A covering theorem for hyperbolic 3-manifolds and its applications

Canary, Richard D.

doi:10.1016/0040-9383(94)00055-7

Cited by 163 publications

(219 citation statements)

References 19 publications

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“…By Minsky's result that the restriction of a primitive stable representation to a proper free factor of the free group is Schottky, we immediately get the following corollary. Note that the the above corollary can be also obtained by Canary's covering theorem ( [9]) directly.…”

Section: Free Groups With Parabolicsmentioning

confidence: 67%

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Primitive stable representations of free Kleinian groups

Jeon

Ким²,

Ohshika

et al. 2013

Isr. J. Math.

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Section: Free Groups With Parabolicsmentioning

confidence: 67%

“…First, we can see that the covering of M corresponding to a free factor of F 2g is convex cocompact by using Canary's covering theorem [9]. This shows that if we restrict ρ to a proper free factor, then the representation is Schottky.…”

Section: Free Groups With Parabolicsmentioning

confidence: 99%

Primitive stable representations of free Kleinian groups

Jeon

Ким²,

Ohshika

et al. 2013

Isr. J. Math.

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“…Let p 2 : H 3 /G S 2 → H 3 /G 2 be the covering associated with the inclusion. By the covering theorem proved in Canary [4], the restriction of p 2 to a small neighbourhood of the end e S 2 is a finite-sheeted covering of its image, and in particular, p 2 |S 2 is homotopic to a finite-sheeted covering of a boundary component S 2 of C 2 . Since k|S is homotopic to p 2 • h S as maps to H 3 /G 2 , hence also in C 2 , the restriction k|S is homotopic in C 2 to a finite-sheeted covering of the boundary component S 2 of C 2 , and the end facing that boundary component S 2 has a neighbourhood homeomorphic to S 2 × R.…”

Section: Conclusion Of the Proof Ofmentioning

confidence: 99%

Rigidity and topological conjugates of topologically tame Kleinian groups

Ohshika

1998

Trans. Amer. Math. Soc.

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Abstract. Minsky proved that two Kleinian groups G 1 and G 2 are quasiconformally conjugate if they are freely indecomposable, the injectivity radii at all points of H 3 /G 1 , H 3 /G 2 are bounded below by a positive constant, and there is a homeomorphism h from a topological core of H 3 /G 1 to that of H 3 /G 2 such that h and h −1 map ending laminations to ending laminations. We generalize this theorem to the case when G 1 and G 2 are topologically tame but may be freely decomposable under the same assumption on the injectivity radii. As an application, we prove that if a Kleinian group is topologically conjugate to another Kleinian group which is topologically tame and not a free group, and both Kleinian groups satisfy the assumption on the injectivity radii as above, then they are quasi-conformally conjugate.

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“…Since M 0 lies in AH(S), it is geometrically tame, and we have inj(x) < R for each x ∈ core(M 0 ) (see [Can2,Thm. 6.2] [Bon1]).…”

Section: 3])mentioning

confidence: 99%

Iteration of mapping classes and limits of hyperbolic 3-manifolds

Brock

2001

Invent. math.

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Let ϕ ∈ Mod(S) be an element of the mapping class group of a surface S. We classify algebraic and geometric limits of sequences {Q(of quasi-Fuchsian hyperbolic 3-manifolds ranging in a Bers slice. When ϕ has infinite order with finite-order restrictions, there is an essential subsurface D ϕ ⊂ S so that the geometric limits have homeomorphism typeically, ϕ has pseudo-Anosov restrictions, and D ϕ has components with negative Euler characteristic; these components correspond to new asymptotically periodic simply degenerate ends of the geometric limit. We show there is an s ≥ 1 depending on ϕ and bounded in terms of S so that {Q(ϕ si X, Y )} ∞ i=1 converges algebraically and geometrically, and we give explicit quasi-isometric models for the limits.

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A covering theorem for hyperbolic 3-manifolds and its applications

Cited by 163 publications

References 19 publications

Primitive stable representations of free Kleinian groups

Primitive stable representations of free Kleinian groups

Rigidity and topological conjugates of topologically tame Kleinian groups

Iteration of mapping classes and limits of hyperbolic 3-manifolds

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