Geometry & Topology

Spherical subcomplexes of spherical buildings

Bernd Schulz

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Let Δ be a thick, spherical building equipped with its natural CAT(1) metric and let M be a proper, convex subset of Δ. If M is open or if M is a closed ball of radius π2, then Λ, the maximal subcomplex supported by ΔM, is dimΛ–spherical and non-contractible.

Article information

Geom. Topol., Volume 17, Number 1 (2013), 531-562.

Received: 22 August 2010
Accepted: 12 June 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51E24: Buildings and the geometry of diagrams
Secondary: 11F75: Cohomology of arithmetic groups

spherical building Cohen–Macaulay connectivity


Schulz, Bernd. Spherical subcomplexes of spherical buildings. Geom. Topol. 17 (2013), no. 1, 531--562. doi:10.2140/gt.2013.17.531.

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  • H Abels, Finiteness properties of certain arithmetic groups in the function field case, Israel J. Math. 76 (1991) 113–128
  • H Abels, P Abramenko, On the homotopy type of subcomplexes of Tits buildings, Adv. Math. 101 (1993) 78–86
  • P Abramenko, Endlichkeitseigenschaften der Gruppen $\mathrm{SL}_n(\mathbb{F}_q[t])$, PhD thesis, Frankfurt am Main (1987)
  • P Abramenko, Twin buildings and applications to S-arithmetic groups, Lecture Notes in Mathematics 1641, Springer, Berlin (1996)
  • P Abramenko, K S Brown, Buildings, Graduate Texts in Mathematics 248, Springer, New York (2008)
  • P Abramenko, H Van Maldeghem, Intersections of apartments, J. Combin. Theory Ser. A 117 (2010) 440–453
  • H Behr, Arithmetic groups over function fields I: A complete characterization of finitely generated and finitely presented arithmetic subgroups of reductive algebraic groups, J. Reine Angew. Math. 495 (1998) 79–118
  • A Bj örner, Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, Adv. in Math. 52 (1984) 173–212
  • A Bj örner, Topological methods, from: “Handbook of combinatorics, Volume II”, (R L Graham, M Grötschel, L Lovász, editors), Elsevier, Amsterdam (1995) 1819–1872
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer, Berlin (1999)
  • K S Brown, Finiteness properties of groups, from: “Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985)”, volume 44 (1987) 45–75
  • F Buekenhout, The basic diagram of a geometry, from: “Geometries and groups (Berlin, 1981)”, (M Aigner, D Jungnickel, editors), Lecture Notes in Math. 893, Springer, Berlin (1981) 1–29
  • K-U Bux, K Wortman, Finiteness properties of arithmetic groups over function fields, Invent. Math. 167 (2007) 355–378
  • K-U Bux, K Wortman, Connectivity properties of horospheres in Euclidean buildings and applications to finiteness properties of discrete groups, Invent. Math. 185 (2011) 395–419
  • K-U Buz, R Gramlich, S Witzel, Finiteness properties of Chevalley groups over a polynomial ring over a finite field
  • K-U Buz, R Köhl, S Witzel, Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem, Ann. of Math. 177 (2013) 311–366
  • R Charney, A Lytchak, Metric characterizations of spherical and Euclidean buildings, Geom. Topol. 5 (2001) 521–550
  • J Dymara, D Osajda, Boundaries of right-angled hyperbolic buildings, Fund. Math. 197 (2007) 123–165
  • A von Heydebreck, Homotopy properties of certain complexes associated to spherical buildings, Israel J. Math. 133 (2003) 369–379
  • P J Hilton, S Wylie, Homology theory: An introduction to algebraic topology, Cambridge Univ. Press, New York (1960)
  • B Kleiner, B Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. (1997) 115–197
  • D Quillen, Homotopy properties of the poset of nontrivial $p$–subgroups of a group, Adv. in Math. 28 (1978) 101–128
  • B Schulz, Sphärische Unterkomplexe sphärischer Gebäude, PhD thesis, Frankfurt am Main (2005)
  • E H Spanier, Algebraic topology, McGraw–Hill Book Co., New York (1966)
  • U Stuhler, Homological properties of certain arithmetic groups in the function field case, Invent. Math. 57 (1980) 263–281
  • J Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics 386, Springer, Berlin (1974)
  • J Tits, Ensembles ordonnés, immeubles et sommes amalgamées, Bull. Soc. Math. Belg. Sér. A 38 (1986) 367–387
  • K Vogtmann, Spherical posets and homology stability for $\mathrm{ O}_{n,n}$, Topology 20 (1981) 119–132
  • S Witzel, Finiteness properties of Chevalley groups over the Laurent polynomial ring over a finite field