Thin surface subgroups in cocompact lattices in SL(3,R)

Let G be a semi-simple Lie group, and Γ < G a lattice. Following Sarnak (see [19]), a finitely generated subgroup ∆ of Γ is called thin if ∆ has infinite index in Γ, but is Zariski dense. There has been a good deal of interest recently in thin groups (see for example [7], [8] and [19] to name a few), and there are many results that give credence to the statement that “generic subgroups of lattices are free and thin” (see [7], [9] and [18]). Our interest is rather more focused on the case where ∆ is freely indecomposable, and in previous work [14], the authors exhibited thin surface subgroups contained in any non-uniform lattice in SL(3,R). This note devotes itself to proving the following theorem.