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Cores of hyperbolic 3-manifolds and limits of Kleinian groups. (English) Zbl 0863.30048
In this work, the relationship between algebraic and geometric limits of Kleinian groups is studied. Previously Jorgensen, Thurston, Kerckhoff and Ohshika found examples of groups where the former is a proper subgroup of the latter. When the geometric limit is finite, a detailed study was done by Jorgensen and Marden. Here the general situation is discussed and the criteria for both limits to be equal are given. It is proven that algebraic limit is the fundamental group of a compact submanifold of the quotient of the geometric limit under some restrictions. When the algebraic limit has non-empty domain of discontinuity, connected limit set and no accidental parabolic elements, it is a precisely embedded subgroup of the geometric limit and therefore will be either a degenerate group or a generalized web group. Combining all these, it is obtained that algebraic limit corresponds to a compact submanifold of the geometric limit. Also shown that any precisely embedded web subgroup of a torsion-free Kleinian group $$\Gamma$$ is a summand of a Klein-Maskit decomposition of $$\Gamma$$. The relations between these limits and the cores of hyperbolic 3-manifolds are established.

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)