Margulis' normal subgroup theorem states that any normal subgroup of a higher rank irreducible lattice is either finite or of finite index. 
What are the known counter-examples in rank $1$ ?


I am especially interested by $PU(2,1)$.



Counterexamples to Margulis Normal subgroup theorem in rank 1
Any cocompact lattice in a rank-one Lie group is *word-hyperbolic*. Olshanskii proved that such groups are SQ-universal, meaning in particular that they have uncountably many normal subgroups. Similar results are known for non-uniform lattices, which are *relatively hyperbolic*.


As Yemon Choi suggests in comments, the simplest example comes from $PSL(2,\mathbb{Z})$. The 2-congruence subgroup $\Gamma(2)$ is free of rank two, and hence surjects every two-generator group.


