Deformation spaces of Kleinian surface groups are not locally connected
The author considers the deformation space \(AH(S \times I)\) of marked hyperbolic \(3\)-manifolds which are homotopy equivalent to a given closed surface \(S\) of genus greater or equal to \(2\). The intriguing main result of the paper is that in this situation \(AH(S\times I)\) is not locally connected. This result generalises a result of \textit{K. Bromberg} [Duke Math. J. 156, No. 3, 387--427 (2011; Zbl 1213.30078)], who obtained non-locally-connectedness for Kleinian punctured torus groups. Bromberg's proof makes essential use of \textit{Y. N. Minsky}'s classification of punctured torus groups [Ann. Math. (2) 149, No. 2, 559--626 (1999; Zbl 0939.30034)], which does not extend to higher genus surfaces. Therefore, the author has to deviate from Bromberg's method and he does this by proving and then employing an improved version of the \textit{Filling Theorem}. The latter is the key technical result of the paper.NEWLINENEWLINEInterestingly, the failure of local connectivity of these deformation spaces adds a nice new entry into \textit{D. Sullivan}'s dictionary [Ann. Math. (2) 122, 401--418 (1985; Zbl 0589.30022)], which collects analogies between Kleinian groups and rational maps. Recall that it is still a major open conjecture that the Mandelbrot set is locally connected and that confirming this conjecture would have important consequences for the classification of quadratic maps. For instance, local connectivity would imply that a quadratic map admits no invariant line field on its Julia set and that hyperbolic maps are dense within the set of all quadratic maps. For Kleinian groups, corresponding complex \(1\)-dimensional deformation spaces are, for example, Maskit slice and Bers' slices in the deformation space of punctured torus groups. The latter were shown to be locally connected by Minsky. Whereas for higher dimensions, Lavaurs [PhD thesis, Université Paris-Sud, Orsay (1989)] and \textit{J. Milnor} [Exp. Math. 1, No. 1, 5--24 (1992; Zbl 0762.58018)] showed that, for instance, the cubic connectivity locus is not locally connected. Clearly, the latter result is very much in line with the main result of the paper under review, which shows that all deformation spaces of Kleinian surface groups are not locally connected.
scientific article; zbMATH DE number 6068625
scientific article