Hyperbolic volume and Heegaard distance

We prove (Theorem 1.5) that there exists a constant Λ > 0 so that if M is a (μ, d)-generic complete hyperbolic 3-manifold of volume Vol(M) < ∞ and Σ ⊂ M is a Heegaard surface of genus g(Σ) > ΛVol(M), then d(Σ) ≤ 2, where d(Σ) denotes the distance of Σ as defined by Hempel. The term (μ, d)-generic is described precisely in Definition 1.3; see also Remark 1.4. The key for the proof of Theorem 1.5 is Theorem 1.8 which is of independent interest. There we prove that if M is a compact 3-manifold that can be triangulated using at most t tetrahedra (possibly with missing or truncated vertices), and Σ is a Heegaard surface for M with g(Σ) ≥ 76t + 26, then d(Σ) ≤ 2