# The visual core of a hyperbolic 3-manifold

### James W. Anderson and Richard D. Canary^{1}

### Faculty of Mathematical Studies, University of
Southampton

Highfield, Southampton, SO17 1BJ, England

Department of Mathematics, University of Michigan, Ann Arbor,
MI 48109

## Abstract

In this note we introduce the notion of the visual core of a hyperbolic
3-manifold N = **H**^{3}/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 **H**^{3} 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(N_{H}) of a cover N_{H} = **H**^{3}/H of N embeds in N, via the
usual covering map. We begin by showing that
the interior of V(N_{H}) 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.

### Footnotes:

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

File translated fromT_{E}Xby T_{T}H,version 2.56.

On 27 Nov 1999, 12:26.