Summary: Math. Res. Lett. 15 (2008), no. 3, 521524 c International Press 2008
FINDING FIBRE FACES IN FINITE COVERS
D. D. Long and A. W. Reid
A well-known conjecture about closed hyperbolic 3-manifolds asserts that the first
Betti number can be increased without bound by passage to finite sheeted covers. If
the manifold is fibred, it is not difficult to see that a strengthening of this conjecture
is that the number of fibred faces (see §2.1 for the definition of a fibred face) of the
unit ball of the Thurston norm can be made arbitrarily large by passage to finite
sheeted covers. The main result of this note is the following.
Theorem 1.1. Suppose that M is a closed arithmetic hyperbolic 3-manifold which
fibres over the circle.
Then given any K N, there is a finite sheeted covering of M for which the unit
ball of the Thurston norm has > K fibred faces.
A consequence of Theorem 1.1 (see §2 for a proof) is:
Corollary 1.2. Let M be a closed arithmetic hyperbolic 3-manifold that fibres over
the circle. Then the rank of its second homology can be increased without bound.
While this follows from a stronger result proved in  (subsequently reproved in
 and ), namely that the conclusion of Corollary 1.2 holds for an arbitrary closed
arithmetic hyperbolic 3-manifold with positive first Betti number, the proof given