(Reprinted from my old ldt blog)

I don’t know about you, but when I read the words “residiually finite rationally solvable group”, my eyes start to glaze over and my mind wanders to thoughts of transverse surfaces and simple closed curves. Now, this is just a personal bias and I know most people experience similar feelings when they read or hear something like “tunnel number one genus one fibered knots”. I’m just saying I prefer my long and convoluted definitions to be of a more geometric nature. Having said that, I will now suggest that you at least skim through the two and a half pages on RFRS groups in Ian Agol’s recent preprint “Criteria for Virtual Fibering” [1] in order to get to the geometry that makes up the rest of it.

Recall that the virtually fibered conjecture says that every hyperbolic 3-manifold has a finite cover that is a surface bundle. (This is not to be confused with the virtually Haken conjecture, that every hyperbolic 3-manifold has a finite cover containing a two sided incompressible surface.) Agol shows that if the fundamental group of a hyperbolic 3-manifold is RFRS then the manifold is virtually fibered. Despite the algebraic beginning to the paper, the argument is in fact very geometric. Quoting from the paper:

“If an oriented surface F ⊂ M is non-separating, then M fibers over S^1 with fiber F if and only if M\\F [is homeomorphic to] I × F. If F is not a fiber of a fibration, then there is a JSJ decomposition of M\\F, which has a product part (window) and a non-product part (guts). The idea is to produce a complexity of the guts, and use the RFRS condition to produce a sequence of covers of M for which we can decrease the complexity of the guts, by “killing” it using non-separating surfaces coming from new homology in these covers.”

It is interesting to note that the rough outline of the argument is reminiscent of Lackenby’s approach [2] to the virtually Haken conjecture: we expect that there will be a sequence of finite covers such that the covers become, in some sense, progressively less complex. If you find the correct definition of complexity then a dropping complexity implies the property that you’re looking for. Lackenby uses Heegaard genus as his complexity. Agol uses a more complicated definition based on foliations and normal surfaces. In either case, proving the conjecture is reduced to showing there is always a sequence of finite covers where the complexity drops.

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