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Trees and discrete subgroups of Lie groups over local fields. (English) Zbl 0676.22007
Let K be a locally compact nonarchimedean field and G the set of K- rational points of an almost simple algebraic group of rank one defined over K. The author sketches two constructions giving an uncountable family of conjugacy classes of lattices in G and announces the result that these give essentially all lattices. In particular, one obtains infinite families of nonarithmetic lattices. As a corollary the author obtains that no arithmetic lattice in G has the congruence subgroup property. His results also give a new proof of the fact that each finitely generated lattice in G is cocompact.
Reviewer: R.Schulze-Pillot

MSC:
22E40 Discrete subgroups of Lie groups
20G25 Linear algebraic groups over local fields and their integers
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[1] Helmut Behr, Finite presentability of arithmetic groups over global function fields, Proc. Edinburgh Math. Soc. (2) 30 (1987), no. 1, 23 – 39. Groups — St. Andrews 1985. · Zbl 0618.20033
[2] H. Bass and A. Lubotzky (in preparation).
[3] F. Bruhat and J. Tits, Groupes algébriques simples sur un corps local, Proc. Conf. Local Fields (Driebergen, 1966) Springer, Berlin, 1967, pp. 23 – 36 (French). · Zbl 0263.14014
[4] P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5 – 89. G. D. Mostow, Generalized Picard lattices arising from half-integral conditions, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 91 – 106. · Zbl 0615.22008
[5] Patrick Eberlein, Lattices in spaces of nonpositive curvature, Ann. of Math. (2) 111 (1980), no. 3, 435 – 476. · Zbl 0401.53015
[6] Lothar Gerritzen and Marius van der Put, Schottky groups and Mumford curves, Lecture Notes in Mathematics, vol. 817, Springer, Berlin, 1980. · Zbl 0442.14009
[7] M. Gromov and I. Piatetski-Shapiro, Nonarithmetic groups in Lobachevsky spaces, Inst. Hautes Études Sci. Publ. Math. 66 (1988), 93 – 103. · Zbl 0649.22007
[8] H. Garland and M. S. Raghunathan, Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups, Ann. of Math. (2) 92 (1970), 279 – 326. · Zbl 0206.03603
[9] Yasutaka Ihara, On discrete subgroups of the two by two projective linear group over \?-adic fields, J. Math. Soc. Japan 18 (1966), 219 – 235. · Zbl 0158.27702
[10] David Kazhdan, Some applications of the Weil representation, J. Analyse Mat. 32 (1977), 235 – 248. · Zbl 0445.22018
[11] Alexander Lubotzky, Group presentation, \?-adic analytic groups and lattices in \?\?\(_{2}\)(\?), Ann. of Math. (2) 118 (1983), no. 1, 115 – 130. · Zbl 0541.20020
[12] A. Lubotzky, Lattices in rank one semisimple Lie groups over local fields (in preparation). · Zbl 0786.22017
[13] G. A. Margulis, Arithmeticity of the irreducible lattices in the semisimple groups of rank greater than 1, Invent. Math. 76 (1984), no. 1, 93 – 120. · Zbl 0551.20028
[14] John J. Millson, On the first Betti number of a constant negatively curved manifold, Ann. of Math. (2) 104 (1976), no. 2, 235 – 247. · Zbl 0364.53020
[15] G. D. Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. Annals of Mathematics Studies, No. 78. · Zbl 0265.53039
[16] G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), no. 1, 171 – 276. · Zbl 0456.22012
[17] M. S. Raghunathan, On the congruence subgroup problem, Inst. Hautes Études Sci. Publ. Math. 46 (1976), 107 – 161. · Zbl 0347.20027
[18] M. S. Raghunathan, On the congruence subgroup problem. II, Invent. Math. 85 (1986), no. 1, 73 – 117. · Zbl 0603.20044
[19] Jean-Pierre Serre, Le problème des groupes de congruence pour SL2, Ann. of Math. (2) 92 (1970), 489 – 527 (French). · Zbl 0239.20063
[20] Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. · Zbl 0548.20018
[21] Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. · Zbl 0571.58015
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