Compact totally disconnected Moufang buildings. (English) Zbl 1269.20024
By the work of J. Tits, [Buildings of spherical types and finite \(BN\)-pairs. Lect. Notes Math. 386. Springer-Verlag (1974; Zbl 0295.20047)], it is possible to classify all spherical buildings of rank \(>3\), and all spherical buildings of rank \(2\) with the Moufang condition. However, many interesting examples of spherical buildings appear with an additional feature: they are topological buildings. For example, the boundary at infinity of a symmetric space, or a Euclidean building, is a topological spherical building. Furthermore, assuming the space is locally compact, this building will be compact. Therefore, one can wonder which spherical buildings admit a compact topology. Several authors worked on this problem in special cases, mostly when the building is connected or locally connected: see for example K. Burns and R. Spatzier [Publ. Math., Inst. Hautes Étud. Sci. 65, 5-34 (1987; Zbl 0643.53036)], and T. Grundhöfer, N. Knarr and L. Kramer [Geom. Dedicata 83, No. 1-3, 1-29 (2000; Zbl 0974.51014)].
The paper under review extends these results to the general case. The authors prove that a compact infinite irreducible spherical building \(\Delta\) which is Moufang is always the boundary of a symmetric space or of a Bruhat-Tits building. Furthermore, the topology of \(\Delta\) is unique (meaning that every automorphism is continuous), except maybe for the conjugation in complex symmetric spaces.
The paper under review extends these results to the general case. The authors prove that a compact infinite irreducible spherical building \(\Delta\) which is Moufang is always the boundary of a symmetric space or of a Bruhat-Tits building. Furthermore, the topology of \(\Delta\) is unique (meaning that every automorphism is continuous), except maybe for the conjugation in complex symmetric spaces.
Reviewer: Jean Lécureux (Orsay)
MSC:
| 20E42 | Groups with a \(BN\)-pair; buildings |
| 51E24 | Buildings and the geometry of diagrams |
| 22F50 | Groups as automorphisms of other structures |
| 51H20 | Topological geometries on manifolds |
| 53C35 | Differential geometry of symmetric spaces |
Keywords:
Moufang buildings; spherical buildings; topological buildings; compact buildings; locally compact groups; symmetric spacesReferences:
| [1] | P. Abramenko and K. S. Brown, Buildings, theory and application, Springer, New York, 2008. · Zbl 1214.20033 |
| [2] | A. Borel and J. Tits, Homomorphismes “abstraits” de groupes algébriques simples, Ann. Math. 97 (1973), 499-571. · Zbl 0272.14013 · doi:10.2307/1970833 |
| [3] | M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Springer, Berlin, Heidelberg, New York, 1999. · Zbl 0988.53001 |
| [4] | K. S. Brown, Buildings, Springer, Berlin, New York, Heidelberg, 1998. |
| [5] | R. H. Bruck and E. Kleinfeld, The structure of alternative division rings, Proc. Amer. Math. Soc. 2 (1951), 878-890. · Zbl 0044.02205 · doi:10.2307/2031702 |
| [6] | F. Bruhat and J. Tits, Groupes algébriques simples sur un corps local, I. Données radicielles valuées, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5-251. · Zbl 0254.14017 · doi:10.1007/BF02715544 |
| [7] | K. Burns and R. Spatzier, On topological Tits buildings and their classification, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 5-34. · Zbl 0643.53036 · doi:10.1007/BF02698933 |
| [8] | J. Dugundji, Topology, Allyn & Bacon, Boston, 1966. |
| [9] | P. B. Eberlein, Geometry of nonpositively curved manifolds, University of Chicago Press, Chicago, 1996. · Zbl 0883.53003 |
| [10] | R. Engelking, General topology, 2nd edition, Heldermann, Berlin, 1989. · Zbl 0684.54001 |
| [11] | H. Freudenthal, Die Topologie der Lieschen Gruppen als algebraisches Phänomen. I, Ann. of Math. (2) 42 (1941), 1051-1074. · Zbl 1378.22007 · doi:10.2307/1970461 |
| [12] | T. Grundhöfer, Compact disconnected Moufang planes are Desarguesian, Arch. Math. (Basel) 49 (1987), 124-126. · Zbl 0624.51009 · doi:10.1007/BF01200475 |
| [13] | T. Grundhöfer, Compact disconnected planes, inverse limits and homomorphisms, Monatsh. Math. 105 (1988), 261-277. · Zbl 0653.51015 · doi:10.1007/BF01318803 |
| [14] | T. Grundhöfer, N. Knarr and L. Kramer, Flag-homogeneous compact connected polygons, Geom. Dedicata 55 (1995), 95-114. · Zbl 0840.51009 · doi:10.1007/BF02179088 |
| [15] | T. Grundhöfer, N. Knarr and L. Kramer, Flag-homogeneous compact connected polygons. II, Geom. Dedicata 83 (2000), 1-29. · Zbl 0974.51014 · doi:10.1023/A:1005287430579 |
| [16] | T. Grundhöfer and H. Van Maldeghem, Topological polygons and affine buildings of rank three, Atti Sem. Mat. Fis. Univ. Modena 38 (1990), 459-479. · Zbl 0725.51003 |
| [17] | S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Amer. Math. Soc., Providence, RI, 2001. · Zbl 0993.53002 |
| [18] | E. Hewitt and K. A. Ross, Abstract harmonic analysis Vol. I, Academic Press, New York, 1963. · Zbl 0115.10603 |
| [19] | J. G. Hocking and G. S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1961. |
| [20] | J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge, 1990. · Zbl 0725.20028 |
| [21] | E. Kleinfeld, Alternative division rings of characterisitic 2, Proc. Nat. Acad. Sci. U. S. 37 (1951), 818-820. · Zbl 0044.02301 · doi:10.1073/pnas.37.12.818 |
| [22] | A. Kolmogoroff, Zur Begründung der projektiven Geometrie, Ann. of Math. (2) 33 (1932), 175-176. · Zbl 0003.07803 · doi:10.2307/1968111 |
| [23] | L. Kramer, Compact polygons, Dissertation, 72 pp., Math. Fak. Univ. Tübingen, 1994. · Zbl 0844.51006 |
| [24] | L. Kramer, Homogeneous spaces, Tits buildings, and isoparametric hypersurfaces, Mem. Amer. Math. Soc. 158 (2002). · Zbl 1027.57002 |
| [25] | L. Kramer, The topology of a semisimple Lie group is essentially unique, Adv. Math. 228 (2011), 2623-2633. · Zbl 1234.22001 · doi:10.1016/j.aim.2011.07.019 |
| [26] | H. P. Petersson, Lokal kompakte Jordan-Divisionsringe, Abh. Math. Sem. Univ. Hamburg 39 (1973), 164-179. · Zbl 0298.17015 · doi:10.1007/BF02992829 |
| [27] | L. Pontrjagin, Über stetige algebraische Körper, Ann. Math. 33 (1932), 163-174. · Zbl 0003.07802 · doi:10.2307/1968110 |
| [28] | M. Ronan, Lectures on buildings, University of Chicago Press, Chicago, 2009. · Zbl 1190.51008 |
| [29] | H. Salzmann, Homogene kompakte projektive Ebenen, Pacific J. Math. 60 (1975), 217-234. · Zbl 0323.50009 · doi:10.2140/pjm.1975.60.217 |
| [30] | H. Salzmann et al., Compact projective planes, de Gruyter, Berlin, 1995. |
| [31] | H. Salzmann, T. Grundhöfer, H. Hähl and R. Löwen, The classical fields, Cambridge University Press, Cambridge, 2007. · Zbl 1173.00006 |
| [32] | J.-P. Serre, Local fields, Springer, Berlin, New York, Heidelberg, 1979. |
| [33] | M. Stroppel, Locally compact groups, EMS Textbk. Math., Europ. Math. Soc., Zurich, 2006. · Zbl 1102.22005 |
| [34] | J. Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Math. 386, Springer, Berlin, New York, Heidelberg, 1974. · Zbl 0295.20047 |
| [35] | J. Tits, Reductive groups over local fields, in Proc. Symp. Pure Math. 33, Part 1 (Automorphic forms, representations and \(L\)-functions, Corvallis 1977), 29-69, Amer. Math. Soc., Providence, 1979. · Zbl 0415.20035 |
| [36] | J. Tits, Immeubles de type affine, in Buildings and the geometry of diagrams (Como, 1984), 159-190, Lecture Notes in Math. 1181, Springer, Berlin, New York, Heidelberg, 1974. · Zbl 0611.20026 · doi:10.1007/BFb0075514 |
| [37] | J. Tits and R. M. Weiss, Moufang polygons, Springer, Berlin, Heidelberg, New York, 2002. · Zbl 1010.20017 |
| [38] | A. R. Wadsworth, Valuation theory on finite dimensional division algebras, in Valuation theory and its applications, Vol. I (Saskatoon, 1999), 385-449, Field Inst. Commun. 32. Amer. Math. Soc., Providence, 2002. · Zbl 1017.16011 |
| [39] | G. Warner, Harmonic analysis on semi-simple Lie groups. I, Springer, New York, Heidelberg, 1972. · Zbl 0265.22020 |
| [40] | S. Warner, Locally compact vector spaces and algebras over discrete fields, Trans. Amer. Math. Soc. 130 (1968), 463-493. · Zbl 0157.20002 · doi:10.2307/1994760 |
| [41] | S. Warner, Topological fields, North-Holland, Amsterdam, 1989. · Zbl 0675.43003 |
| [42] | A. Weil, Basic number theory, Springer, Berlin, Heidelberg, New York, 1995. · Zbl 0823.11001 |
| [43] | R. M. Weiss, The structure of spherical buildings, Princeton University Press, Princeton, 2003. · Zbl 1061.51011 |
| [44] | R. M. Weiss, Quadrangular algebras, Math. Notes 46, Princeton University Press, Princeton, 2006. · Zbl 1129.17001 · doi:10.1007/b105592 |
| [45] | R. M. Weiss, The structure of affine buildings, Ann. Math. Stud. 168, Princeton University Press, Princeton, 2009. · Zbl 1166.51001 |
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.
