At the end of his monumental preprint addressing the Virtually Fibred Conjecture for Haken 3-manifolds [7], Wise makes a remarkably bold conjecture. (Nathan Dunfield blogged about Wise’s work here.) The purpose of this post is to highlight that conjecture and explain what it means. It’s such a remarkable conjecture that it’s difficult to believe it’s true, but it’s also a win-win in the sense that either a positive or a negative answer would be a huge advance in geometric group theory.

**Wise’s Conjecture (Conjecture 20.5 of [7]):** Let be a word-hyperbolic group which is also the fundamental group of a compact, non-positively curved cube complex . Then has a finite-sheeted covering space which is special.

Most of the rest of this post will be an attempt to explain what ‘special’ means, but let me first whet your appetite by giving some consequences.