Algebraic and geometric convergence of Kleinian groups.

Jørgensen, Tanja Schjødt; Marden, Albert

doi:10.7146/math.scand.a-12292

Cited by 74 publications

(88 citation statements)

References 8 publications

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“…We use this theorem to produce a sequence { n } of convex co-compact uniformizations of M converging to ∈ D( 1 (M )) such that N is homeomorphic to the interior of M , and {N n } converges "geometrically" to a geometrically ÿnite hyperbolic 3-manifold N which is homeomorphic to the interior of M − , where is the core curve of the solid torus. (This type of phenomenon was ÿrst discovered by J rgensen [12,15] and was subsequently investigated by Marden [18], Thurston [27], and others, e.g. see [3, 8, 16, 24, and 28].)…”

Section: Introductionmentioning

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: 99%

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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: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Anderson

Canary

1996

Inventiones Mathematicae

118

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“…With these normalizations, f α → I uniformly on compact subsets of Ω( G), and G α → G algebraically. We remark that it follows from the Jørgensen-Marden criterion [5] that G α → G geometrically.…”

Section: Vertical Projectionmentioning

confidence: 93%

On neoclassical Schottky groups

Hidalgo

Maskit

2005

Trans. Amer. Math. Soc.

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Abstract. The goal of this paper is to describe a theoretical construction of an infinite collection of non-classical Schottky groups. We first show that there are infinitely many non-classical noded Schottky groups on the boundary of Schottky space, and we show that infinitely many of these are "sufficiently complicated". We then show that every Schottky group in an appropriately defined relative conical neighborhood of any sufficiently complicated noded Schottky group is necessarily non-classical. Finally, we construct two examples; the first is a noded Riemann surface of genus 3 that cannot be uniformized by any neoclassical Schottky group (i.e., classical noded Schottky group); the second is an explicit example of a sufficiently complicated noded Schottky group in genus 3.

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“…This phenomenon occurs when we have a cyclic loxodromic subgroup which is geometrically close to a parabolic group of rank 2. (See the discussion of this phenomenon in Thurston [39] or Jorgensen-Marden [20].) (2) The proof works equally well for any pleated surface p : S -> N (into any hyperbolic 3-manifold N) with a maximal finite-leaved pleating locus A such that (S,dS,p)…”

Section: Ai-i N2mentioning

confidence: 97%

Algebraic Convergence of Schottky Groups

Canary¹

1993

Transactions of the American Mathematical Society

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Abstract. A discrete faithful representation of the free group on g generators Fg into Isom+(H3) is said to be a Schottky group if (H3 u Di-)/T is homeomorphic to a handlebody Hg (where ZJr-is the domain of discontinuity for T 's action on the sphere at infinity for H3). Schottky space ¿fg , the space of all Schottky groups, is parameterized by the quotient of the Teichmüller spacê (Sg) of the closed surface of genus g by Modo(//f) where Modo(//Ä) is the group of (isotopy classes of) homeomorphisms of Sg which extend to homeomorphisms of Hg which are homotopic to the identity. Masur exhibited a domain ff(Hg) of discontinuity for Modo(//f)'s action on PL(Sg) (the space of projective measured laminations on Sg), so 3 §(Hg) = tf(Hg)/Modo(Hg) may be appended to 5?g as a boundary. Thurston conjectured that if a sequence {p¡: Fg -y Isom+(H3)} of Schottky groups converged into â §(Hg), then it converged as a sequence of representations, up to subsequence and conjugation. In this paper, we prove Thurston's conjecture in the case where Hg is homeomorphic to 5 x / and the length ¡^((dS)*) in N¡ = H3//?,(Fg) of the closed geodesic(s) in the homotopy class of the boundary of S is bounded above by some constant K.

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Algebraic and geometric convergence of Kleinian groups.

Cited by 74 publications

References 8 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

On neoclassical Schottky groups

Algebraic Convergence of Schottky Groups

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