Is the braid group hyperbolic?

Question

The braid groups satisfy a number of properties that one would expect of a hyperbolic group, liking having a solvable word problem, and having exponential growth. Are the braid groups hyperbolic groups? If not, is there any obvious property of hyperbolic groups showing that they are not?

Answer 1

Outside of the one and two strand case they are not hyperbolic. One obstruction is that braid groups on $n>2$ strands, $B_n$, contain $\mathbb Z^2$ subgroups which can not happen in hyperbolic groups.

First note that $B_3$ is a subgroup of $B_n$ for $n>2$. Order the strands and name them $b_1,b_2,b_3,...,b_n$. Let $\sigma$ be the element which braids $b_1$ around $b_2,b_3$ where $b_1$ goes around the back of those two and then cross in front and let $\rho$ be the element which braids $b_2,b_3$ (basically an element in $B_2$). These two elements "obviously" commute so we get $\langle \sigma, \rho \rangle \cong \mathbb{Z}^2$ as a subgroup of $B_n$ for $n>2$ (if you want a formal proof there is enough detail in A Primer on Mapping Class Groups by Farb and Margalit). In the case of $B_1,B_2$ we have the trivial group and $\mathbb{Z}$ respectively and these are hyperbolic but not in an "interesting" way.

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You mention that one of your motivations is that there do seems to be some similarities to hyperbolic groups and it turns out this is a really good observation which gets into some interesting math. I won't be giving much details but it seems good to mention some of these ideas.

There is a different way to describe braid groups and it is as mapping class groups of punctured disk (basically the group of topological symmetries up to a natural identification). Very roughly you can think about punctures moving around each other in the disk as braids moving around each other. If you "plot" this with a time axis you can see the punctures moving "drawing" the braid.

This becomes important because now you get tools from surface theory and their mapping class groups. The mapping class group acts naturally on something called the curve complex which, somewhat surprisingly, is $\delta$-hyperbolic, proved by Masur and Minsky, and infinite diameter (you can choose $\delta =19$). Lots of the hyperbolic behavior can actually be seen in this action on the curve complex. Further, Masur and Minsky developed a way of studying the geometry of mapping class groups through the curve complex of the surface and curve complexes of all its (essential) subsurfaces by "piecing together" the geometric information of the curve complexes in some consistent way.