Surface bundles over surfaces with arbitrarily many fiberings

In this paper we give the first example of a surface bundle over a surface with at least three fiberings. In fact, for each $n \ge 3$ we construct $4$-manifolds $E$ admitting at least $n$ distinct fiberings $p_i: E \to \Sigma_{g_i}$ as a surface bundle over a surface with base and fiber both closed surfaces of negative Euler characteristic. We give examples of surface bundles admitting multiple fiberings for which the monodromy representation has image in the Torelli group, showing the necessity of all of the assumptions made in the main theorem of our recent paper [arXiv:1404.0066]. Our examples show that the number of surface bundle structures that can be realized on a $4$-manifold $E$ with Euler characteristic $d$ grows exponentially with $d$.