Bounded geometry for Kleinian groups

Details Bounded geometry for Kleinian groups Bounded geometry for Kleinian groups

2001-05-10

arxiv.org/abs/math/0105078v1

Mathematics

Geometric Topology

49 pages, 13 figures. Revised from IMS preprint version, with additional introductory material. To appear in Invent. Math

Scientific paper

10.1007/s002220100163

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. Applications include an improvement to the bounded geometry versions of Thurston's ending lamination conjecture, and of Bers' density conjecture.

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