Geometric Behaviour of Kleinian Groups on Boundaries for Deformation Spaces

Ohshika, Ken’ichi

doi:10.1093/qmath/43.1.97

Cited by 18 publications

(22 citation statements)

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“…We will use a construction outlined by Kerckho and Thurston [16] (and later generalized by Ohshika [24], Bonahon-Otal [3] and Comar [8]) which was originally used to produce a sequence of discrete, faithful representations of a surface group whose geometric limit properly contains its algebraic limit.…”

Section: The Examplesmentioning

confidence: 99%

“…(See [7,13,14] for more information about algebraic convergence of Kleinian groups.) In many situations (see [1,6,15,24,25,27]), it has been shown that N n must be homeomorphic to N for all large enough n, and we had suspected that this would always be the case. In this paper, we give a collection of examples where N n is not homeomorphic to N for any n. Our sequences are quite well-behaved: the n ( 1 (M )) are convex co-compact and mutually quasiconformally conjugate, and the algebraic limit ( 1 (M )) is geometrically ÿnite.…”

Section: Introductionmentioning

confidence: 98%

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Algebraic limits of Kleinian groups which rearrange the pages of a book

Anderson

Canary

1996

Inventiones Mathematicae

118

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Section: The Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Algebraic limits of Kleinian groups which rearrange the pages of a book

Anderson

Canary

1996

Inventiones Mathematicae

118

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“…Since .M / 0 has only one end by Lemma 6.11, which is topologically tame, q must be finite-sheeted by Canary's covering theorem [17]. This implies that D G 1 by the argument in Section 9.3 in Thurston [58] (see Lemma 2.3 in Ohshika [46] for a detailed proof).…”

Section: Claim 613mentioning

confidence: 99%

Realising end invariants by limits of minimally parabolic, geometrically finite groups

Ohshika

2011

Geom. Topol.

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We shall show that for a given homeomorphism type and a set of end invariants (including the parabolic locus) with necessary topological conditions which a topologically tame Kleinian group with that homeomorphism type must satisfy, there is an algebraic limit of minimally parabolic, geometrically finite Kleinian groups which has exactly that homeomorphism type and the given end invariants. This shows that the Bers-Sullivan-Thurston density conjecture follows from Marden's conjecture proved by Agol and Calegari-Gabai combined with Thurston's uniformisation theorem and the ending lamination conjecture proved by Minsky, partially collaborating with Masur, Brock and Canary.

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“…By the covering theorem due to Thurston [28] (see also Canary [3], Ohshika [23]), there is a neighbourhoodẼ ∞ ofẽ ∞ homeomorphic to S × R such that q|Ẽ ∞ is a finite-sheeted covering. In our situation, this finite-sheeted covering must be one-toone, for the following reason.…”

Section: The First Alternative Holds If and Only If ι(ν) Is An Endingmentioning

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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Geometric Behaviour of Kleinian Groups on Boundaries for Deformation Spaces

Cited by 18 publications

References 0 publications

Algebraic limits of Kleinian groups which rearrange the pages of a book

Algebraic limits of Kleinian groups which rearrange the pages of a book

Realising end invariants by limits of minimally parabolic, geometrically finite groups

Rigidity and topological conjugates of topologically tame Kleinian groups

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