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Inventiones mathematicae

, Volume 167, Issue 2, pp 355–378 | Cite as

Finiteness properties of arithmetic groups over function fields

  • Kai-Uwe Bux
  • Kevin Wortman
Article

Abstract

We determine when an arithmetic subgroup of a reductive group defined over a global function field is of type FP by comparing its large-scale geometry to the large-scale geometry of lattices in real semisimple Lie groups.

Keywords

Semisimple Group Finiteness Property Zariski Closure Arithmetic Group Spherical Building 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abels, H.: Finiteness properties of certain arithmetic groups in the function field case. Isr. J. Math. 76, 113–128 (1991)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Abels, H., Tiemeyer, A.: Compactness properties of locally compact groups. Transform. Groups 2, 119–135 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Abramenko, P.: Endlichkeitseigenschaften der Gruppen \(SL_n(\mathbb{F}_q[t])\). Thesis, Frankfurt (1987)Google Scholar
  4. 4.
    Abramenko, P.: Finiteness properties of Chevalley groups over F q[t]. Isr. J. Math. 87, 203–223 (1994)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Abramenko, P.: Twin Buildings and Applications to S-arithmetic Groups. Lecture Notes in Mathematics, vol. 1641. Springer, Berlin (1996)zbMATHGoogle Scholar
  6. 6.
    Behr, H.: Endliche Erzeugbarkeit arithmetischer Gruppen über Funktionenkörpern. Invent. Math. 7, 1–32 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Behr, H.: SL3(F q[t]) is not finitely presentable. Proc. Sympos. “Homological group theory” (Durham 1977). Lecture Notes Ser., vol. 36, pp. 213–224. London Math. Soc. Cambridge Univ. Press, Cambridge New York 1979Google Scholar
  8. 8.
    Behr, H.: 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, 79–118 (1998)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Behr, H.: Higher finiteness properties of S-arithmetic groups in the function field case I. In: Müller, T.W. (ed.), Groups: Topological, Combinatorial, Arithmetic Aspects. London Mathematical Society Lecture Notes, vol. 311, pp. 27–42 (2004)Google Scholar
  10. 10.
    Bestvina, M., Brady, N.: Morse theory and finiteness properties of groups. Invent. Math. 129, 445–470 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Borel, A.: Linear algebraic groups. Graduate Texts in Mathematics, no. 126. New York: Springer (1991)Google Scholar
  12. 12.
    Borel, A., Serre, J.P.: Cohomologie d’immeubles et de groupes S-arithmétiques. Topology 15, 211–232 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Borel, A., Springer, T.A.: Rationality properties of linear algebraic groups II. Tohoku Math. J., II. Ser. 20, 443–497 (1968)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Brown, K.: Finiteness properties of groups. J. Pure Appl. Algebra 44, 45–75 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Brown, K.: Buildings. Springer, New York (1989)zbMATHGoogle Scholar
  16. 16.
    Bux, K.-U., Wortman, K.: A geometric proof that SL 2(ℤ[t,t -1]) is not finitely presented. Algebr. Geom. Topol. 6, 839–852 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    DeBacker, S.: Parameterizing conjugacy classes of maximal unramified tori via Bruhat-Tits theory. Mich. Math. J. 54, 157–178 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Druţu, C.: Nondistorsion des horosphéres dans des immeubles euclidiens et dans des espaces symétriques. Geom. Funct. Anal. 7, 712–754 (1997)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Druţu, C.: Remplissage dans des réseaux de Q-rang 1 et dans des groupes résolubles. Pac. J. Math. 185, 269–305 (1998)CrossRefGoogle Scholar
  20. 20.
    Epstein, D.B.A., Cannon, J., Holt, D., Levy, S., Paterson, M., Thurston, W.: Word Processing in Groups. Jones and Bartlett Publishers, Boston (1992)zbMATHGoogle Scholar
  21. 21.
    Gromov, M.: Asymptotic invariants of infinite groups. Geometric Group Theory, Vol. 2 (Sussex, 1991). London Math. Soc. Lecture Note Ser., vol. 182. Cambridge Univ. Press, Cambridge (1993)Google Scholar
  22. 22.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  23. 23.
    Hattori, T.: Non-combability of Hilbert modular groups. Commun. Anal. Geom. 3, 223–251 (1995)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Hurrelbrink, J.: Endlich präsentierte arithmetische Gruppen und K 2 über Laurent-Polynomringen. Math. Ann. 225, 123–129 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Keller, K.: Nicht endlich erzeugbare arithmetische Gruppen über Funktionenkörpern. Thesis, Frankfurt (1980)Google Scholar
  26. 26.
    Krstić, S., McCool, J.: The non-finite presentability of IA(F 3) and GL2(Z[t,t -1]). Invent. Math. 129, 595–606 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Leuzinger, E., Pittet, C.: Isoperimetric inequalities for lattices in semisimple Lie groups of rank 2. Geom. Funct. Anal. 6, 489–511 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Leuzinger, E., Pittet, C.: On quadratic Dehn functions. Math. Z. 248, 725–755 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Lubotzky, A.: Lattices in rank one Lie groups over local fields. Geom. Funct. Anal. 1, 406–431 (1991)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Lubotzky, A., Mozes, S., Raghunathan, M.S.: The word and Riemannian metrics on lattices of semisimple groups. Publ. Math., Inst. Hautes Étud. Sci. 91, 5–53 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Macdonald, I.D.: The Theory of Groups. Malabar, FL: Robert E. Krieger Publishing (1988)Google Scholar
  32. 32.
    Margulis, G.A.: Discrete Subgroups of Semisimple Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebeite, Springer, Berlin Heidelberg New York (1991)Google Scholar
  33. 33.
    McHardy, G.: Endliche und fast-endliche Präsentierbarkeit einiger arithmetischer Gruppen. Thesis, Frankfurt (1982)Google Scholar
  34. 34.
    Nagao, H.: On GL(2,K[X]). J. Inst. Polytech. Osaka City Univ. Ser. A 10, 117–121 (1959)MathSciNetGoogle Scholar
  35. 35.
    Noskov, G.: Multidimensional isoperimetric inequalities and the “non-combability” of the Hilbert modular group. St. Petersbg. Math. J. 11, 535–542 (2000)MathSciNetGoogle Scholar
  36. 36.
    O’Meara, O.T.: On the finite generation of linear groups over Hasse domains. J. Reine Angew. Math. 217, 79–108 (1965)zbMATHMathSciNetGoogle Scholar
  37. 37.
    Pittet, C.: Hilbert modular groups and isoperimetric inequalities. Combinatorial and geometric group theory (Edinburgh 1993), Lecture Note Ser. vol. 204, pp. 259–268. London Math. Soc. (1993)Google Scholar
  38. 38.
    Platonov, V., Rapinchuk, A.: Algebraic Groups and Number Theory. Pure and Applied Mathematics, no. 139. Academic Press, Boston (1994)Google Scholar
  39. 39.
    Prasad, G.: Strong approximation for semi-simple groups over function fields. Ann. Math. 105, 553–572 (1977)CrossRefGoogle Scholar
  40. 40.
    Raghunathan, M.S.: A Note on quotients of real algebraic groups by arithmetic subgroups. Invent. Math. 4, 318–335 (1968)CrossRefMathSciNetGoogle Scholar
  41. 41.
    Raghunathan, M.S.: Discrete Subgroups of Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 68. Springer, New York Heidelberg (1972)zbMATHGoogle Scholar
  42. 42.
    Rehmann, U., Soulé, C.: Finitely presented groups of matrices. Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976). Lecture Notes in Math. vol. 551, pp. 164–169. Springer, Berlin (1976)Google Scholar
  43. 43.
    Serre, J.-P.: Cohomologie des groupes discrets. Prospects in mathematics, pp. 77–169. Princeton Univ. Press, Princeton, N.J. (1971)Google Scholar
  44. 44.
    Serre, J.-P.: Trees. Springer, Berlin (2003)zbMATHGoogle Scholar
  45. 45.
    Splitthoff, S.: Finite presentability of Steinberg groups and related Chevalley groups. Thesis, Bielefeld (1985)Google Scholar
  46. 46.
    Stuhler, U.: Zur Frage der endlichen Präsentierbarkeit gewisser arithmetischer Gruppen im Funktionenkörperfall. Math. Ann. 224, 217–232 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Stuhler, U.: Homological properties of certain arithmetic groups in the function field case. Invent. Math. 57, 263–281 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Taback, J.: The Dehn function of PSL2(ℤ[1/p]). Geom. Dedicata 102, 179–195 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Tits, J.: Buildings of Spherical Type and Finite BN-pairs. Lecture Notes in Math., vol. 386. Springer, New York (1974)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of VirginiaCharlottevilleUSA
  2. 2.Mathematics DepartmentYale UniversityNew HavenUSA

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