Arithmeticity of rank-1 lattices with dense commensurators in positive characteristic.
Let \(G\) be a cartesian product of finitely many groups, each of which is the group of \(k\)-points of a semisimple algebraic group defined over a local field \(k\). Let \(\Gamma\) be an irreducible lattice in \(G\), such that \(C(\Gamma)/\Gamma\) is infinite, where \(C(\Gamma)\) denotes the commensurator of \(\Gamma\). It was proved by G. A. Margulis that if either \(G/\Gamma\) is compact, or one of the components in the product is defined over a field of characteristic \(0\), or the rank of \(G\) is at least two, then \(\Gamma\) is an arithmetic lattice. The paper under review completes the picture by showing that the same conclusion holds also in the remaining case, namely when \(G\) is the group of \(k\)-points of a simple algebraic group of \(k\)-rank 1, over a local field \(k\) of positive characteristic, and \(\Gamma\) is a nonuniform lattice in \(G\). The proof is based on a description of lattices in such groups, due to \textit{A. Lubotzky} [Geom. Funct. Anal. 405--431 (1991; Zbl 0786.22017)].
scientific article; zbMATH DE number 1901955
scientific article