It is natural to ask what geometric/dynamical restrictions on discrete subgroups of Lie groups produce restrictions on the isomorphism type of the group. For example, Canary and Tsouvalas showed that certain growth conditions on singular values of group elements give rise to bounds on the cohomological dimension of the group. Farre, Pozzetti and Viaggi introduced a restriction on subgroups of PSL(d,C) which guaranteed that they must be isomorphic to convex cocompact subgroups of PSL(2,C).
We introduce a slightly stronger condition which guarantees that the subgroup of PSL(d,C) is isomorphic to a uniform lattice in PSL(2,C). If, in addition, the subgroup is strongly irreducible, then we show that it is the image of a uniform lattice in PSL(2,C) by an irreducible representation of PSL(2,C) into PSL(d,C). We may regard this as a global version of a classical local rigidity result of Ragunathan.
This is joint work with Tengren Zhang and Andy Zimmer.
