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The sphericity of the Phan geometries of type $ B_n$ and $ C_n$ and the Phan-type theorem of type $ F_4$


Authors: Ralf Köhl (né Gramlich) and Stefan Witzel
Journal: Trans. Amer. Math. Soc. 365 (2013), 1577-1602
MSC (2010): Primary 51E24; Secondary 20G30, 20E42, 51A50
DOI: https://doi.org/10.1090/S0002-9947-2012-05694-7
Published electronically: September 25, 2012
MathSciNet review: 3003275
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Abstract | References | Similar Articles | Additional Information

Abstract: We adapt and refine the methods developed by Abramenko and Devillers-Köhl-Mühlherr in order to establish the sphericity of the Phan geometries of types $ B_n$ and $ C_n$ and their generalizations.

As an application we determine the finiteness length of the unitary form of certain hyperbolic Kac-Moody groups. We also reproduce the finiteness length of the unitary form of the groups $ \mathrm {Sp}_{2n}(\mathbb{F}_{q^2}[t,t^{-1}])$.

Another application is the first published proof of the Phan-type theorem of type $ F_4$. Within the revision of the classification of the finite simple groups this concludes the revision of Phan's theorems and their extension to the non-simply laced diagrams. We also reproduce the Phan-type theorems of types $ B_n$ and $ C_n$.


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  • [AA93] Herbert Abels and Peter Abramenko, On the homotopy type of subcomplexes of Tits buildings, Adv. Math. 101 (1993), 78-86. MR 1239453 (95a:55013)
  • [AB08] Peter Abramenko and Kenneth S. Brown, Buildings: Theory and applications, Springer, 2008. MR 2439729 (2009g:20055)
  • [Abe91] Herbert Abels, Finiteness properties of certain arithmetic groups in the function field case, Israel J. Math. 76 (1991), 113-128. MR 1177335 (94a:20077)
  • [Abr96] Peter Abramenko, Twin buildings and applications to $ {S}$-arithmetic groups, Springer, 1996. MR 1624276 (99k:20060)
  • [Art57] Emil Artin, Geometric algebra, Interscience, New York, 1957. MR 0082463 (18:553e)
  • [Beh98] Helmut Behr, Arithmetic groups over function fields I, J. Reine Angew. Math. 495 (1998), 79-118. MR 1603845 (99g:20088)
  • [BGHS07] Curt Bennett, Ralf Gramlich, Corneliu Hoffman, and Sergey Shpectorov, Odd-dimensional orthogonal groups as amalgams of unitary groups. part 1: General simple connectedness, J. Algebra 312 (2007), 426-444. MR 2320467 (2008d:20087)
  • [BGW] Kai-Uwe Bux, Ralf Gramlich, and Stefan Witzel, Higher finiteness properties of reductive arithmetic groups in positive characteristic. The Rank Theorem, arXiv:1102.0428v1. Annals of Math., to appear. http://annals.math.princeton.edu/articles/5670.
  • [Bjö95] Anders Björner, Topological methods, Handbook of Combinatorics, Volume 2 (R.L. Graham, M. Grötschel, and L. Lovãlsz, eds.), Elsevier, 1995, pp. 1819-1872. MR 1373690 (96m:52012)
  • [Bro87] Kenneth S. Brown, Finiteness properties of groups, J. Pure Appl. Algebra 44 (1987), 45-75. MR 885095 (88m:20110)
  • [Bro89] -, Buildings, Springer, 1989. MR 969123 (90e:20001)
  • [BS04] Curtis D. Bennett and Sergey Shpectorov, A new proof of a theorem of Phan, J. Group Theory 7 (2004), 287-310. MR 2062999 (2005k:57004)
  • [Cam] Peter J. Cameron, Notes on classical groups, http://www.maths.qmw.ac.uk/˜pjc/
    class_gps/.
  • [Cam91] -, Projective and polar spaces, QMW Math. Notes, vol. 13, University of London, 1991.
  • [CG99] Lisa Carbone and Howard Garland, Lattices in Kac-Moody groups, Math. Res. Lett. 6 (1999), 439-447. MR 1713142 (2000k:22026)
  • [CM06] Pierre-Emmanuel Caprace and Bernhard Mühlherr, Isomorphisms of Kac-Moody groups which preserve bounded subgroups, Adv. Math. 206 (2006), 250-278. MR 2261755 (2007k:20060)
  • [CR09] Pierre-Emmanuel Caprace and Bertrand Rémy, Groups with a root group datum, Innov. Incidence Geom. 9 (2009), 5-77. MR 2658894 (2011j:20074)
  • [DGM09] Alice Devillers, Ralf Gramlich, and Bernhard Mühlherr, The sphericity of the complex of non-degenerate subspaces, J. Lond. Math. Soc. 79 (2009), 684-700. MR 2506693 (2010c:51018)
  • [DM07] Alice Devillers and Bernhard Mühlherr, On the simple connectedness of certain subsets of buildings, Forum Math. 19 (2007), 955-970. MR 2367950 (2008i:20042)
  • [DMGH09] Tom De Medts, Ralf Gramlich, and Max Horn, Iwasawa decompositions of split Kac-Moody groups, J. Lie Theory 19 (2009), no. 2, 311-337. MR 2572132 (2010j:20075)
  • [DMS09] Tom De Medts and Yoav Segev, A course on Moufang sets, Innov. Incidence Geom. 9 (2009), 79-122. MR 2658895 (2011h:20058)
  • [Dun05] Jonathan R. Dunlap, Uniqueness of Curtis-Phan-Tits amalgams, Ph.D. thesis, Bowling Green State University, 2005.
  • [GHN06] Ralf Gramlich, Max Horn, and Werner Nickel, The complete Phan-type theorem for $ {Sp}(2n,q)$, J. Group Theory 9 (2006), 603-626. MR 2253955 (2007j:20073)
  • [GHN07] -, Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 2: Machine computations, J. Algebra 316 (2007), 591-607. MR 2356846 (2008g:20111)
  • [GHS03] Ralf Gramlich, Corneliu Hoffman, and Sergey Shpectorov, A Phan-type theorem for $ {Sp}(2n,q)$, J. Algebra 264 (2003), 358-384. MR 1981410 (2004i:20088)
  • [GLS98] Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 3, vol. 40, Mathematical Surveys and Monographs, no. 3, American Mathematical Society, 1998. MR 1490581 (98j:20011)
  • [GLS05] -, The classification of the finite simple groups, number 6, vol. 40, Mathematical Surveys and Monographs, no. 6, American Mathematical Society, 2005. MR 2104668 (2005m:20039)
  • [GM08] Ralf Gramlich and Bernhard Mühlherr, Lattices from involutions of Kac-Moody groups, Oberwolfach Rep. 5/2008 (2008), 139-140. MR 2568660
  • [Gra04] Ralf Gramlich, Weak Phan systems of type $ {C}_n$, J. Algebra 280 (2004), 1-9. MR 2081918 (2005h:20027)
  • [Hor08] Max Horn, Involutions of Kac-Moody groups, Ph.D. thesis, TU Darmstadt, 2008.
  • [Hua49] Loo-Keng Hua, On the automorphisms of a sfield, Proc. Natl. Acad. Sci. USA 35 (1949), 386-389. MR 0029886 (10:675d)
  • [Kno05] Rafael Knop, The geometry of Moufang sets, ULB Sachsen-Anhalt, 2005.
  • [Pha77a] Kok-Wee Phan, On groups generated by three-dimensional special unitary groups. I, J. Aust. Math. Soc. 23 (1977), 67-77. MR 0435247 (55:8207)
  • [Pha77b] -, On groups generated by three-dimensional special unitary groups. II, J. Aust. Math. Soc. 23 (1977), 129-146. MR 0447427 (56:5739)
  • [Rém99] Bertrand Rémy, Construction de réseaux en théorie de Kac-Moody, C. R. Acad. Sc. Paris Ser. I Math. 329 (1999), 475-478. MR 1715140 (2001d:20028)
  • [Sol69] Louis Solomon, The Steinberg character of a finite group with BN-pair, Theory of finite groups, Benjamin, 1969, pp. 213-221. MR 0246951 (40:220)
  • [Spa66] Edwin H. Spanier, Algebraic topology, Springer, 1966. MR 0210112 (35:1007)
  • [Ste77] Robert Steinberg, On theorems of Lie-Kolchin, Borel, and Lang, Contributions to algebra (collection of papers dedicated to Ellis Kolchin), Academic Press, 1977, pp. 349-354. MR 0466336 (57:6216)
  • [Tit74] Jacques Tits, Buildings of spherical type and finite BN-pairs, Springer, 1974. MR 0470099 (57:9866)
  • [Tit81] -, A local approach to buildings, The geometric vein: The Coxeter Festschrift, Springer, 1981, pp. 519-547. MR 661801 (83k:51014)
  • [Tit86] Jacques Tits, Ensembles ordonnés, immeubles et sommes amalgamées, Bull. Soc. Math. Belg., Sér. A 38 (1986), 367-387. MR 885545 (88j:20041)

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Additional Information

Ralf Köhl (né Gramlich)
Affiliation: Mathematisches Institut, Universität Gießen, Arndtstraße 2, 35392 Gießen, Germany
Email: ralf.koehl@math.uni-giessen.de

Stefan Witzel
Affiliation: Mathematisches Institut, Universität Münster, Einsteinstraße 62, 48149 Münster, Germany
Email: s.witzel@uni-muenster.de

DOI: https://doi.org/10.1090/S0002-9947-2012-05694-7
Received by editor(s): January 5, 2009
Received by editor(s) in revised form: August 11, 2011
Published electronically: September 25, 2012
Article copyright: © Copyright 2012 Ralf Köhl and Stefan Witzel

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