Hostname: page-component-89b8bd64d-x2lbr Total loading time: 0 Render date: 2026-05-08T07:34:27.257Z Has data issue: false hasContentIssue false
  • English
  • Français

Coarse flow spaces for relatively hyperbolic groups

Part of: Special aspects of infinite or finite groups Categories and geometry $K$-theory of forms

Published online by Cambridge University Press:  09 March 2017

A. Bartels
A. Bartels*
Affiliation:
Westfälische Wilhelms-Universität Münster, Mathematisches Institut, Einsteinstr. 62, D-48149 Münster, Germany email a.bartels@wwu.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce coarse flow spaces for relatively hyperbolic groups and use them to verify a regularity condition for the action of relatively hyperbolic groups on their boundaries. As an application the Farrell–Jones conjecture for relatively hyperbolic groups can be reduced to the peripheral subgroups (up to index-2 overgroups in the $L$ -theory case).

Keywords

Farrell–Jones conjectureK- and L-theory of group rings

MSC classification

Primary: 18F25: Algebraic $K$-theory and $L$-theory 19G24: $L$-theory of group rings 20F67: Hyperbolic groups and nonpositively curved groups

Information

Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 4 , April 2017 , pp. 745 - 779
Copyright
© The Author 2017 

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.)

Article purchase

Temporarily unavailable

References

Antolín, Y., Coulon, R. and Gandini, G., Farrell–Jones via Dehn fillings, Preprint (2015),arXiv:1510.08113v1.Google Scholar
Bartels, A., Farrell, F. T. and Lück, W., The Farrell–Jones conjecture for cocompact lattices in virtually connected Lie groups , J. Amer. Math. Soc. 27 (2014), 339–388.Google Scholar
Bartels, A. and Lück, W., The Borel conjecture for hyperbolic and CAT(0)-groups , Ann. of Math. (2) 175 (2012), 631–689.CrossRefGoogle Scholar
Bartels, A. and Lück, W., Geodesic flow for CAT(0)-groups , Geom. Topol. 16 (2012), 1345–1391.CrossRefGoogle Scholar
Bartels, A., Lück, W. and Reich, H., Equivariant covers for hyperbolic groups , Geom. Topol. 12 (2008), 1799–1882.CrossRefGoogle Scholar
Bartels, A., Lück, W. and Reich, H., The K-theoretic Farrell–Jones conjecture for hyperbolic groups , Invent. Math. 172 (2008), 29–70.Google Scholar
Bartels, A., Lück, W. and Reich, H., On the Farrell–Jones conjecture and its applications , J. Topol. 1 (2008), 57–86.CrossRefGoogle Scholar
Bartels, A., Lück, W., Reich, H. and Rüping, H., K- and L-theory of group rings over GL n (Z) , Publ. Math. Inst. Hautes Études Sci. 119 (2014), 97–125.CrossRefGoogle Scholar
Bartels, A. and Reich, H., Coefficients for the Farrell–Jones conjecture , Adv. Math. 209 (2007), 337–362.Google Scholar
Bass, H., The degree of polynomial growth of finitely generated nilpotent groups , Proc. Lond. Math. Soc. (3) 25 (1972), 603–614.Google Scholar
Bestvina, M. and Mess, G., The boundary of negatively curved groups , J. Amer. Math. Soc. 4 (1991), 469–481.CrossRefGoogle Scholar
Borsuk, K., Theory of retracts, Monografie Matematyczne, vol. 44 (Państwowe Wydawnictwo Naukowe, Warsaw, 1967).Google Scholar
Bowditch, B. H., Relatively hyperbolic groups , Internat. J. Algebra Comput. 22 (2012), 1250016, 66.Google Scholar
Bridson, M. R. and Haefliger, A., Metric spaces of non-positive curvature, Die Grundlehren der mathematischen Wissenschaften, vol. 319 (Springer, Berlin, 1999).Google Scholar
Dahmani, F., Classifying spaces and boundaries for relatively hyperbolic groups , Proc. Lond. Math. Soc. (3) 86 (2003), 666–684.Google Scholar
Davis, J. F., Quinn, F. and Reich, H., Algebraic K-theory over the infinite dihedral group: a controlled topology approach , J. Topol. 4 (2011), 505–528.CrossRefGoogle Scholar
Elsner, T., Systolic groups with isolated flats, Preprint (2008).Google Scholar
Farb, B., Relatively hyperbolic groups , Geom. Funct. Anal. 8 (1998), 810–840.CrossRefGoogle Scholar
Farrell, F. T. and Jones, L. E., K-theory and dynamics. I , Ann. of Math. (2) 124 (1986), 531–569.Google Scholar
Farrell, F. T. and Jones, L. E., Isomorphism conjectures in algebraic K-theory , J. Amer. Math. Soc. 6 (1993), 249–297.Google Scholar
Farrell, F. T. and Roushon, S. K., The Whitehead groups of braid groups vanish , Int. Math. Res. Not. IMRN 10 (2000), 515–526.Google Scholar
Gromov, M., Hyperbolic groups , in Essays in group theory (Springer, New York, 1987), 75–263.CrossRefGoogle Scholar
Gromov, M., Asymptotic invariants of infinite groups , in Geometric group theory, Vol. 2 (Sussex, 1991) (Cambridge University Press, Cambridge, 1993), 1–295.Google Scholar
Groves, D. and Manning, J. F., Dehn filling in relatively hyperbolic groups , Israel J. Math. 168 (2008), 317–429.Google Scholar
Hurewicz, W. and Wallman, H., Dimension theory, Princeton Mathematical Series, vol. 4. (Princeton University Press, Princeton, NJ, 1941).Google Scholar
Kasprowski, D. and Rüping, H., Long and thin covers for cocompact flow spaces, Preprint (2015), arXiv:1502.05001.Google Scholar
Lück, W., K- and L-theory of group rings , in Proceedings of the international congress of mathematicizans, Vol. II (Hindustan Book Agency, New Delhi, 2010), 1071–1098.Google Scholar
Lück, W. and Reich, H., The Baum–Connes and the Farrell–Jones conjectures in K- andL-theory , in Handbook of K-theory, Vols. 1, 2 (Springer, Berlin, 2005), 703–842.Google Scholar
Mineyev, I., Flows and joins of metric spaces , Geom. Topol. 9 (2005), 403–482 (electronic).CrossRefGoogle Scholar
Mineyev, I. and Yaman, A., Relative hyperbolicity and bounded cohomology, Preprint,http://www.math.uiuc.edu/∼mineyev/math/art/rel-hyp.pdf.Google Scholar
Mole, A., Extending a metric on a simplicial complex, Preprint (2013), arXiv:1309.0981.Google Scholar
Mole, A. and Rüping, H., Equivariant refinements, Preprint (2013), arXiv:1308.2799.Google Scholar
Osin, D., Asymptotic dimension of relatively hyperbolic groups , Int. Math. Res. Not. IMRN 35 (2005), 2143–2161.CrossRefGoogle Scholar
Ozawa, N., Amenable actions and applications , in International congress of mathematicians, Vol. II (European Mathematical Society, Zürich, 2006), 1563–1580.Google Scholar
Ozawa, N., Boundary amenability of relatively hyperbolic groups , Topology Appl. 153 (2006), 2624–2630.CrossRefGoogle Scholar
Pears, A. R., Dimension theory of general spaces (Cambridge University Press, Cambridge, 1975).Google Scholar
Quinn, F., Algebraic K-theory over virtually abelian groups , J. Pure Appl. Algebra 216 (2012), 170–183.CrossRefGoogle Scholar
Sauer, R., Amenable covers, volume and L 2 -Betti numbers of aspherical manifolds , J. Reine Angew. Math. 636 (2009), 47–92.Google Scholar
Szabó, G., Wu, J. and Zacharias, J., Rohklin dimension for actions of residually finite groups, Preprint (2014), arXiv:1408.6096.Google Scholar
Szczepański, A., Relatively hyperbolic groups , Michigan Math. J. 45 (1998), 611–618.Google Scholar
Wegner, C., The K-theoretic Farrell–Jones conjecture for CAT(0)-groups , Proc. Amer. Math. Soc. 140 (2012), 779–793.CrossRefGoogle Scholar