= Solution
A <Kleinian group> is a discrete subgroup of the <orientation>-preserving isometry group of $\mathbb H^3$, identified with $\operatorname{PSL}_2(\mathbb C)$. For an abstract finitely generated group this means a faithful representation into $\operatorname{PSL}_2(\mathbb C)$ whose image is discrete; the chosen image is the geometric realization. Its boundary action is by <Möbius transformations> on $S^2_\infty\cong\mathbb C\cup\{\infty\}$.
Discreteness means the identity is isolated in the group topology, and for a subgroup it also excludes an infinite sequence of distinct elements converging anywhere in the isometry group. Torsion-freeness is not part of this definition: <elliptic Möbius transformations> of finite order are allowed. Therefore a Kleinian quotient need not be a smooth hyperbolic manifold.
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