On the complexity of the Heegaard structure of hyperbolic 3-manifolds

Abstract

A well known result of Waldhausen [22] states that all Heegaard splittings of S 3 and of S 2 × S 1 of a given genus g are homeomorphic (and even isotopic). Recent work of various authors (see [20], [19], [2]) shows that there are more “simple” 3-manifolds where such a uniqueness holds. However, this is not true for most 3-manifolds (see [12], [15]). Pitts-Rubinstein [18] have announced a proof of the Waldhausen conjecture [23] that for any 3-manifold M the maximal number hM (g) of non-homeomorphic Heegaard splittings is finite for any given genus g ∈ N. (This was shown by Hass [10] for genus 2 manifolds and g = 2 and by Johannson [11] for Haken manifolds.) It is expected that the number hM (g) grows for “generic” M (i.e. not the above mentioned “simple” ones) with the “complexity” of M . In this note we consider Heegaard splittings of minimal genus gmin(M ) and show that for a large class of hyperbolic 3-manifolds hM (gmin (M )) grows exponentially. Theorem A Let K ⊂ S 3 be the Montesinos knot m(e; (α1, β1), . . . , (αg, βg) such that all βi are mutually distinct odd primes, and let α = g.c.d .(α1, . . . αg) > 22g−1(β1 · . . . · βg). Let M be the closed 3-manifold obtained from K by m/n-surgery, where m is even. Then for each m-tupel (e1, . . . , eg) ∈ {1,−1}g \ {±(1, . . . , 1)} there exists a minimal genus Heegaard splitting Σy (e1, . . . , eg), and these are pairwise non-homeomorphic.