It is shown that for each integer n > 1 there exists a constant R n > 0 such that if M is a closed hyperbolic 3-inanifold with Rank 7Ti(M) = n, then the injectivity radius of M is bounded above by R n .

We show if M is a closed, connected, orientable, hyperbolic 3-manifold with Heegaard genus g then g ≥ . We also give an upper bound on the volume in terms of the flip distance of a Heegaard splitting, and describe isoperimetric surfaces in hyperbolic balls.

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