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Algebraic convergence of function groups. (English) Zbl 1008.30026
In this article the authors consider a non-elementary torsion-free geometrically finite function group \(\Gamma_{0}\), which is not a quasifuchsian one, without parabolic transformations and so that any quasifuchian component is of genus \(2\) (convex co-compact). If we denote by \(\Omega\) the region of discontinuity of \(\Gamma_{0}\), then there is a unique invariant component \(\Omega_{0}\) and a finite collection of non-invariant ones \(\Omega_{1}\),..., \(\Omega_{s}\), which are non-equivalents, so that \(\Omega/\Gamma_{0}=\Omega_{0}/\Gamma_{0} + \Omega_{1}/\Gamma_{1} + \cdots + \Omega_{s}/\Gamma_{s}\), where \(\Gamma_{j}\) is the stabilizer of the component \(\Omega_{j}\) in \(\Gamma_{0}\). The quasiconformal deformation space \(QH(\Gamma_{0})\) is an open subset of \(D(\Gamma_{0})\), the deformation space of \(\Gamma_{0}\). The universal covering of \(QH(\Gamma_{0})\) is given by the product of the Teichm├╝ller spaces \({\mathcal T}_{j}\) of the surfaces \(S_{j}=\Omega_{j}/\Gamma_{j}\), for \(j=0,1,..,s\). The non-compactness of \(QH(\Gamma_{0})\) in \(D(\Gamma_{0})\) makes natural to ask when a divergent sequence in \(QH(\Gamma_{0})\) converges in \(D(\Gamma_{0})\). R.D.Canary has shown that if \(K \subset {\mathcal T}_{0}\) is compact, then its image in \(QH(\Gamma_{0})\) has compact closure in \(D(\Gamma_{0})\). The authors consider the case of sequences in \(QH(\Gamma_{0})\) for which the corresponding sequence in \({\mathcal T}_{0}\) diverges. A conjecture due to W.P. Thurston asserts that such a sequence will converge in \(D(\Gamma_{0})\) if the corresponding sequence in \({\mathcal T}_{0}\) has a convergent subsequence in the Thurston’s compactification given by projective measured laminations of \(S_{0}\). There is an open subset of such a compactification, called the Masur domain, on which the extended action of the mapping class group acts discontinuously. The main two results of the authors asserts that (i) if the sequence in \(QH(\Gamma_{0})\) has a corresponding sequence in \({\mathcal T}_{0}\) that converges to a rational lamination in Masur domain, then it has a convergent subsequence in \(D(\Gamma_{0})\) and also, (ii) if \(\Gamma_{0}\) is a Schottky group of genus at least \(2\) and \(\lambda\) is a measured lamination in Masur domain, then the set of convex co-compact representations of \(\Gamma_{0}\) for which there is a fixed upper bound on the corresponing length of \(\lambda\) is a precompact subset of \(D(\Gamma_{0})\).

MSC:
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20E08 Groups acting on trees
57M50 General geometric structures on low-dimensional manifolds
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