a little question about Heegaard splitting

Question

for any compact orientable hyperbolic 3-manifold with totally geodesic boundary, is there a strongly irreducible heegaard splitting?

Answer 1

I believe this is true for a small 3-manifold, one which contains no closed incompressible surfaces which are not boundary parallel. I think the argument of Casson and Gordon carries over to this case (if there is a weakly reducible yet irreducible splitting, then this should give a closed incompressible surface). Many simple examples of manifolds with totally geodesic boundary are small, and therefore should have a strongly irreducible splitting.

There are many Haken 3-manifolds which contain strongly irreducible surfaces. There are pretzel knot examples due to Casson-Gordon. I expect one may be able to construct analogous examples of manifolds with totally geodesic boundary which have this property too, e.g. by taking careful branched covers, and which are not small.

Answer 2

Casson and Gordon's argument goes through for minimal genus splittings of manifolds with boundary. See Moriah's paper "On boundary primitive manifolds and a theorem of Casson–Gordon". Now we follow Agol's idea: A small manifold is an (irreducible, connected, compact, orientable) three-manifold where all incompressible surfaces are boundary parallel. By Moriah's version of Casson-Gordon, any minimal genus splitting of a small manifold must be strongly irreducible.

Now, a manifold (irreducible, connected, compact, orientable) that is not small is called large. So your question reduces to asking "does every large hyperbolic manifold with totally geodesic boundary admit a strongly irreducible splitting?"

I would conjecture that the answer is "No". I guess this is suggested by the following construction: take two small manifolds $X$ and $Y$ with a homeomorphic incompressible boundary components $F_X$ and $F_Y$. Glue $F_X$ to $F_Y$ by a sufficiently complicated map $f$. Then (hopefully) all splittings of $X \cup_f Y$ should be amalgamations and thus none are strongly irreducible. This is just conjecture, as nobody knows how to rule out the possibility of very high genus strongly irreducible splittings. (If that has changed recently I'd be really interested to hear about it!)

I would next conjecture that a careful modification of the above discussion can give $Y$ an extra boundary component which is totally geodesic.

There are many papers in this general area. See "The Heegaard genus of amalgamated 3-manifolds" by Lackenby and "On the Heegaard splittings of amalgamated 3-manifolds" by Tao Li, for example.