Small subgroups of SL(3,Z)

The study of representations of the fundamental groups of finite volume hyperbolic surfaces and finite volume hyperbolic 3-manifolds into Lie groups has been long studied. Classical cases such as the case of the Lie groups SL(2,R), SL(2,C) and SU(2) have provided powerful tools to bring to bear in the study of these groups, and the geometry and topology of the manifolds. More recently, this has been pursued in other Lie groups (see [6], [9], and [20] to name a few). In particular, [6] provides a powerful method for the construction of representations into the groups SL(n,R) for n ≥ 3. This paper was motivated by an examination of the integral points of such representations with a view to addressing questions about the subgroup structure of SL(3,Z). In particular, we can answer a question of Lubotzky which we now describe. The group SL(3,Z) has the Congruence Subgroup Property, and in this sense its finite index subgroup structure is much simpler than that of a lattice in SL(2,C). However, some interesting questions about the structure of subgroups of finite index had remained. For example, in [11], Lubotzky asked the following question: