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We examine the internal geometry of a Kleinian surface group and its relations to the
asymptotic geometry of its ends, using the combinatorial structure of the complex of
curves on the surface. Our main results give necessary conditions for the Kleinian
group to have ‘bounded geometry’ (lower bounds on injectivity radius) in terms of a
sequence of coefficients (subsurface projections) computed using the ending invariants
of the group and the complex of curves.
These results are directly analogous to those obtained in the case of
punctured-torus surface groups. In that setting the ending invariants are points in the
closed unit disk and the coefficients are closely related to classical continued-fraction
coefficients. The estimates obtained play an essential role in the solution of
Thurston’s ending lamination conjecture in that case.
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