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.
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Oblatum 31-VII-2000 & 9-V-2001¶Published online: 20 July 2001
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Minsky, Y. Bounded geometry for Kleinian groups. Invent. math. 146, 143–192 (2001). https://doi.org/10.1007/s002220100163
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DOI: https://doi.org/10.1007/s002220100163