I suppose I'm obligated to make a stab at answering this :)

First, the virtual fibering question was asked by Thurston: "Does every hyperbolic 3-manifold have a finite-sheeted cover which fibers over the circle? This dubious-sounding question seems to have a definite chance for a positive answer."

When I originally read this, I found the question intriguing, partly because it is surprising as made evident in the phrasing. Also, certainly questions and conjectures of Fields medalists receive sometimes an inordinate amount of attention (Thurston has been correct for most of his conjectures and questions). However, from the perspective of Kleinian groups, this question is quite interesting, since doubly degenerate Kleinian groups have a nice structure (e.g. they give rise to group-invariant Peano curves in $S^2$).

There are some group-theoretic consequences of virtual fibering:

• If a hyperbolic 3-manifold is virtually fibered, its fundamental group does not have the finitely generated intersection property, which seems to be of interest to some group theorists.

• Virtual fibered 3-manifold fundamental groups have property FD introduced by Lubotzky and Shalom (finite reps. are dense in the unitary dual with respect to the Fell topology).

• Virtual fibering or virtual Haken implies the group is "good", which roughly means its cohomology is reflected in that of its profinite completion.

I think there may be some other applications, and certainly will be more in the future.

Finally, the proof of virtual fibering in full generality relies on the group-theoretic property I call the "RFRS" condition (for residually finite rational solvable). This condition has other consequences, such as the twisted Alexander polynomials detect the Thurston norm, which has been applied by Friedl and Vidussi to study minimal genus surfaces in 4-manifolds which fiber over 3-manifolds.

I should also mention that recently Przytycki and Wise have proved that a 3-manifold with torus boundary components is virtually RFRS if and only if it admits a non-positively curved metric, which greatly generalizes Thurston's question.

OK, I think the elevator has gone all the way up to the top of the Sears tower and come back down.

1

I suppose I'm obligated to make a stab at answering this :)

First, the virtual fibering question was asked by Thurston: "Does every hyperbolic 3-manifold have a finite-sheeted cover which fibers over the circle? This dubious-sounding question seems to have a definite chance for a positive answer."

When I originally read this, I found the question intriguing, partly because it is surprising as made evident in the phrasing. Also, certainly questions and conjectures of Fields medalists receive sometimes an inordinate amount of attention (Thurston has been correct for most of his conjectures and questions). However, from the perspective of Kleinian groups, this question is quite interesting, since doubly degenerate Kleinian groups have a nice structure (e.g. they give rise to group-invariant Peano curves in $S^2$).

There are some group-theoretic consequences of virtual fibering:

• If a hyperbolic 3-manifold is virtually fibered, its fundamental group does not have the finitely generated intersection property, which seems to be of interest to some group theorists.

• Virtual fibered 3-manifold fundamental groups have property FD introduced by Lubotzky and Shalom (finite reps. are dense in the unitary dual with respect to the Fell topology).

• Virtual fibering or virtual Haken implies the group is "good", which roughly means its cohomology is reflected in that of its profinite completion.

I think there may be some other applications, and certainly will be more in the future.

Finally, the proof of virtual fibering in full generality relies on the group-theoretic property I call the "RFRS" condition (for residually finite rational solvable). This condition has other consequences, such as the twisted Alexander polynomials detect the Thurston norm, which has been applied by Friedl and Vidussi to study minimal genus surfaces in 4-manifolds which fiber over 3-manifolds.

I should also mention that recently Przytycki and Wise have proved that a 3-manifold with torus boundary components is virtually RFRS if and only if it admits a non-positively curved metric, which greatly generalizes Thurston's question.

OK, I think the elevator has gone all the way up to the top of the Sears tower and come back down.