×

Buildings have finite asymptotic dimension. (English) Zbl 1177.20049

Summary: It is proved that the asymptotic dimension of any building is finite and equal to the asymptotic dimension of an apartment in that building.

MSC:

20F69 Asymptotic properties of groups
20E42 Groups with a \(BN\)-pair; buildings
51E24 Buildings and the geometry of diagrams
54F45 Dimension theory in general topology
55M10 Dimension theory in algebraic topology
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] G. Bell and A. Dranishnikov, ”Asymptotic Dimension in Bedlęvo,” arXiv: math.GR/0507570 (2005).
[2] G. C. Bell and A.N. Dranishnikov, ”A Hurewicz-Type Theorem for Asymptotic Dimension and Applications to Geometric Group Theory,” Trans. Amer. Math. Soc. 358(11), 4749–4764 (2006). · Zbl 1117.20032 · doi:10.1090/S0002-9947-06-04088-8
[3] M. Bestvina, ”The Virtual Cohomological Dimension of Coxeter Groups,” in Geometric Group Theory (Sussex, 1991), Vol. 1, London Math. Soc. Lecture Note Ser. 181 (Cambridge Univ. Press, Cambridge, 1993), pp. 19–23. · Zbl 0826.20034
[4] M. Bestvina and G. Mess, ”The Boundary of Negatively Curved Groups,” J. Amer. Math. Soc. 4(3), 469–481 (1991). · Zbl 0767.20014 · doi:10.1090/S0894-0347-1991-1096169-1
[5] N. Bourbaki, ”Groupes et algèbres de Lie, chapitres IV–VI” (Hermann, Paris, 1968).
[6] S. V. Buyalo and N. D. Lebedeva, ”Dimensions of Locally and Asymptotically Self-Similar Spaces,” Algebra i Analiz 19(1), 60–92 (2007) [St. Petersburg Math. J. 19 (1), 45–65 (2008)]; arXiv: math.MG/0601744. · Zbl 1145.54029
[7] G. Carlsson and B. Goldfarb, ”On Homological Coherence of Discrete Groups,” J. Algebra 276(2), 502–514 (2004). · Zbl 1057.22013 · doi:10.1016/j.jalgebra.2004.02.006
[8] M. W. Davis, ”Buildings Are CAT(0),” in Geometry and Cohomology in Group Theory, ed. by P. Kropholler, G. Niblo, R. and Stohr, London Math. Soc. Lecture Note Ser. 252 (Cambridge Univ. Press, Cambridge, 1998), pp. 108–123.
[9] A. Dranishnikov, ”Cohomological Approach to Asymptotic Dimension,” arXiv: math.MG/0608215. · Zbl 1175.20035
[10] A. Dranishnikov and T. Januszkiewicz, ”Every Coxeter Group Acts Amenably on a Compact Space,” in Proceedings of the 1999 Topology and Dynamics Conference (Salt Lake City, UT), Topology Proc. 24 Spring, 135–141 (1999). · Zbl 0973.20029
[11] P. Garrett, Buildings and Classical Groups (Chapman & Hall, London, 1997). · Zbl 0933.20019
[12] B. Grave, Coarse Geometry and Asymptotic Dimension, Mathematica Gottingensis (2006); arXiv: math.MG/0601744. · Zbl 1111.20038
[13] M. Gromov, ”Asymptotic Invariants of Infinite Groups,” in Geometric Group Theory (Sussex, 1991), Vol. 2, London Math. Soc. Lecture Note Ser. 182 (Cambridge Univ. Press, Cambridge, 1993), pp. 1–295. · Zbl 0841.20039
[14] D. Matsnev, ”The Baum-Connes Conjecture and Group Actions on Affine Buildings” (PhD Thesis, Pennsylvania State University, 2005).
[15] M. Ronan, Lectures on Buildings, Perspect. Math. 7 (Academic Press, Boston, 1989). · Zbl 0694.51001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.