Mj, Mahan Cannon-Thurston maps for pared manifolds of bounded geometry. (English) Zbl 1166.57009 Geom. Topol. 13, No. 1, 189-245 (2009). The author uses the following terminology:A pared manifold is a pair \((M,P)\) where \(M\) is a \(3\)-manifold with boundary and \(P\) is a (possibly empty) \(2\)-dimensional submanifold with boundary of \(\partial M\) such that: 1) the fundamental group of each component of \(P\) injects into the fundamental group of \(M\) and it contains an abelian subgroup of finite index; 2) any cylinder \(C:(S^1\times I,\partial(S^1\times I))\to (M,P)\) with \(C_*:\pi_1(S^1\times I)\to \pi_1(M)\) injective is homotopic rel. boundary to \(P\); 3) \(P\) contains every component of \(\partial M\) which has an abelian subgroup of finite index. The definition is due to Thurston.A pared manifold \((M,P)\) is said to have incompressible boundary if each component of \(M\setminus P\) is incompressible in \(M\).A manifold \(M\) equipped with a metric is said to have bounded geometry if in a complement of the cusps the injectivity radius is bounded below by some positive number.The main result of the paper is the following Theorem: Let \(N^h\) be a hyperbolic structure of bounded geometry on a pared manifold \((M,P)\) with incompressible boundary \(\partial_0M=(\partial M-P)\) and let \(M_{gf}\) be a geometrically finite hyperbolic structure adapted to \((M,P)\). Then the map \(i:\widetilde{M_{gf}}\to \widetilde{N^h}\) between the universal covers extends continuously to a map between the Gromov boundaries \(\widehat{M_{gf}}\to\widehat{N^h}\). Furthermore, the limit set of \(\widetilde{M}\) is locally connected. This extension map between the boundaries is an example of a Cannon-Thurston map. The result of this paper is one of several results by the author on similar problems of existence of Cannon-Thurston maps, written in a series of papers. Reviewer: Athanase Papadopoulos (Strasbourg) Cited in 1 ReviewCited in 13 Documents MSC: 57M50 General geometric structures on low-dimensional manifolds 20F67 Hyperbolic groups and nonpositively curved groups 57N16 Geometric structures on manifolds of high or arbitrary dimension Keywords:Cannon-Thurston map; pared manifold; hyperbolic structure; bounded geometry PDFBibTeX XMLCite \textit{M. Mj}, Geom. Topol. 13, No. 1, 189--245 (2009; Zbl 1166.57009) Full Text: DOI arXiv References: [1] W Abikoff, Kleinian groups-geometrically finite and geometrically perverse, Contemp. Math. 74, Amer. Math. Soc. 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