Spherical subcomplexes of spherical buildings

Schulz, Bernd¹

¹ Tulpenhofstraße 31, 63067 Offenbach, Germany

Geometry & topology, Tome 17 (2013) no. 1, pp. 531-562.

Résumé

Let $\Delta$ be a thick, spherical building equipped with its natural CAT(1) metric and let $M$ be a proper, convex subset of $\Delta$. If $M$ is open or if $M$ is a closed ball of radius $\pi ∕2$, then $\Lambda$, the maximal subcomplex supported by $\Delta \setminus M$, is $dim\Lambda$–spherical and non-contractible.

Détail

DOI : 10.2140/gt.2013.17.531

Keywords: spherical building, Cohen–Macaulay, connectivity

Bibliographie

[1] H Abels, Finiteness properties of certain arithmetic groups in the function field case, Israel J. Math. 76 (1991) 113

[2] H Abels, P Abramenko, On the homotopy type of subcomplexes of Tits buildings, Adv. Math. 101 (1993) 78

[3] P Abramenko, Endlichkeitseigenschaften der Gruppen SLn(Fq[t]), PhD thesis, Frankfurt am Main (1987)

[4] P Abramenko, Twin buildings and applications to S-arithmetic groups, 1641, Springer (1996)

[5] P Abramenko, K S Brown, Buildings, 248, Springer (2008)

[6] P Abramenko, H Van Maldeghem, Intersections of apartments, J. Combin. Theory Ser. A 117 (2010) 440

[7] 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

[8] A Björner, Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, Adv. in Math. 52 (1984) 173

[9] A Björner, Topological methods, from: "Handbook of combinatorics, Volume II" (editors R L Graham, M Grötschel, L Lovász), Elsevier (1995) 1819

[10] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, 319, Springer (1999)

[11] K S Brown, Finiteness properties of groups, from: "Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985)" (1987) 45

[12] F Buekenhout, The basic diagram of a geometry, from: "Geometries and groups (Berlin, 1981)" (editors M Aigner, D Jungnickel), Lecture Notes in Math. 893, Springer (1981) 1

[13] K U Bux, K Wortman, Finiteness properties of arithmetic groups over function fields, Invent. Math. 167 (2007) 355

[14] 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

[15] K U Buz, R Gramlich, S Witzel, Finiteness properties of Chevalley groups over a polynomial ring over a finite field

[16] 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

[17] R Charney, A Lytchak, Metric characterizations of spherical and Euclidean buildings, Geom. Topol. 5 (2001) 521

[18] J Dymara, D Osajda, Boundaries of right-angled hyperbolic buildings, Fund. Math. 197 (2007) 123

[19] A Von Heydebreck, Homotopy properties of certain complexes associated to spherical buildings, Israel J. Math. 133 (2003) 369

[20] P J Hilton, S Wylie, Homology theory : An introduction to algebraic topology, Cambridge Univ. Press (1960)

[21] B Kleiner, B Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. (1997) 115

[22] D Quillen, Homotopy properties of the poset of nontrivial p–subgroups of a group, Adv. in Math. 28 (1978) 101

[23] B Schulz, Sphärische Unterkomplexe sphärischer Gebäude, PhD thesis, Frankfurt am Main (2005)

[24] E H Spanier, Algebraic topology, McGraw–Hill Book Co. (1966)

[25] U Stuhler, Homological properties of certain arithmetic groups in the function field case, Invent. Math. 57 (1980) 263

[26] J Tits, Buildings of spherical type and finite BN-pairs, 386, Springer (1974)

[27] J Tits, Ensembles ordonnés, immeubles et sommes amalgamées, Bull. Soc. Math. Belg. Sér. A 38 (1986) 367

[28] K Vogtmann, Spherical posets and homology stability for On,n, Topology 20 (1981) 119

[29] S Witzel, Finiteness properties of Chevalley groups over the Laurent polynomial ring over a finite field