The visual core of a hyperbolic 3-manifold

James W. Anderson and Richard D. Canary1

Faculty of Mathematical Studies, University of Southampton
Highfield, Southampton, SO17 1BJ, England
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109


In this note we introduce the notion of the visual core of a hyperbolic 3-manifold N = H3/G. One may think of the visual core as a harmonic analysis analogue of the convex core. Explicitly, the visual core V(N) is the projection to N of all the points in H3 at which no component of the domain of discontinuity of G has visual (equivalently harmonic) measure greater than half that of the entire sphere at infinity.

We investigate circumstances under which the visual core V(NH) of a cover NH = H3/H of N embeds in N, via the usual covering map. We begin by showing that the interior of V(NH) embeds in N when H1 is a precisely QF-embedded subgroup of G, while the visual core itself embeds whenever H is a nicely QF-embedded subgroup.

Applying the results from an earlier paper, we are able to conclude that if the algebraic limit of a sequence of isomorphic Kleinian groups is a generalized web group, then the visual core of the algebraic limit manifold embeds in the geometric limit manifold. This result is part of our ongoing investigation of the relationship between algebraic and geometric limits of sequences of isomorphic Kleinian groups.

We also discuss the relationship between the visual core and Klein-Maskit combination along component subgroups. Klein-Maskit combination gives a geometric realization of the topological operation of gluing hyperbolizable 3-manifolds together along incompressible surfaces in their boundaries. While the topology underlying Klein-Maskit combination is well-understood, the geometry is more mysterious. For example, the convex core of a summand of a Klein-Maskit combination need not embed in the resulting manifold. However, we show that the (interior of the) visual core of a summand does embed in the resulting manifold.


1Research supported in part by grants from the National Science Foundation

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On 27 Nov 1999, 12:26.