the geometric invariants of a group. J. Pure Appl. Algebra 208 (2007), no. 3, 951–959, With an appendix by Pascal Schweitzer.10.1016/j.jpaa.2006.02.003CrossRefGoogle Scholar
Bieri, R., Neumann, W. D., and Strebel, R.. A geometric invariant of discrete groups. Invent. Math. 90 (1987), no. 3, 451–477.10.1007/BF01389175CrossRefGoogle Scholar
Bieri, R. and Renz, B.. Valuations on free resolutions and higher geometric invariants of groups. Comment. Math. Helv. 63 (1988), no. 3, 464–497.10.1007/BF02566775CrossRefGoogle Scholar
Brown, K. S.. Trees, valuations, and the Bieri–Neumann–Strebel invariant. Invent. Math. 90 (1987), no. 3, 479–504.10.1007/BF01389176CrossRefGoogle Scholar
Bieri, R. and Strebel, R.. Valuations and finitely presented metabelian groups. Proc. London Math. Soc. (3) 41 (1980), no. 3, 439–464.10.1112/plms/s3-41.3.439CrossRefGoogle Scholar
Gehrke, R.. The higher geometric invariants for groups with sufficient commutativity. Comm. Algebra 26 (1998), no. 4, 1097–1115.10.1080/00927879808826186CrossRefGoogle Scholar
Geoghegan, R. and Ontaneda, P.. Boundaries of cocompact proper CAT(0) spaces. Topology 46 (2007), no. 2, 129–137.10.1016/j.top.2006.12.002CrossRefGoogle Scholar