Linear representations and arithmeticity of lattices in product of trees, preprint, p

In this paper we continue our study of lattices in the automorphisms groups of products of trees initiated in [BM97], [BM00a], [BM00b], [Moz98] (see also [Gla03], [BG02], [Rat04]). We concentrate here on the interplay between the linear representation theory and the structure of these lattices. Before turning to the main results of this paper it may be worthwhile to put certain concepts and results from [BM00a], [BM00b] in perspective, and explain the motivations for our approach. A lattice Γ in a locally compact group G is a discrete subgroup such that the quotient space G/Γ carries a finite G-invariant measure. If in addition G/Γ is compact the lattice is called cocompact or uniform. Consider the following special setting: let Qp be the field of p-adic numbers and let G = PSL(2,Qp) × PSL(2,Qq). Lattices in products fall into two natural classes: reducible and irreducible. A lattice Γ < G is called reducible if it has a subgroup of finite index which is a direct product Γp × Γq where Γp < PSL(2,Qp) and Γq < PSL(2,Qq) ar