Abstract
We prove that if\(G = \underline G (K)\) is theK-rational points of aK-rank one semisimple group\(\underline G \) over a non archimedean local fieldK, thenG has cocompact non-arithmetic lattices and if char(K)>0 also non-uniform ones. We also give a general structure theorem for lattices inG, from which we confirm Serre's conjecture that such arithmetic lattices do not satisfy the congruence subgroup property.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[BK]H. Bass, R. Kulkarni, Uniform tree lattices, J. of the A.M.S. 3 (1990), 843–902.
[Be1]H. Behr, Finite presentability of arithmetic groups over global function fields, Proc. Edinburgh Math. Soc. 30 (1987) 23–39.
[Be2]H. Behr, Nicht endlich erzeugte arithmetische gruppen ülner functionenkörpern, unpublished manuscript.
[BH]A. Borel, G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math. 298 (1978), 53–64.
[BoT1]A. Borel, J. Tits, Groupes reductifs, Publ. Math. I.H.E.S. 27 (1965), 55–150.
[BoT2]A. Borel, J. Tits, Éléments unipotents et sous-groupes paraboliques de groupes reductifs I, Invent. Math. 12 (1971), 95–104.
[BT1]F. Bruhat, J. Tits, Groupes algébrique simples sur un corps local, Proc. Conf. on Local Fields (ed: T.A. Springer) (Driebergen) Springer-Verlag, New York, 1967, pp. 23–36.
[BT2]F. Bruhat, J. Tits, Groupes réductifs sur un corps local I. Données radicielles valuées, Publ. Math. I.H.E.S. 41 (1972), 5–251.
[Eb]P. Eberlein, Lattices in spaces of non positive curvature, Ann. of Math. 111 (1980), 435–476.
[Ef]I. Efrat, On the existence of cusp forms over function fields, J. für die reine und angewandte Math. 399 (1989), 173–187.
[GR]H. Garland, M.S. Raghunathan, Fundamental domains for lattices inR-rank 1 semisimple Lie groups, Ann. of Math. 92 (1970), 279–326.
[GP]L. Gerritzen, M. van der Put, Schottky Groups and Mumford Curves, Springer Lecture Notes in Mathematics 817, Springer-Verlag, New York 1980.
[G]M. Gromov, Hyperbolic groups, in “Essays in Group Theory” (Ed: S.M. Gersten) 75–264, MSRI publications No. 8, Springer-Verlag, New York 1987.
[I]Y. Ihara, On discrete subgroups of the two by two projective linear groups overp-adic fields, J. Math. Soc. Japan 18 (1966), 219–235.
[L1]A. Lubotzky, Free quotients and the congruence kernel ofSL 2, J. of Alg. 77 (1982), 411–418.
[L2]A. Lubotzky, Trees and discrete subgroups of Lie groups over local fields, Bull. A.M.S. 20 (1989), 27–30.
[L3]A. Lubotzky, Lattices of minimal covolume inSL 2: A non Archimedean analogue of Siegel's Theorem\(\mu \geqslant \tfrac{\pi }{{21}}\), J. of the A.M.S. 3 (1990), 961–975.
[LZ]A. Lubotzky, R.J. Zimmer, Variants of Kazhdan's property for subgroups of semi-simple groups, Israel J. of Math. 66 (1989), 289–299.
[R1]M.S. Raghunathan, Discrete Subgroups of Lie Groups, Springer Verlag, New York, 1972.
[R2]M.S. Raghunathan, Discrete subgroups of algebraic groups over, local fields of positive characteristics, Proc. Indian Acad. Sci. (Math. Sci.) 99 (1989), 127–146.
[S1]J.P. Serre, Le problème des groupes de congruence pour SL2, Ann. of Math. 92 (1970), 489–527.
[S2]J.P. Serre, Trees, Springer Verlag, New York, 1980.
[T1]J. Tits, Sur le groupe des automorphismes d'un arbre, in “Essays in Topology and Related Toplics, Mémoires dédicés à Georges de Rham”, Springer-Verlag (1970), 188–211.
[T2]J. Tits, Unipotent elements and parabolic subgroups of reductive groups II, in “Algebraic Groups-Utrecht 1985”, Springer Lecture Notes in Math. 1271 (1987), 265–284.
[T3]J. Tits, Travaux de Margulis sur les sous-groupes discrete de groupes de Lie, in “Sém. Bourbaki 28e année, 1975/6, Exp. 482, Springer Lecture Notes in Math. 576 (1977).
[V]T.N. Venkataramana, On superrigidity and arithmeticity of lattices in semi-simple groups over local fields of arbitrary characteristic, Invent. Math. 92 (1988), 255–306.
Author information
Authors and Affiliations
Additional information
Partially supported by a grant from the Bi-national Science Foundation U.S.-Israel.
Rights and permissions
About this article
Cite this article
Lubotzky, A. Lattices in rank one Lie groups over local fields. Geometric and Functional Analysis 1, 405–431 (1991). https://doi.org/10.1007/BF01895641
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01895641