Heegaard splittings, the virtually Haken conjecture and Property (τ)

Rank gradient, cost of groups and the rank versus Heegaard genus problem

“…See [17], Section 3 for more details. The Heegaard genus g(M ) of M is defined to be the minimal genus of the surfaces indentified in some decomposition for M c as above.…”

Surface subgroups of Kleinian groups with torsion

“…But the Heegaard genus is at least the rank of H 1 (O i ; Z/p). So, if, in addition, we can arrange that h(O i ) → 0, then Theorem 1.7 in [11] implies that O is virtually Haken. In fact, Theorem 1.7 in [13] gives much more: under these hypotheses, π 1 (O) is large.…”

“…In addition, a theorem of the author [11] states that the Lubotzky-Sarnak conjecture, plus another conjecture called the Heegaard gradient conjecture, together imply the virtually Haken conjecture. Unfortunately, the Lubotzky-Sarnak conjecture remains wide open at present, despite a considerable amount of positive evidence [8].…”

“…As mentioned above, the author proposed an approach in [11] to the virtually Haken conjecture. If one can find a nested sequence of finite regular covers M i of a closed orientable irreducible 3-manifold such that h(M i ) → 0 but where the Heegaard genus of M i grows linearly as a function of the covering degree, then Theorem 1.7 of [11] states that the manifolds M i are eventually Haken. (The Heegaard genus of a closed orientable 3-manifold is the minimal integer g such that the manifold can be expressed as the union of two genus g handlebodies glued homeomorphically along their boundaries.)…”

Homology-genericity, horizontal Dehn surgeries and ubiquity of rational homology 3-spheres

“…There has been little progress on this conjecture for more than 20 years (see, for example, [1], [2] and [7]: a small list from the huge literature on this topic).…”