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Cannon-Thurston maps for pared manifolds of bounded geometry. (English) Zbl 1166.57009
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.

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
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