Constructing thin subgroups in SL(4,R)

Let G be a semi-simple Lie group and Γ < G be a lattice. This paper is motivated by the attempt to understand the infinite index subgroup structure of Γ . In particular, to understand the possibilities for infinite index, finitely generated, freely indecomposable, Zariski dense subgroups of Γ . The study of Zariski dense subgroups of semi-simple Lie groups has a long and rich history. Some highlights are the Borel Density Theorem, which establishes that a lattice in a semi-simple Lie group is Zariski dense, and works of Oh [17] and Venkataramana [23], which establish in certain cases that Zariski dense subgroups of a nonuniform lattice in a high rank Lie group are themselves lattices. In addition, there are many constructions (based on ping-pong) of free subgroups (or more generally subgroups that are free products), of semi-simple Lie groups, and lattices that are Zariski dense (see [16, 22] and references therein). Indeed, it is shown in [20] that, in a precise