The topology of deformation spaces of Kleinian groups

James W. Anderson, Richard D. Canary and Darryl McCullough

Review from Zentralblatt MATH:

For a compact hyperbolizable 3-manifold \$M\$ (i.e. the interior is hyperbolic, or uniformized by a Kleinian group), with nonempty incompressible boundary, the deformation space \$AH(\pi_1(M))\$ is the space of conjugacy classes of discrete faithful representations of \$\pi_1(M)\$ into \$\text{ PSL}_2(\Bbb C)\$ (or of marked hyperbolic 3-manifolds homotopy equivalent to \$M\$). In the present paper the global topology of deformation spaces of Kleinian groups is studied. The interior \$MH(\pi_1(M))\$ of \$AH(\pi_1(M))\$ is very well understood by the work of various authors, in particular its components are enumerated by topological data, namely the set \${\cal A} (M)\$ of marked compact 3-manifolds homotopy equivalent to \$M\$, while each component is parametrized by analytic data coming from the conformal boundaries of the hyperbolic 3-manifolds.

\`\`Thurston's Ending Lamination Conjecture provides a conjectural classification for elements of \$AH(\pi_1(M))\$ by data which are partially topological, specifically the marked homeomorphism type of the marked hyperbolic 3-manifold \$N\$ as an element of \${\cal A} (M)\$, and partially geometric, coming from the conformal boundary of \$N\$ and the geodesic laminations which encode the asymptotic geometry of any geometrically infinite end of \$N\$. However, the data in this conjectural classification do not vary continuously so they do not provide a clear conjectural picture of the global topology."

\`\`In the case that \$\pi_1(M)\$ is freely indecomposable, the present investigation allows us to give an enumeration of the components of the closure of \$MH(\pi_1(M))\$. Since it is conjectured that this closure equals \$AH(\pi_1(M))\$, this gives a conjectural enumeration of the components of \$AH(\pi_1(M))\$. In particular, we charaterize exactly which components of \$MH(\pi_1(M))\$ have intersecting closures, by analyzing exactly which changes in marked homeomorphism type can occur in the algebraic limit of a sequence of homotopy equivalent marked hyperbolic 3-manifolds."

\`\`One can think of our work as a study of how the topological data in the Ending Lamination Conjecture vary over \$AH(\pi_1(M))\$. It follows from our earlier work [{\it J. W. Anderson} and {\it R. D. Canary}, Invent. Math. 126, No. 2, 205-214 (1996; Zbl 0874.57012)] that these topological data, the marked homeomorphism type, need not be locally constant in general. In this paper, we show that marked homeomorphism type is locally constant modulo primitive shuffles. Roughly, primitive shuffles are homotopy equivalences obtained by shuffling or rearranging the way in which the manifold is glued together along the solid torus components of its characteristic submanifold."

Manifolds \$M\$ are exhibited for which \$AH(\pi_1(M))\$ has infinitely many components.

Reviewed by Bruno Zimmermann

Keywords: hyperbolic 3-manifold; Kleinian group; deformation space; ending lamination conjecture

Classification (MSC2000): 57M50 30F40 57N10

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