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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.

MSC:
22E40 Discrete subgroups of Lie groups
20G25 Linear algebraic groups over local fields and their integers
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