# Low Dimensional Topology

## March 6, 2012

### Wise’s Conjecture

Filed under: 3-manifolds,Geometric Group Theory — Henry Wilton @ 6:13 am
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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 $G$ be a word-hyperbolic group which is also the fundamental group of a compact, non-positively curved cube complex $X$.  Then $X$ has a finite-sheeted covering space $X'$ 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.