Discrete subgroups generated by lattices in opposite horospherical subgroups
This paper is essentially the author's Yale thesis. The principal question dealt with in the paper is the following: if \(\Gamma \subset G(\mathbb{R})\) \((G\) a linear algebraic group over \(\mathbb{R}\), which is semisimple and of real \(\text{rank}\geq 2\), [for example: \[ G(\mathbb{R}) = SL_n (\mathbb{R}) (n\geq 3),\;S_{p_{2n}} (\mathbb{R}) (n\geq 2),\;SO(p,q), \] \((2\leq p\leq q)])\) is a discrete subgroup whose intersections \((U\cap \Gamma, V\cap \Gamma)\) in two opposing horospherical subgroups \((U,V)\) have finite covolume in \(U\) and \(V\) respectively, then is \(\Gamma\) itself of finite covolume? [If the answer is yes, then it turns out (by a famous theorem of Margulis) that \(\Gamma\) is commensurable with \(G(\mathbb{Z})\), where \(G(\mathbb{Z}) = \rho^{-1} (SL_N (\mathbb{Z}))\cap G\) for some algebraic representation \(\rho: G\to SL_N\) (one then says that \(\Gamma\) is an arithmetic (non-cocompact) lattice of \(G(\mathbb{R}))\). Conversely, a non-cocompact irreducible arithmetic lattice \(\Gamma\) does have the property that it intersects two opposing horospherical subgroups in lattices.] The author shows (Theorem (0.2)) that if \(G\) is a group split over the reals (and \(G\neq SL_3 (\mathbb{R}))\), then the answer to the above question is yes. (In Theorem (0.3) the author proves a more general statement.) The proof of this theorem involves a theorem of Ratner concerning Raghunathan's conjecture.
scientific article; zbMATH DE number 1200015
scientific article