Zariski dense surface subgroups in SL(3,Z)

The nature of finitely generated infinite index subgroups of SL(3,Z) remains extremely mysterious. It follows from the famous theorem of Tits [12] that free groups abound and, moreover, Zariski dense free groups abound. Less trivially, classical arithmetic considerations (see for example §6.1 of [9]) can be used to construct surface subgroups of SL(3,Z) of every genus ≥ 2. However these are constructed using the theory of quadratic forms and their Zariski closures in SL(3,R) are SO(f,R) for some appropriate ternary quadratic form f ; in particular these surface groups are not Zariski dense in SL(3,R). The purpose of this note is two-fold. First, we aim to shed new light on the structure of Zariski dense faithful representations of surface groups into SL(3,Z), and second we will use this to compare and contrast the surface subgroup structure of SL(3,Z) with many other examples of groups arising naturally in geometry and topology; for example Kleinian groups, word hyperbolic groups, Mapping Class groups and SL(n,Z) with n > 3. The main result is the following.