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

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

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
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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

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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