Finiteness of arithmetic surface bundles over the circle

In joint work with Homin Lee and Nick Miller, we show that for a fixed genus g there are at most finitely many closed genus g surface bundles over the circle up to taking finite-sheeted cyclic fibered covers. We also show the more general result that for a fixed genus g there are at most finitely many (admissible) genus g surface group representations living in arithmetic lattices in PSL(2,C) up to conjugation. Our discussion will focus on motivation, examples, and how large tubes from the Arithmetic Margulis Lemma of Fraczyk-Hurtado-Raimbault help us establish the key results.