Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-25T19:31:01.651Z Has data issue: false hasContentIssue false
  • English
  • Français

COARSE COHERENCE OF METRIC SPACES AND GROUPS AND ITS PERMANENCE PROPERTIES

Part of: Metric geometry Homological algebra Homological methods

Published online by Cambridge University Press:  12 September 2018

BORIS GOLDFARB Open the ORCID record for BORIS GOLDFARB [Opens in a new window]  and
JONATHAN L. GROSSMAN Open the ORCID record for JONATHAN L. GROSSMAN [Opens in a new window]
BORIS GOLDFARB*
Affiliation:
Department of Mathematics and Statistics, SUNY, Albany, NY 12222, USA email bgoldfarb@albany.edu
JONATHAN L. GROSSMAN
Affiliation:
Department of Mathematics and Statistics, SUNY, Albany, NY 12222, USA email algrossman@albany.edu
*
✉bgoldfarb@albany.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.

We introduce properties of metric spaces and, specifically, finitely generated groups with word metrics, which we call coarse coherence and coarse regular coherence. They are geometric counterparts of the classical algebraic notion of coherence and the regular coherence property of groups defined and studied by Waldhausen. The new properties can be defined in the general context of coarse metric geometry and are coarse invariants. In particular, they are quasi-isometry invariants of spaces and groups. The new framework allows us to prove structural results by developing permanence properties, including the particularly important fibering permanence property, for coarse regular coherence.

Keywords

coherencecoarse coherenceregular coherenceweak regular coherencemetric spacepermanence

MSC classification

Primary: 16E65: Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
Secondary: 18G99: None of the above, but in this section 51F99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 3 , December 2018 , pp. 422 - 433
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Bell, G. C. and Dranishnikov, A. N., ‘A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory’, Trans. Amer. Math. Soc. 358 (2006), 4749–4764.Google Scholar
Carlsson, G. and Goldfarb, B., ‘The integral K-theoretic Novikov conjecture for groups with finite asymptotic dimension’, Invent. Math. 157 (2004), 405–418.Google Scholar
Carlsson, G. and Goldfarb, B., ‘On homological coherence of discrete groups’, J. Algebra 276 (2004), 502–514.Google Scholar
Carlsson, G. and Goldfarb, B., ‘Controlled algebraic G-theory, I’, J. Homotopy Relat. Struct. 6 (2011), 119–159.Google Scholar
Carlsson, G. and Goldfarb, B., ‘ $K$ -theory with fibred control’, Preprint, 2015, arXiv:1404.5606.Google Scholar
Carlsson, G. and Goldfarb, B., ‘On modules over infinite group rings’, Internat. J. Algebra Comput. 26 (2016), 451–466.Google Scholar
Dranishnikov, A. and Zarichnyi, M., ‘Asymptotic dimension, decomposition complexity, and Haver’s property C’, Topology Appl. 169 (2014), 99–107.Google Scholar
Goldfarb, B., ‘Weak coherence and the K-theory of groups with finite decomposition complexity’, Int. Math. Res. Not. IMRN, to appear, arXiv:1307.5345.Google Scholar
Grossman, J. L., Coarse Coherence of Metric Spaces and Groups, PhD Thesis, University at Albany, SUNY, 2018.Google Scholar
Guentner, E., ‘Permanence in coarse geometry’, in: Recent Progress in General Topology III (eds. Hart, K. P., van Mill, J. and Simon, P.) (Atlantis Press, Paris, 2014), 507–533.Google Scholar
Kasprowski, D., Nicas, A. and Rosenthal, D., ‘Regular finite decomposition complexity’, J. Topol. Anal., to appear, arXiv:1608.04516.Google Scholar
Kropholler, P. H., ‘On groups of type (FP) ∞ ’, J. Pure Appl. Algebra 90 (1993), 55–67.Google Scholar
Kropholler, P. H., ‘Modules possessing projective resolutions of finite type’, J. Algebra 216 (1999), 40–55.Google Scholar
Waldhausen, F., ‘Algebraic K-theory of generalized free products’, Ann. of Math. (2) 108 (1978), 135–256.Google Scholar