Finiteness of closed arithmetic hyperbolic surface bundles

Abstract

For each $g\ge 2$, we show that there are finitely many cyclic commensurability classes of closed, fibered arithmetic hyperbolic $3$-manifolds with genus $g$ fiber. More generally, we show that there are only finitely many conjugacy classes of admissible surface subgroups of PSL$_2(\mathbb{C})$ whose image is contained in the fundamental group of some arithmetic hyperbolic $3$-manifold. We also give an effectively computable upper bound with effective asymptotic rate on both finiteness statements. This affirms a conjecture of Bowditch, Maclachlan, and Reid.