Higher finiteness properties of reductive arithmetic groups in positive characteristic: The Rank Theorem

Abstract

We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in a noncommutative isotropic absolutely almost simple group $\mathcal{G}$ over a global function field is one less than the sum of the local ranks of $\mathcal{G}$ taken over the places in $S$. This determines the finiteness properties for $S$-arithmetic subgroups in isotropic reductive groups, confirming the conjectured finiteness properties for this class of groups.

Our main tool is Behr–Harder reduction theory which we recast in terms of the metric structure of euclidean buildings.

  • [Abels:1991] Go to document H. Abels, "Finiteness properties of certain arithmetic groups in the function field case," Israel J. Math., vol. 76, iss. 1-2, pp. 113-128, 1991.
    @article {Abels:1991, MRKEY = {1177335},
      AUTHOR = {Abels, Herbert},
      TITLE = {Finiteness properties of certain arithmetic groups in the function field case},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {76},
      YEAR = {1991},
      NUMBER = {1-2},
      PAGES = {113--128},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {20G25 (20F05)},
      MRNUMBER = {1177335},
      MRREVIEWER = {J. Matthew Douglass},
      DOI = {10.1007/BF02782847},
      ZBLNUMBER = {0819.20051},
      }
  • [Abramenko:1987] P. Abramenko, Endlichkeitseigenschaften der Gruppen ${\mathrm{{SL}}}_n({\mathbb{{F}}}_q[t])$, 1987.
    @misc{Abramenko:1987,
      author={Abramenko, P.},
      TITLE={Endlichkeitseigenschaften der {G}ruppen {${\mathrm{{SL}}}_n({\mathbb{{F}}}_q[t])$}},
      NOTE={Ph.D. thesis, Frankfurt am Main},
      YEAR={1987},
     }
  • [Abramenko:1996] Go to document P. Abramenko, Twin Buildings and Applications to $S$-Arithmetic Groups, New York: Springer-Verlag, 1996, vol. 1641.
    @book {Abramenko:1996, MRKEY = {1624276},
      AUTHOR = {Abramenko, Peter},
      TITLE = {Twin Buildings and Applications to {$S$}-Arithmetic Groups},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1641},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1996},
      PAGES = {x+123},
      ISBN = {3-540-61973-9},
      MRCLASS = {20E42 (20G25 51E24)},
      MRNUMBER = {1624276},
      MRREVIEWER = {Curtis D. Bennett},
      ZBLNUMBER = {0908.20003},
      DOI = {10.1007/BFb0094079},
     }
  • [Abramenko.Brown:2008] Go to document P. Abramenko and K. S. Brown, Buildings: Theory and Applications, New York: Springer-Verlag, 2008, vol. 248.
    @book {Abramenko.Brown:2008, MRKEY = {2439729},
      AUTHOR = {Abramenko, Peter and Brown, Kenneth S.},
      TITLE = {Buildings: Theory and Applications},
      SERIES = {Graduate Texts in Math.},
      VOLUME = {248},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2008},
      PAGES = {xxii+747},
      ISBN = {978-0-387-78834-0},
      MRCLASS = {20E42 (20F55 20J06 51E24 51F15)},
      MRNUMBER = {2439729},
      MRREVIEWER = {Ralf Gramlich},
      DOI = {10.1007/978-0-387-78835-7},
      ZBLNUMBER = {1214.20033},
      }
  • [Alonso:1994] Go to document J. M. Alonso, "Finiteness conditions on groups and quasi-isometries," J. Pure Appl. Algebra, vol. 95, iss. 2, pp. 121-129, 1994.
    @article {Alonso:1994, MRKEY = {1293049},
      AUTHOR = {Alonso, Juan M.},
      TITLE = {Finiteness conditions on groups and quasi-isometries},
      JOURNAL = {J. Pure Appl. Algebra},
      FJOURNAL = {Journal of Pure and Applied Algebra},
      VOLUME = {95},
      YEAR = {1994},
      NUMBER = {2},
      PAGES = {121--129},
      ISSN = {0022-4049},
      CODEN = {JPAAA2},
      MRCLASS = {20J99 (20F32)},
      MRNUMBER = {1293049},
      MRREVIEWER = {Thomas Brady},
      DOI = {10.1016/0022-4049(94)90069-8},
      ZBLNUMBER = {0823.20034},
      }
  • [Bass.Lubotzky:2001] H. Bass and A. Lubotzky, Tree Lattices, Boston, MA: Birkhäuser, 2001, vol. 176.
    @book {Bass.Lubotzky:2001, MRKEY = {1794898},
      AUTHOR = {Bass, Hyman and Lubotzky, Alexander},
      TITLE = {Tree Lattices},
      SERIES = {Progr. Math.},
      VOLUME = {176},
      NOTE = {With appendices by Bass, L. Carbone, Lubotzky, G. Rosenberg and J. Tits},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {2001},
      PAGES = {xiv+233},
      ISBN = {0-8176-4120-3},
      MRCLASS = {20E08 (20E06 20F65 22D05 22E40 57M07)},
      MRNUMBER = {1794898},
      MRREVIEWER = {Alain Valette},
      ZBLNUMBER = {1053.20026},
      }
  • [Behr:1968] H. Behr, "Zur starken Approximation in algebraischen Gruppen über globalen Körpern," J. Reine Angew. Math., vol. 229, pp. 107-116, 1968.
    @article {Behr:1968, MRKEY = {0223371},
      AUTHOR = {Behr, Helmut},
      TITLE = {Zur starken {A}pproximation in algebraischen {G}ruppen über globalen {K}örpern},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {229},
      YEAR = {1968},
      PAGES = {107--116},
      ISSN = {0075-4102},
      MRCLASS = {14.50},
      MRNUMBER = {0223371},
      MRREVIEWER = {E. Abe},
      ZBLNUMBER = {0184.24404},
     }
  • [Behr:1969] Go to document H. Behr, "Endliche Erzeugbarkeit arithmetischer Gruppen über Funktionenkörpern," Invent. Math., vol. 7, pp. 1-32, 1969.
    @article {Behr:1969, MRKEY = {0238853},
      AUTHOR = {Behr, Helmut},
      TITLE = {Endliche {E}rzeugbarkeit arithmetischer {G}ruppen über {F}unktionenkörpern},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {7},
      YEAR = {1969},
      PAGES = {1--32},
      ISSN = {0020-9910},
      MRCLASS = {14.50},
      MRNUMBER = {0238853},
      MRREVIEWER = {T. Kubota},
      ZBLNUMBER = {0169.34802},
      DOI = {10.1007/BF01418772},
     }
  • [Behr:1998] Go to document 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., vol. 495, pp. 79-118, 1998.
    @article {Behr:1998, MRKEY = {1603845},
      AUTHOR = {Behr, Helmut},
      TITLE = {Arithmetic groups over function fields. {I}. {A} complete characterization of finitely generated and finitely presented arithmetic subgroups of reductive algebraic groups},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {495},
      YEAR = {1998},
      PAGES = {79--118},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {20G30 (20F05)},
      MRNUMBER = {1603845},
      MRREVIEWER = {Herbert Abels},
      DOI = {10.1515/crll.1998.023},
      ZBLNUMBER = {0923.20038},
     }
  • [Behr:2004] Go to document H. Behr, "Higher finiteness properties of $S$-arithmetic groups in the function field case I," in Groups: Topological, Combinatorial and Arithmetic Aspects, Cambridge: Cambridge Univ. Press, 2004, vol. 311, pp. 27-42.
    @incollection {Behr:2004, MRKEY = {2073345},
      AUTHOR = {Behr, H.},
      TITLE = {Higher finiteness properties of {$S$}-arithmetic groups in the function field case {I}},
      BOOKTITLE = {Groups: Topological, Combinatorial and Arithmetic Aspects},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {311},
      PAGES = {27--42},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2004},
      MRCLASS = {20G30 (20G25 20J05)},
      MRNUMBER = {2073345},
      MRREVIEWER = {B. Sury},
      DOI = {10.1017/CBO9780511550706.004},
      ZBLNUMBER = {1099.20022},
      }
  • [Behrend.Dhillon:2009] Go to document K. Behrend and A. Dhillon, "Connected components of moduli stacks of torsors via Tamagawa numbers," Canad. J. Math., vol. 61, iss. 1, pp. 3-28, 2009.
    @article {Behrend.Dhillon:2009, MRKEY = {2488447},
      AUTHOR = {Behrend, Kai and Dhillon, Ajneet},
      TITLE = {Connected components of moduli stacks of torsors via {T}amagawa numbers},
      JOURNAL = {Canad. J. Math.},
      FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
      VOLUME = {61},
      YEAR = {2009},
      NUMBER = {1},
      PAGES = {3--28},
      ISSN = {0008-414X},
      CODEN = {CJMAAB},
      MRCLASS = {14G15 (11E72 11R34 14D20 14H60 14L15)},
      MRNUMBER = {2488447},
      MRREVIEWER = {Takao Yamazaki},
      DOI = {10.4153/CJM-2009-001-5},
      ZBLNUMBER = {1219.14030},
      }
  • [Bestvina.Eskin.Wortman:2011] M. Bestvina, A. Eskin, and K. Wortman, Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups.
    @misc{Bestvina.Eskin.Wortman:2011,
      author={Bestvina, M. and Eskin, A. and Wortman, K.},
      TITLE={Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups},
      ARXIV={1106.0162},
      NOTE={to appear in {\it J. Eur. Math. Soc.}},
      SORTYEAR={2011},
     }
  • [Borel.Serre:1976] Go to document A. Borel and . J-P. Serre, "Cohomologie d’immeubles et de groupes $S$-arithmétiques," Topology, vol. 15, iss. 3, pp. 211-232, 1976.
    @article {Borel.Serre:1976, MRKEY = {0447474},
      AUTHOR = {Borel, A. and Serre, {\relax J-P}},
      TITLE = {Cohomologie d'immeubles et de groupes {$S$}-arithmétiques},
      JOURNAL = {Topology},
      FJOURNAL = {Topology. An International Journal of Mathematics},
      VOLUME = {15},
      YEAR = {1976},
      NUMBER = {3},
      PAGES = {211--232},
      ISSN = {0040-9383},
      MRCLASS = {22E40},
      MRNUMBER = {0447474},
      MRREVIEWER = {H. Bass},
      ZBLNUMBER = {0338.20055},
      DOI = {10.1016/0040-9383(76)90037-9},
     }
  • [Borel.Tits:1965] Go to document A. Borel and J. Tits, "Groupes réductifs," Inst. Hautes Études Sci. Publ. Math., iss. 27, pp. 55-150, 1965.
    @article {Borel.Tits:1965, MRKEY = {0207712},
      AUTHOR = {Borel, Armand and Tits, Jacques},
      TITLE = {Groupes réductifs},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {27},
      YEAR = {1965},
      PAGES = {55--150},
      ISSN = {0073-8301},
      MRCLASS = {14.50 (20.75)},
      MRNUMBER = {0207712},
      MRREVIEWER = {F. D. Veldkamp},
      ZBLNUMBER = {0145.17402},
      DOI = {10.1007/BF02684375},
     }
  • [Bridson.Haefliger:1999] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, New York: Springer-Verlag, 1999, vol. 319.
    @book {Bridson.Haefliger:1999, MRKEY = {1744486},
      AUTHOR = {Bridson, Martin R. and Haefliger, Andr{é}},
      TITLE = {Metric Spaces of Non-Positive Curvature},
      SERIES = {Grundl. Math. Wissen.},
      VOLUME = {319},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1999},
      PAGES = {xxii+643},
      ISBN = {3-540-64324-9},
      MRCLASS = {53C23 (20F65 53C70 57M07)},
      MRNUMBER = {1744486},
      MRREVIEWER = {Athanase Papadopoulos},
      ZBLNUMBER = {0988.53001},
      }
  • [Brown:1987] Go to document K. S. Brown, "Finiteness properties of groups. Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985)," J. Pure Appl. Algebra, vol. 44, pp. 45-75, 1987.
    @article{Brown:1987, MRKEY = {0885095},
      AUTHOR = {Brown, Kenneth S.},
      TITLE = {Finiteness properties of groups. {P}roceedings of the {N}orthwestern conference on cohomology of groups ({E}vanston, {I}ll., 1985)},
      JOURNAL = {J. Pure Appl. Algebra},
      FJOURNAL = {Journal of Pure and Applied Algebra},
      VOLUME = {44},
      YEAR = {1987},
      PAGES = {45--75},
      ISSN = {0022-4049},
      CODEN = {JPAAA2},
      MRCLASS = {20J05 (11F75 20F05 22E40)},
      MRNUMBER = {0885095},
      MRREVIEWER = {Ralph Strebel},
      DOI = {10.1016/0022-4049(87)90015-6},
      ZBLNUMBER = {0613.20033},
      }
  • [Brown:1989] K. S. Brown, Buildings, New York: Springer-Verlag, 1989.
    @book {Brown:1989, MRKEY = {0969123},
      AUTHOR = {Brown, Kenneth S.},
      TITLE = {Buildings},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1989},
      PAGES = {viii+215},
      ISBN = {0-387-96876-8},
      MRCLASS = {20-02 (20E32 20G15 22E99 51B25)},
      MRNUMBER = {0969123},
      MRREVIEWER = {W. M. Kantor},
      ZBLNUMBER = {0715.20017},
      }
  • [Bux.Gramlich.Witzel:2009] K. -U. Bux, R. Gramlich, and S. Witzel, Finiteness properties of Chevalley Groups over a Polynomial Ring over a Finite Field, 2009.
    @misc{Bux.Gramlich.Witzel:2009,
      author={Bux, K.-U. and Gramlich, R. and Witzel, S.},
      TITLE={Finiteness properties of {C}hevalley Groups over a Polynomial Ring over a Finite Field},
      YEAR={2009},
      ARXIV={0908.4531},
     }
  • [Bux.Wortman:2007] Go to document K. Bux and K. Wortman, "Finiteness properties of arithmetic groups over function fields," Invent. Math., vol. 167, iss. 2, pp. 355-378, 2007.
    @article {Bux.Wortman:2007, MRKEY = {2270455},
      AUTHOR = {Bux, Kai-Uwe and Wortman, Kevin},
      TITLE = {Finiteness properties of arithmetic groups over function fields},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {167},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {355--378},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F75 (11F06 20G35 22E40 57M07)},
      MRNUMBER = {2270455},
      MRREVIEWER = {Herbert Abels},
      DOI = {10.1007/s00222-006-0017-y},
      ZBLNUMBER = {1126.20030},
      }
  • [Bux.Wortman:2008] Go to document K. Bux and K. Wortman, "Connectivity properties of horospheres in Euclidean buildings and applications to finiteness properties of discrete groups," Invent. Math., vol. 185, iss. 2, pp. 395-419, 2011.
    @article {Bux.Wortman:2008, MRKEY = {2819164},
      AUTHOR = {Bux, Kai-Uwe and Wortman, Kevin},
      TITLE = {Connectivity properties of horospheres in {E}uclidean buildings and applications to finiteness properties of discrete groups},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {185},
      YEAR = {2011},
      NUMBER = {2},
      PAGES = {395--419},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {20E42 (22F50)},
      MRNUMBER = {2819164},
      MRREVIEWER = {Anne Thomas},
      DOI = {10.1007/s00222-011-0311-1},
      ZBLNUMBER = {1237.20041},
      }
  • [Chevalley:1955] C. Chevalley, "Sur certains groupes simples," Tôhoku Math. J., vol. 7, pp. 14-66, 1955.
    @article {Chevalley:1955, MRKEY = {0073602},
      AUTHOR = {Chevalley, C.},
      TITLE = {Sur certains groupes simples},
      JOURNAL = {Tôhoku Math. J.},
      FJOURNAL = {The Tohoku Mathematical Journal. Second Series},
      VOLUME = {7},
      YEAR = {1955},
      PAGES = {14--66},
      ISSN = {0040-8735},
      MRCLASS = {20.0X},
      MRNUMBER = {0073602},
      MRREVIEWER = {F. I. Mautner},
      ZBLNUMBER = {0066.01503},
      }
  • [Devillers.Gramlich.Muehlherr:2009] Go to document A. Devillers, R. Gramlich, and B. Mühlherr, "The sphericity of the complex of non-degenerate subspaces," J. Lond. Math. Soc., vol. 79, iss. 3, pp. 684-700, 2009.
    @article {Devillers.Gramlich.Muehlherr:2009, MRKEY = {2506693},
      AUTHOR = {Devillers, Alice and Gramlich, Ralf and M{ü}hlherr, Bernhard},
      TITLE = {The sphericity of the complex of non-degenerate subspaces},
      JOURNAL = {J. Lond. Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {79},
      YEAR = {2009},
      NUMBER = {3},
      PAGES = {684--700},
      ISSN = {0024-6107},
      MRCLASS = {51E24 (20E42 51A50)},
      MRNUMBER = {2506693},
      MRREVIEWER = {Theo Grundh{ö}fer},
      DOI = {10.1112/jlms/jdn088},
      ZBLNUMBER = {1170.51006},
     }
  • [Dymara.Osajda:2007] Go to document J. Dymara and D. Osajda, "Boundaries of right-angled hyperbolic buildings," Fund. Math., vol. 197, pp. 123-165, 2007.
    @article {Dymara.Osajda:2007, MRKEY = {2365885},
      AUTHOR = {Dymara, Jan and Osajda, Damian},
      TITLE = {Boundaries of right-angled hyperbolic buildings},
      JOURNAL = {Fund. Math.},
      FJOURNAL = {Fundamenta Mathematicae},
      VOLUME = {197},
      YEAR = {2007},
      PAGES = {123--165},
      ISSN = {0016-2736},
      MRCLASS = {20E42 (20F67 57M07)},
      MRNUMBER = {2365885},
      MRREVIEWER = {Anne Thomas},
      DOI = {10.4064/fm197-0-6},
      ZBLNUMBER = {1177.20042},
      }
  • [Gai] Go to document D. Gaitsgory, Contractibility of the space of rational maps.
    @misc{Gai,
      author={Gaitsgory, D.},
      TITLE={Contractibility of the space of rational maps},
      NOTE={{\it Invent. Math.},
      published online March 7, 2012},
      SORTYEAR={2012},
      DOI={10.1007/s00222-012-0392-5},
     }
  • [Gandini:2011] Go to document G. Gandini, Bounding the homological finiteness length.
    @misc{Gandini:2011,
      author={Gandini, G.},
      TITLE={Bounding the homological finiteness length},
      NOTE={{\it Bull. London Math. Soc.},
      published online July 1, 2012},
      DOI = {10.1112/blms/bds047},
      SORTYEAR={2012},
     }
  • [Godement:1963] R. Godement, "Domaines Fondamentaux des Groupes Arithmétiques," in Séminaire Bourbaki, 1962/63. Fasc. 3, No. 257, Paris: Secrétariat mathématique, 1964, vol. 15.
    @incollection{Godement:1963,
      author={Godement, R.},
      TITLE={Domaines Fondamentaux des Groupes Arithmétiques},
      BOOKTITLE = {Séminaire {B}ourbaki, 1962/63. {F}asc. 3, {N}o. 257},
      VOLUME = {15},
      PUBLISHER = {Secrétariat mathématique},
      ADDRESS = {Paris},
      YEAR = {1964},
      MRNUMBER = {0191899},
      ZBLNUMBER = {0136.30101},
     }
  • [Harder:1969] Go to document G. Harder, "Minkowskische Reduktionstheorie über Funktionenkörpern," Invent. Math., vol. 7, pp. 33-54, 1969.
    @article {Harder:1969, MRKEY = {0284441},
      AUTHOR = {Harder, G.},
      TITLE = {Minkowskische {R}eduktionstheorie über {F}unktionenkörpern},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {7},
      YEAR = {1969},
      PAGES = {33--54},
      ISSN = {0020-9910},
      MRCLASS = {14.35 (10.00)},
      MRNUMBER = {0284441},
      MRREVIEWER = {B. Ve{\u\i}sfe{\u\i}ler},
      ZBLNUMBER= {0242.20046},
      DOI = {10.1007/BF01418773},
     }
  • [Harder:1974] Go to document G. Harder, "Chevalley groups over function fields and automorphic forms," Ann. of Math., vol. 100, pp. 249-306, 1974.
    @article {Harder:1974, MRKEY = {0563090},
      AUTHOR = {Harder, G.},
      TITLE = {Chevalley groups over function fields and automorphic forms},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {100},
      YEAR = {1974},
      PAGES = {249--306},
      ISSN = {0003-486X},
      MRCLASS = {10D99 (14G25 22E55)},
      MRNUMBER = {0563090},
      MRREVIEWER = {S. Gelbart},
      ZBLNUMBER = {0309.14041},
      DOI = {10.2307/1971073},
     }
  • [Harder:1977] Go to document G. Harder, "Die Kohomologie $S$-arithmetischer Gruppen über Funktionenkörpern," Invent. Math., vol. 42, pp. 135-175, 1977.
    @article {Harder:1977, MRKEY = {0473102},
      AUTHOR = {Harder, G.},
      TITLE = {Die {K}ohomologie {$S$}-arithmetischer {G}ruppen über {F}unktionenkörpern},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {42},
      YEAR = {1977},
      PAGES = {135--175},
      ISSN = {0020-9910},
      MRCLASS = {22E40},
      MRNUMBER = {0473102},
      MRREVIEWER = {Avner Ash},
      DOI = {10.1007/BF01389786},
      }
  • [Heydebreck:2003] Go to document A. von Heydebreck, "Homotopy properties of certain complexes associated to spherical buildings," Israel J. Math., vol. 133, pp. 369-379, 2003.
    @article {Heydebreck:2003, MRKEY = {1968436},
      AUTHOR = {von Heydebreck, Anja},
      TITLE = {Homotopy properties of certain complexes associated to spherical buildings},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {133},
      YEAR = {2003},
      PAGES = {369--379},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {20E42 (55U10)},
      MRNUMBER = {1968436},
      MRREVIEWER = {Tom De Medts},
      DOI = {10.1007/BF02773075},
      ZBLNUMBER = {1046.55011},
      }
  • [Kac:1990] Go to document V. G. Kac, Infinite-dimensional Lie algebras, Third ed., Cambridge: Cambridge Univ. Press, 1990.
    @book {Kac:1990, MRKEY = {1104219},
      AUTHOR = {Kac, Victor G.},
      TITLE = {Infinite-dimensional {L}ie algebras},
      EDITION = {Third},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1990},
      PAGES = {xxii+400},
      ISBN = {0-521-37215-1; 0-521-46693-8},
      MRCLASS = {17B65 (17B67 17B68 58F07)},
      MRNUMBER = {1104219},
      DOI = {10.1017/CBO9780511626234},
      ZBLNUMBER = {0716.17022},
      }
  • [Kneser:1965] Go to document M. Kneser, "Starke Approximation in algebraischen Gruppen. I," J. Reine Angew. Math., vol. 218, pp. 190-203, 1965.
    @article {Kneser:1965, MRKEY = {0184945},
      AUTHOR = {Kneser, Martin},
      TITLE = {Starke {A}pproximation in algebraischen {G}ruppen. {I}},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {218},
      YEAR = {1965},
      PAGES = {190--203},
      ISSN = {0075-4102},
      MRCLASS = {14.50 (20.75)},
      MRNUMBER = {0184945},
      MRREVIEWER = {T. Ono},
      ZBLNUMBER = {0143.04701},
      DOI = {10.1515/crll.1965.218.190},
     }
  • [Gramlich.Witzel:2009] Go to document R. Köhl and S. Witzel, The sphericity of the Phan geometries of type ${\text{B}}_n$ and ${\text{C}}_n$ and the Phan-type theorem of type ${\text{F}}_4$.
    @misc{Gramlich.Witzel:2009,
      author={Köhl, R. and Witzel, S.},
      TITLE={The sphericity of the {P}han geometries of type {${\text{B}}_n$ and ${\text{C}}_n$} and the {P}han-type theorem of type {${\text{F}}_4$}},
      NOTE={{\it Trans. Amer. Math. Soc.},
      posted online September 25, 2012},
      DOI = {10.1090/S0002-9947-2012-05694-7},
     }
  • [Margulis:1991] G. A. Margulis, Discrete subgroups of semisimple Lie groups, New York: Springer-Verlag, 1991.
    @book {Margulis:1991, MRKEY = {1090825},
      AUTHOR = {Margulis, G. A.},
      TITLE = {Discrete subgroups of semisimple {L}ie groups},
      SERIES = {Ergeb. Math. Grenzgeb.},
      NUMBER = {17},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1991},
      PAGES = {x+388},
      ISBN = {3-540-12179-X},
      MRCLASS = {22E40 (20Hxx 22-02 22D40)},
      MRNUMBER = {1090825},
      MRREVIEWER = {Gopal Prasad},
      ZBLNUMBER = {0732.22008},
      }
  • [Maldeghem.Muehlherr:2009] B. Mühlherr and H. Van Maldeghem, "Codistances in buildings," Innov. Incidence Geom., vol. 10, pp. 81-91, 2009.
    @article {Maldeghem.Muehlherr:2009, MRKEY = {2665195},
      AUTHOR = {M{ü}hlherr, Bernhard and Van Maldeghem, Hendrik},
      TITLE = {Codistances in buildings},
      JOURNAL = {Innov. Incidence Geom.},
      FJOURNAL = {Innovations in Incidence Geometry},
      VOLUME = {10},
      YEAR = {2009},
      PAGES = {81--91},
      ISSN = {1781-6475},
      MRCLASS = {51E24},
      MRNUMBER = {2665195},
      MRREVIEWER = {Petra Schwer},
      }
  • [Nagao:1959] H. Nagao, "On ${ GL}(2,\,K[x])$," J. Inst. Polytech. Osaka City Univ. Ser. A, vol. 10, pp. 117-121, 1959.
    @article {Nagao:1959, MRKEY = {0114866},
      AUTHOR = {Nagao, Hirosi},
      TITLE = {On {${\rm GL}(2,\,K[x])$}},
      JOURNAL = {J. Inst. Polytech. Osaka City Univ. Ser. A},
      VOLUME = {10},
      YEAR = {1959},
      PAGES = {117--121},
      MRCLASS = {20.00},
      MRNUMBER = {0114866},
      MRREVIEWER = {A. Rosenberg},
      ZBLNUMBER = {0092.02504},
      }
  • [Pressley.Segal:1986] A. Pressley and G. Segal, Loop Groups, New York: The Clarendon Press Oxford University Press, 1986.
    @book {Pressley.Segal:1986, MRKEY = {0900587},
      AUTHOR = {Pressley, Andrew and Segal, Graeme},
      TITLE = {Loop Groups},
      SERIES = {Oxford Math. Monogr.},
      NOTE = {Oxford Science Publications},
      PUBLISHER = {The Clarendon Press Oxford University Press},
      ADDRESS = {New York},
      YEAR = {1986},
      PAGES = {viii+318},
      ISBN = {0-19-853535-X},
      MRCLASS = {22E65 (58D15 81D15)},
      MRNUMBER = {0900587},
      MRREVIEWER = {Jouko Mickelsson},
      ZBLNUMBER = {0618.22011},
      ZBLNUMBER = {0603.17012},
      }
  • [Raghunathan:1968] Go to document M. S. Raghunathan, "A note on quotients of real algebraic groups by arithmetic subgroups," Invent. Math., vol. 4, pp. 318-335, 1967/1968.
    @article {Raghunathan:1968, MRKEY = {0230332},
      AUTHOR = {Raghunathan, M. S.},
      TITLE = {A note on quotients of real algebraic groups by arithmetic subgroups},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {4},
      YEAR = {1967/1968},
      PAGES = {318--335},
      ISSN = {0020-9910},
      MRCLASS = {57.45},
      MRNUMBER = {0230332},
      MRREVIEWER = {H. Garland},
      ZBLNUMBER = {0218.22015},
      DOI = {10.1007/BF01425317},
     }
  • [Remy:2002] B. Rémy, "Groupes de Kac–Moody déployés et presque déployés," Astérisque, vol. 277, 2002.
    @article{Remy:2002,
      author={Rémy, B.},
      TITLE={Groupes de {K}ac--{M}oody déployés et presque déployés},
      JOURNAL={Astérisque},
      VOLUME={277},
      YEAR={2002},
     }
  • [Rousseau:1977] G. Rousseau, Immeubles des groupes réductives sur les corps locaux, 1977.
    @misc{Rousseau:1977,
      author={Rousseau, G.},
      TITLE={Immeubles des groupes réductives sur les corps locaux},
      NOTE={Ph.D. thesis, Paris},
      YEAR={1977},
     }
  • [Schulz:2005] B. Schulz, Sphärische Unterkomplexe sphärischer Gebäude, 2005.
    @misc{Schulz:2005,
      author={Schulz, B.},
      TITLE={Sphärische {U}nterkomplexe sphärischer {G}ebäude},
      NOTE={Ph.D. thesis, Frankfurt am Main},
      YEAR={2005},
     }
  • [Schulz:2010] B. Schulz, Spherical subcomplexes of spherical buildings.
    @misc{Schulz:2010,
      author={Schulz, B.},
      TITLE={Spherical subcomplexes of spherical buildings},
      NOTE={to appear in {\it Geometry and Topology}},
      SORTYEAR={2010},
      ARXIV={1007.2407},
     }
  • [Serre:1971] . J-P. Serre, "Cohomologie des groupes discrets," in Prospects in Mathematics, Princeton, N.J.: Princeton Univ. Press, 1971, vol. 70, pp. 77-169.
    @incollection {Serre:1971, MRKEY = {0385006},
      AUTHOR = {Serre, {\relax J-P}},
      TITLE = {Cohomologie des groupes discrets},
      BOOKTITLE = {Prospects in Mathematics},
      VENUE={{P}roc. {S}ympos., {P}rinceton {U}niv., {P}rinceton, {N}.{J}., 1970},
      PAGES = {77--169},
      SERIES={Ann. of Math. Studies},
      VOLUME={70},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, N.J.},
      YEAR = {1971},
      MRCLASS = {22E40 (12A70 20G10 32N99)},
      MRNUMBER = {0385006},
      MRREVIEWER = {H. Bass},
      ZBLNUMBER = {0235.22020},
      }
  • [Serre:1979] . J-P. Serre, "Arithmetic groups," in Homological Group Theory, Cambridge: Cambridge Univ. Press, 1979, vol. 36, pp. 105-136.
    @incollection {Serre:1979, MRKEY = {0564421},
      AUTHOR = {Serre, {\relax J-P}},
      TITLE = {Arithmetic groups},
      BOOKTITLE = {Homological Group Theory},
      VENUE={{P}roc. {S}ympos., {D}urham, 1977},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {36},
      PAGES = {105--136},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1979},
      MRCLASS = {22E40 (20G99 22E41)},
      MRNUMBER = {82b:22021},
      MRREVIEWER = {Gopal Prasad},
      ZBLNUMBER = {0432.20042},
      }
  • [Serre:1980] . J-P. Serre, Trees, New York: Springer-Verlag, 1980.
    @book{Serre:1980,
      author={Serre, {\relax J-P}},
      TITLE = {Trees},
      PUBLISHER={Springer-Verlag},
      ADDRESS={New York},
      YEAR={1980},
      MRNUMBER = {0607504},
      ZBLNUMBER = {0548.20018},
     }
  • [Springer:1994] Go to document T. A. Springer, "Reduction theory over global fields," Proc. Indian Acad. Sci. Math. Sci., vol. 104, iss. 1, pp. 207-216, 1994.
    @article {Springer:1994, MRKEY = {1280068},
      AUTHOR = {Springer, T. A.},
      TITLE = {Reduction theory over global fields},
      JOURNAL = {Proc. Indian Acad. Sci. Math. Sci.},
      FJOURNAL = {Indian Academy of Sciences. Proceedings. Mathematical Sciences},
      VOLUME = {104},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {207--216},
      ISSN = {0253-4142},
      MRCLASS = {11E57 (20G30)},
      MRNUMBER = {1280068},
      MRREVIEWER = {Jasbir Singh Chahal},
      DOI = {10.1007/BF02830884},
      ZBLNUMBER = {0838.20057},
      }
  • [Springer:1998] Go to document T. A. Springer, Linear Algebraic Groups, Second ed., Boston, MA: Birkhäuser, 1998, vol. 9.
    @book {Springer:1998, MRKEY = {1642713},
      AUTHOR = {Springer, T. A.},
      TITLE = {Linear Algebraic Groups},
      SERIES = {Progr. Math.},
      VOLUME = {9},
      EDITION = {Second},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1998},
      PAGES = {xiv+334},
      ISBN = {0-8176-4021-5},
      MRCLASS = {20G15 (14L10)},
      MRNUMBER = {1642713},
      MRREVIEWER = {Andy R. Magid},
      DOI = {10.1007/978-0-8176-4840-4},
      ZBLNUMBER = {0927.20024},
      }
  • [Stuhler:1980] Go to document U. Stuhler, "Homological properties of certain arithmetic groups in the function field case," Invent. Math., vol. 57, iss. 3, pp. 263-281, 1980.
    @article {Stuhler:1980, MRKEY = {0568936},
      AUTHOR = {Stuhler, Ulrich},
      TITLE = {Homological properties of certain arithmetic groups in the function field case},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {57},
      YEAR = {1980},
      NUMBER = {3},
      PAGES = {263--281},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {22E40 (20G99)},
      MRNUMBER = {0568936},
      MRREVIEWER = {Avner Ash},
      DOI = {10.1007/BF01418929},
      ZBLNUMBER = {0432.14026},
      }
  • [Tits:1987] Go to document J. Tits, "Uniqueness and presentation of Kac-Moody groups over fields," J. Algebra, vol. 105, iss. 2, pp. 542-573, 1987.
    @article {Tits:1987, MRKEY = {0873684},
      AUTHOR = {Tits, Jacques},
      TITLE = {Uniqueness and presentation of {K}ac-{M}oody groups over fields},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {105},
      YEAR = {1987},
      NUMBER = {2},
      PAGES = {542--573},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {17B67 (17B20 20G15 20H10)},
      MRNUMBER = {0873684},
      MRREVIEWER = {James F. Hurley},
      DOI = {10.1016/0021-8693(87)90214-6},
      ZBLNUMBER = {0626.22013},
      }
  • [Weil:1982] A. Weil, Adeles and Algebraic Groups, Boston: Birkhäuser, 1982.
    @book {Weil:1982, MRKEY = {0670072},
      AUTHOR = {Weil, Andr{é}},
      TITLE = {Adeles and Algebraic Groups},
      SERIES = {Progr. Math.},
      NUMBER = {23},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston},
      YEAR = {1982},
      PAGES = {iii+126},
      ISBN = {3-7643-3092-9},
      MRCLASS = {10C30 (12A82 12A85 20G35)},
      MRNUMBER = {0670072},
      MRREVIEWER = {K. F. Lai},
      ZBLNUMBER = {0493.14028},
      }
  • [Weiss:2008] R. M. Weiss, The structure of affine buildings, Princeton, NJ: Princeton Univ. Press, 2009.
    @book {Weiss:2008, MRKEY = {2468338},
      AUTHOR = {Weiss, Richard M.},
      TITLE = {The structure of affine buildings},
      SERIES = {Annals of Mathematics Studies},
      NUMBER = {168},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {2009},
      PAGES = {xii+368},
      ISBN = {978-0-691-13881-7},
      MRCLASS = {51E24 (20E42)},
      MRNUMBER = {2468338},
      MRREVIEWER = {Theo Grundh{ö}fer},
      ZBLNUMBER = {1166.51001},
      }
  • [Witzel:2011] S. Witzel, Finiteness Properties of Chevalley Groups over the ring of (Laurent) polynomials over a Finite Field, 2011.
    @misc{Witzel:2011,
      author={Witzel, S.},
      TITLE={Finiteness Properties of {C}hevalley Groups over the ring of ({L}aurent) polynomials over a Finite Field},
      YEAR={2011},
      ARXIV={1112.2961},
     }

Authors

Kai-Uwe Bux

Fakultä fü Mathematik
Universität Bielefeld
Universitästrasse 25
33501 Bielefeld
Germany

Ralf Köhl

Mathematisches Institut
Justus-Liebig-Universität Giessen
Arndstrasse 2
35392 Giessen
Germany

Stefan Witzel

Mathematisches Institute
Universitä t Münster
Einsteinstrasse 62
48148 Münster
Germany