Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds

Abstract

We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold $M$ is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried's dynamical characterization of the fibered faces. The origin of the basic fibration $M\to S^1$ is the modular elliptic curve $E=X_0(49)$, which admits multiplication by the ring of integers of ${\Bbb Q}[\sqrt{-7}]$. We first base change the holomorphic differential on $E$ to a cusp form on ${{\rm GL}(2)}$ over $K={\Bbb Q}[\sqrt{-3}]$, and then transfer over to a quaternion algebra $D/K$ ramified only at the primes above $7$; the fundamental group of $M$ is a quotient of the principal congruence subgroup of ${\cal O}_D^\ast$ of level $7$. To analyze the topological properties of $M$, we use a new practical method for computing the Thurston norm, which is of independent interest. We also give a noncompact finite-volume hyperbolic 3-manifold with the same properties by using a direct topological argument.