In the 1970's, Thurston found hyperbolic 3-manifolds which fibered over the circle (\(S^1\)) and his results prompted a natural follow-up question: are there higher-dimensional hyperbolic manifolds which also fiber over \(S^1\)? In 2023, Italiano-Martelli-Migliorini used a combinatorial result about right-angled Coxeter groups (RACGs) to construct a hyperbolic 5-manifold which also fibered over the circle. Might we finally have a way forward in tackling the big question??

In the immortal words of Vizzini (The Princess Bride) "You'd like to think that, wouldn't you?" Alas, the crucial ingredient in the aformentioned construction was a nice right-angled hyperbolic Coxeter polytope, and few things are even known about the existence of such objects in higher dimensions, so answering this question with geometric methods seems impractical at this juncture. In this talk, we'll look at recent results to the analogous group theoretic question about fibering and virtual cohomological dimension (VCD) of these RACGs.

This work is joint with Lafont, Minemyer, Sorcar, and Stover.