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

Kai-Uwe Bux; Ralf Köhl; Stefan Witzel

doi:https://doi.org/10.4007/annals.2013.177.1.6

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.

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@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.

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@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}, }
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@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.

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@misc{Rousseau:1977, author={Rousseau, G.}, TITLE={Immeubles des groupes réductives sur les corps locaux}, NOTE={Ph.D. thesis, Paris}, YEAR={1977}, }
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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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.

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@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}, }
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@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.

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@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.

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@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.

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@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}, }

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

Abstract

Authors