  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.
