Bounded geometry for Kleinian groups

By Yair N. Minsky. 2001

84 citations recorded by the source corpus.

Abstract.We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a positive lower bound on the lengths of closed geodesics. When the surface is a once-punctured torus, the coefficients coincide with the continued fraction coefficients associated to the ending laminations.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/00c1674955624d6f0385be5b2e916d86d2c07a95

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Heegaard Splittings and Pseudo-Anosov Maps (2009, methodology): “However the method of construction of the negatively curved metric is dierent from here and involves a more elaborate study of the deformation space of hyperbolic structures on a handlebody similar to work of Minsky in [Min94, Min01 ] in the case of hyperbolic structures on an interval bundle over ”
Projective structures with degenerate holonomy and the Bers density conjecture (2002, background): “nerate then Γ ∈ BX where X is the conformal boundary of M. There are some special cases where the conjecture is known. Abikoﬀ [Ab] proved the conjecture when M is geometrically ﬁnite. Recently Minsky [Min3] has proved the conjecture in the case where there is a lower bound on the length of any close”
The classification of Kleinian surface groups I : Models and bounds : preprint (2003, background): “We prove, in Theorem 6.1 (Relative Coarse Projection), that the composition π Y @BULLETΠ ρ,L for a subsurface Y has contraction properties generalizing those shown in [46].”
On the growth of $L^2$-invariants for sequences of lattices in Lie groups (2012, background): “ be the endpoints of αgiven by Theorem 12.12. Then by Theorem 12.14, there is a doubly degenerate hyperbolic 3-manifold (N,µ) ∈ DD(Σ) with ending laminations {|λ+|,|λ−|}. Combining theorems of Minsky [83] and Raﬁ [97], we see that as αlies in the ǫ-thick part of T(Σ) there is a lower bound inj(N) ≥ǫ”
Tameness of hyperbolic 3-manifolds (2004, background): “1 last year for tame manifolds by Brock, Canary and Minsky (with major contributions from Masur-Minsky) [50, 47, 49, 43, 44, 48, 16].”
Quasiconvexity in the curve complex (2003, background): “G T ] 7 J ul 2 00 3 QUASICONVEXITY IN THE CURVE COMPLEX HOWARD MASUR AND YAIR MINSKY”
On the density of geometrically finite Kleinian groups (2002, methodology): “While such a reworking is now essentially straightforward after the techniques introduced in [ Min4 ], we have chosen in a similar spirit to defer these technicalities to a later paper in the interest of conveying the main ideas.”