Skip to main content
SpringerLink
Log in

Finiteness properties of arithmetic groups over function fields

  • Published:
Inventiones mathematicae Aims and scope

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abels, H.: Finiteness properties of certain arithmetic groups in the function field case. Isr. J. Math. 76, 113–128 (1991)

    MATH  MathSciNet  Google Scholar 

  2. Abels, H., Tiemeyer, A.: Compactness properties of locally compact groups. Transform. Groups 2, 119–135 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Abramenko, P.: Endlichkeitseigenschaften der Gruppen \(SL_n(\mathbb{F}_q[t])\). Thesis, Frankfurt (1987)

  4. Abramenko, P.: Finiteness properties of Chevalley groups over F q [t]. Isr. J. Math. 87, 203–223 (1994)

    MATH  MathSciNet  Google Scholar 

  5. Abramenko, P.: Twin Buildings and Applications to S-arithmetic Groups. Lecture Notes in Mathematics, vol. 1641. Springer, Berlin (1996)

    MATH  Google Scholar 

  6. Behr, H.: Endliche Erzeugbarkeit arithmetischer Gruppen über Funktionenkörpern. Invent. Math. 7, 1–32 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  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 1979

  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)

    MATH  MathSciNet  Google Scholar 

  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)

  10. Bestvina, M., Brady, N.: Morse theory and finiteness properties of groups. Invent. Math. 129, 445–470 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Borel, A.: Linear algebraic groups. Graduate Texts in Mathematics, no. 126. New York: Springer (1991)

  12. Borel, A., Serre, J.P.: Cohomologie d’immeubles et de groupes S-arithmétiques. Topology 15, 211–232 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  13. Borel, A., Springer, T.A.: Rationality properties of linear algebraic groups II. Tohoku Math. J., II. Ser. 20, 443–497 (1968)

    MATH  MathSciNet  Google Scholar 

  14. Brown, K.: Finiteness properties of groups. J. Pure Appl. Algebra 44, 45–75 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. Brown, K.: Buildings. Springer, New York (1989)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  17. DeBacker, S.: Parameterizing conjugacy classes of maximal unramified tori via Bruhat-Tits theory. Mich. Math. J. 54, 157–178 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Druţu, C.: Nondistorsion des horosphéres dans des immeubles euclidiens et dans des espaces symétriques. Geom. Funct. Anal. 7, 712–754 (1997)

    Article  MathSciNet  Google Scholar 

  19. Druţu, C.: Remplissage dans des réseaux de Q-rang 1 et dans des groupes résolubles. Pac. J. Math. 185, 269–305 (1998)

    Article  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

  22. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  23. Hattori, T.: Non-combability of Hilbert modular groups. Commun. Anal. Geom. 3, 223–251 (1995)

    MATH  MathSciNet  Google Scholar 

  24. Hurrelbrink, J.: Endlich präsentierte arithmetische Gruppen und K 2 über Laurent-Polynomringen. Math. Ann. 225, 123–129 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  25. Keller, K.: Nicht endlich erzeugbare arithmetische Gruppen über Funktionenkörpern. Thesis, Frankfurt (1980)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Leuzinger, E., Pittet, C.: Isoperimetric inequalities for lattices in semisimple Lie groups of rank 2. Geom. Funct. Anal. 6, 489–511 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  28. Leuzinger, E., Pittet, C.: On quadratic Dehn functions. Math. Z. 248, 725–755 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Lubotzky, A.: Lattices in rank one Lie groups over local fields. Geom. Funct. Anal. 1, 406–431 (1991)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  31. Macdonald, I.D.: The Theory of Groups. Malabar, FL: Robert E. Krieger Publishing (1988)

  32. Margulis, G.A.: Discrete Subgroups of Semisimple Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebeite, Springer, Berlin Heidelberg New York (1991)

  33. McHardy, G.: Endliche und fast-endliche Präsentierbarkeit einiger arithmetischer Gruppen. Thesis, Frankfurt (1982)

  34. Nagao, H.: On GL(2,K[X]). J. Inst. Polytech. Osaka City Univ. Ser. A 10, 117–121 (1959)

    MathSciNet  Google Scholar 

  35. Noskov, G.: Multidimensional isoperimetric inequalities and the “non-combability” of the Hilbert modular group. St. Petersbg. Math. J. 11, 535–542 (2000)

    MathSciNet  Google Scholar 

  36. O’Meara, O.T.: On the finite generation of linear groups over Hasse domains. J. Reine Angew. Math. 217, 79–108 (1965)

    MATH  MathSciNet  Google Scholar 

  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)

  38. Platonov, V., Rapinchuk, A.: Algebraic Groups and Number Theory. Pure and Applied Mathematics, no. 139. Academic Press, Boston (1994)

  39. Prasad, G.: Strong approximation for semi-simple groups over function fields. Ann. Math. 105, 553–572 (1977)

    Article  Google Scholar 

  40. Raghunathan, M.S.: A Note on quotients of real algebraic groups by arithmetic subgroups. Invent. Math. 4, 318–335 (1968)

    Article  MathSciNet  Google Scholar 

  41. Raghunathan, M.S.: Discrete Subgroups of Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 68. Springer, New York Heidelberg (1972)

    MATH  Google Scholar 

  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)

  43. Serre, J.-P.: Cohomologie des groupes discrets. Prospects in mathematics, pp. 77–169. Princeton Univ. Press, Princeton, N.J. (1971)

  44. Serre, J.-P.: Trees. Springer, Berlin (2003)

    MATH  Google Scholar 

  45. Splitthoff, S.: Finite presentability of Steinberg groups and related Chevalley groups. Thesis, Bielefeld (1985)

  46. Stuhler, U.: Zur Frage der endlichen Präsentierbarkeit gewisser arithmetischer Gruppen im Funktionenkörperfall. Math. Ann. 224, 217–232 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  47. Stuhler, U.: Homological properties of certain arithmetic groups in the function field case. Invent. Math. 57, 263–281 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  48. Taback, J.: The Dehn function of PSL2(ℤ[1/p]). Geom. Dedicata 102, 179–195 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  49. Tits, J.: Buildings of Spherical Type and Finite BN-pairs. Lecture Notes in Math., vol. 386. Springer, New York (1974)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Virginia, Charlotteville, VA, 22903-4137, USA

    Kai-Uwe Bux

  2. Mathematics Department, Yale University, New Haven, CT, 06520-8283, USA

    Kevin Wortman

Authors
  1. Kai-Uwe Bux
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kevin Wortman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Kai-Uwe Bux or Kevin Wortman.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bux, KU., Wortman, K. Finiteness properties of arithmetic groups over function fields. Invent. math. 167, 355–378 (2007). https://doi.org/10.1007/s00222-006-0017-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-006-0017-y

Keywords

Search

Navigation