Arithmeticity of discrete subgroups containing horospherical lattices. (English) Zbl 1440.22019

There is a longstanding conjecture, formulated by G. Margulis (but not published by him). Let \(G\) be a semisimple real algebraic Lie group of real rank \(>1\) (this assumption is necessary here) and \(U\) be a non-trivial horospherical subgroup of \(G\) (i.e. the unipotent radical of some parabolic subgroup in \(G\)). Let \(\Gamma\) be a discrete Zariski dense subgroup of \(G\) that contains an irreducible lattice \(\Delta\) of \(U\). Then \(\Gamma\) is a non-cocompact irreducible arithmetic lattice of \(G\). It is quite rare to find statements that state about a discrete subgroup that it is a lattice, rather than assuming that it is a lattice.
This conjecture was previously proved in some special cases. In this article, this conjecture is proved in the general case. More precisely, it is supposed here that \(\Delta\) is an indecomposable lattice. For lattices \(\Delta \subset U\) included in discrete Zariski dense subset \(\Gamma\) of \(G\), it is proved that these two notions (irreducibility and indecomposability) are equivalent.
The article provides all basic necessary definitions and provides many examples. This makes it rather easy to read. The proof uses numerous auxiliary statements. At the end of the article, the authors for the convenience of the reader list the main steps of their proof. Then two open questions for the cases of groups \(G=\mathrm{SL}(3, \mathbb{R})\) and \(G=\mathrm{SL}(2,\mathbb{R}) \times\mathrm{SL}(2,\mathbb{R})\) are formulated.


22E40 Discrete subgroups of Lie groups
11F06 Structure of modular groups and generalizations; arithmetic groups
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
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