Hausdorff dimension and limits of Kleinian groups
The authors consider the space \({\mathcal D}(\pi_1(M))\) of discrete faithful representations of \(\pi_1(M)\) into \(\text{PSL}_2({\mathcal C})\), where \(M\) is a compact hyperbolizable 3-manifold, and also two invariants defined on this space, namely, the Hausdorff dimension \(D\) of the limit set of the representation and the bottom \(\Lambda\) of the spectrum of the Laplacian of the quotient manifold. At issue is the question of whether these two invariants are continuous as functions on the representation space \({\mathcal D}(\pi_1(M))\). The continuity depends on the topology with which \({\mathcal D}(\pi_1(M))\) is endowed. Neither invariant is continuous in the compact-open (algebraic) topology on \({\mathcal D}(\pi_1(M))\). The authors review the known results. The authors' new main result is the following: Main Theorem. Let \(M\) be a compact, hyperbolizable 3-manifold which is not homeomorphic to a handlebody. Then \(D\) is continuous on \({\mathcal D}(\pi_1(M))\) in the strong topology. This main theorem follows from known results and the authors' new theorem, Theorem 4.1. Let \(M\) be a compact, hyperbolizable 3-manifold. If \(\{\rho_i\}\) is a sequence in \({\mathcal D}(\pi_1(M))\) converging strongly to \(\rho\), where \(\rho\) is geometrically finite, then \(\{\Lambda(\rho_i)\}\) converges to \(\Lambda(\rho)\). The proof uses spectral theory.
scientific article; zbMATH DE number 1309444