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Heegaard Floer homology and graph manifolds.

Between you and me, I always wanted to be a geometric group theorist. On that note, there is a conjectural relationship between Heegaard Floer homology and left-orderable three-manifold groups, which is now known to hold for all graph manifolds. That is, a closed orientable graph manifold Y is an L-space if and only if the fundamental group of Y is not-left-orderable. The latter condition can be hard to check in general, but in this setting the inequality dimHF(Y)>|H_1(Y)| provides a certificate that the group pi_1(Y) is left-orderable. I'll explain what these symbols mean in an attempt to sell the machinery to geometric group theorists.