Q:

Purely parabolic Kleinian groups

What can be said about a discrete finitely generated subgroup $G$ of $PSL(2,\mathbb C)$ whose
nontrivial elements are parabolic? If $G$ is geometrically finite, one can show that $G$ must be elementary so the real question is: Can $G$ be geometrically infinite? 

A:

Suppose that $G$ is a (not necessarily discrete) nonelementary group of isometries of the hyperbolic $n$-space. Then pairs of fixed points of loxodromic elements of $G$ are dense in $L(G)\times L(G)$, where $L(G)$ is the limit set of $G$. See Lemma 3.24 in "Hyperbolic Manifolds and Discrete Groups". I am quite sure that this is also proven in Maskit's book and in Beardon's book. The same argument works for arbitrary convergence groups acting on compacts, e.g. on ideal boundaries of proper geodesic Gromov-hyperbolic spaces. 
Just as a curiosity: There are 2-generated infinite purely elliptic (nondiscrete, of course) isometry groups of the hyperbolic $4$-space, but not of the hyperbolic 3-space. The existence is a cute application of the Tits alternative for $SU(2)$. 

