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Lattices in rank one Lie groups over local fields. (English) Zbl 0786.22017
From the text: Let \(K\) be a local field, \(\underline {G}\) a semi-simple algebraic \(K\)-group, \(G=\underline {G}(K)\). In this paper we study lattices (i.e., discrete subgroups of finite covolume) in \(G\) when the \(K\)-rank of \(\underline{G}\) is equal to one and \(K\) is a non-archimedean field. When \(K\text{-rank}(\underline{G})\geq 2\), Margulis (and Venkataramana) showed that every lattice is arithmetic. In contrast we have:
Theorem A. Assuming \(K\) is non-archimedean and \(K\text{-rank}(\underline {G})=1\), then (i) \(G\) has a moduli space of cocompact lattices and, in particular, non-arithmetic ones. (ii) If \(\text{char}(K)>0\), the same holds also with non-uniform lattices.
Recall that when \(\text{char}(K)=0\) every lattice of \(G\) is cocompact. For rank one groups over \(\mathbb{R}\) the question of existence of non- arithmetic lattices is meanwhile only still open for \(SU(n,1)\), \(n\geq 4\). For a survey of known results the reader is referred to the author’s paper [Bull. Am. Math. Soc. 20, 27-30 (1989; Zbl 0676.22007)] where the results of this paper were announced.
The proof of Theorem A gives lattices which are free products of hyperbolic cyclic groups and “cusp subgroups”. Moreover we give a general structure theorem for lattices in \(G\), from which Serre’s conjecture that such arithmetic lattices do not satisfy the congruence subgroup property is confirmed.

22E40 Discrete subgroups of Lie groups
20G25 Linear algebraic groups over local fields and their integers
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[1] [BK]H. Bass, R. Kulkarni, Uniform tree lattices, J. of the A.M.S. 3 (1990), 843–902. · Zbl 0734.05052
[2] [Be1]H. Behr, Finite presentability of arithmetic groups over global function fields, Proc. Edinburgh Math. Soc. 30 (1987) 23–39. · Zbl 0618.20033
[3] [Be2]H. Behr, Nicht endlich erzeugte arithmetische gruppen ülner functionenkörpern, unpublished manuscript.
[4] [BH]A. Borel, G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math. 298 (1978), 53–64. · Zbl 0385.14014
[5] [BoT1]A. Borel, J. Tits, Groupes reductifs, Publ. Math. I.H.E.S. 27 (1965), 55–150. · Zbl 0145.17402
[6] [BoT2]A. Borel, J. Tits, Éléments unipotents et sous-groupes paraboliques de groupes reductifs I, Invent. Math. 12 (1971), 95–104. · Zbl 0238.20055
[7] [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.
[8] [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.
[9] [Eb]P. Eberlein, Lattices in spaces of non positive curvature, Ann. of Math. 111 (1980), 435–476. · Zbl 0432.53023
[10] [Ef]I. Efrat, On the existence of cusp forms over function fields, J. für die reine und angewandte Math. 399 (1989), 173–187. · Zbl 0667.10014
[11] [GR]H. Garland, M.S. Raghunathan, Fundamental domains for lattices inR-rank 1 semisimple Lie groups, Ann. of Math. 92 (1970), 279–326. · Zbl 0206.03603
[12] [GP]L. Gerritzen, M. van der Put, Schottky Groups and Mumford Curves, Springer Lecture Notes in Mathematics 817, Springer-Verlag, New York 1980. · Zbl 0442.14009
[13] [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.
[14] [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. · Zbl 0158.27702
[15] [L1]A. Lubotzky, Free quotients and the congruence kernel ofSL 2, J. of Alg. 77 (1982), 411–418. · Zbl 0495.20021
[16] [L2]A. Lubotzky, Trees and discrete subgroups of Lie groups over local fields, Bull. A.M.S. 20 (1989), 27–30. · Zbl 0676.22007
[17] [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. · Zbl 0731.22009
[18] [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. · Zbl 0706.22010
[19] [R1]M.S. Raghunathan, Discrete Subgroups of Lie Groups, Springer Verlag, New York, 1972. · Zbl 0254.22005
[20] [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. · Zbl 0689.22004
[21] [S1]J.P. Serre, Le problème des groupes de congruence pour SL2, Ann. of Math. 92 (1970), 489–527. · Zbl 0239.20063
[22] [S2]J.P. Serre, Trees, Springer Verlag, New York, 1980.
[23] [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.
[24] [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.
[25] [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).
[26] [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. · Zbl 0649.22008
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