Hausdorff dimension and limits of Kleinian groups
type: Item
instance of: scholarly article
title: Hausdorff dimension and limits of Kleinian groups
zbMATH Open document ID: 0935.57021
published in: Geometric and Functional Analysis. GAFA
publication date: 2000-05-09T00:00:00Z
full work available at URL: http://hdl.handle.net/2027.42/41849
review text: 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.
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 1309444
zbMATH Keywords: hyperbolic 3-manifold
zbMATH Keywords: eigenvalues of the Laplacian
zbMATH Keywords: representation of fundamental group
author: Richard D. Canary
author: Edward C. Taylor
reviewed by: James W. Cannon
MaRDI profile type: Publication
OpenAlex ID: W1963983852
DOI: 10.1007/S000390050088
description: scientific article; zbMATH DE number 1309444