Braid groups have an infinite cyclic group center, generated by the square of the fundamental braid.
Geometrically, the fundamental braid has the property that any two strands cross positively exactly once.
For a braid group of about four strands it is easy to show that the square of the fundamental braid commutes with the generators, using the braid relations (I have used the Artin presentation). How can this be generalized? Is there an intuitive reasoning of why this property holds for all braid groups? Maybe an inductive proof or appeal to the geometric nature of every brand crossing twice in the square of the fundamental braid?
Also, is there a way of showing that there are not any other word that commutes with every generator?
Best Answer
A little late, but I happened to see this, and hope the following is still of use to someone. Let's think of the strands as lying in a ribbon. The fundamental braid is just a twist of this ribbon over $180^\circ$ around its long axis. Its square is a twist of the ribbon over $360^\circ$, so that all strands are back where they started. This operation clearly commutes with any braid group element; except for a full twist in the ambient space nothing really happened. It generates a copy of $\mathbb{Z}$ in the center of the braid group. While I don't have a simple proof that it's the whole center, I find it intuitively quite clear from this picture that that should be the case.