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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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##### References:
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