So tubular! Err… cubular

10 Dec

What I’m reading right now: Special Cube Complexes, a 2008 paper by Frederic Haglund and Dani Wise.  Recently Ian Agol proved the Virtual Haken Conjecture , which was a Big Deal in math (this link is LONG but a very well written non-math-person friendly summary of 30 years of math).  In fact, one of my professors from undergraduate, Jesse Johnson, wrote a nice little blog post on what it might mean for the future of low dimensional topology.  Basically, Agol used this special cube complex stuff to prove  this Big Deal, which means that we might be able to use these to prove Lots of Big Deal and Little Deal theorems.  So let’s get into what these guys are.

Update: I just found out that it’s my turn to give the talk in our little colloquium this week.  There’s seven of us, four are my advisor’s students (I guess he’s not technically my adviser yet) and three are in related fields.  So every two months or so, you have to give a half hour talk on some math you’re learning about.  We aren’t supposed to talk about our research, but I think I get a pass since it’s still my first year.  So this post is a prelude to my talk!

A cube complex is an object built by gluing a whole bunch of Euclidean cubes together.  So a one-dimensional cube complex is built by gluing a bunch of lines together; that is, it’s a (mathematical) graph.  And a two-dimensional cube complex is built by gluing a bunch of squares together.  The gluings can happen in funky ways though, and special cube complexes are objects where these pathologies don’t happen.

We’ll define these topologies in terms of hyperplanes.  So if I have a square [-1,1]\times [-1,1],  I’ll have two hyperplanes running through it: one at 0\times [-1,1] and one at $[-1,1]\times 0$.  In the picture below, which is taken from the 11th page of this paper, the red lines are hyperplanes, and the gray lines represent the cubes they’re cutting through.  To be special, we need our cube complex to a) not self-intersect, b) have no one-sided hyperplanes, c) not directly self-osculate, and e) have no two hyperplanes that inter-osculate.  Turns out that case d) hyperplanes indirectly self-osculate is OK.

Pathologies cube complexes (these guys are not special, but don't tell their parents I said that)

Pathologies of cube complexes (these guys are not special, but don’t tell their parents I said that).

Really quick, notice that a cube complex is special if and only if its two skeleton is (the part made up of filled-in squares).  That’s why we can just use this picture.

So what’s so special about special cube complexes?  The ultimate idea is that given a cube complex, if none of these funky things happens, I’ll be able to cut along the hyperplanes and have nice things happen.  That’s how we get to the Big Deal.  But that’s neither here nor there; this post is about a Smaller Deal: that a cube complex is special if and only if it corresponds to a right angled Artin group, that is, that there’s some graph so that our cube complex has an isometry into the Salvetti complex of that graph.

Turns out google image searching “Salvetti complex” is utterly useless, so I’ll just describe it.  Start with a single vertex.  Now given a graph \Gamma, we add one loop to this vertex for each vertex x_i in \Gamma.  For any edge in \Gamma, say it’s (x_i,x_j), we attach a square that looks like this:

Pub crawl that winds up back at the first bar invite idea: be there or be a torus!

Pub crawl that winds up back at the first bar invite idea: be there or be a torus!

So far we have the standard 2-complex for the group A(\Gamma) = \langle x_i: i\in\text{Vertices}(\Gamma): [x_i,x_j]: (x_i,x_j)\in\text{Edges}(\Gamma)\rangle, which is the definition of a right angled Artin group.  Now to make the Salvetti complex, we attach an n-torus for every n-cycle in our graph.  So if there was a triangle in our graph, we get a corresponding 3-torus in the Salvetti complex.  And the fundamental group of our Salvetti complex is that right angled Artin group.

To restate our deal, we’re saying that special cube complexes always have some graph \Gamma so that we can see our cube complex somewhere in the corresponding Salvetti complex, and everything still looks nice.  More formally and rigidly, we’re saying that X is special if and only if there’s an immersion from X into some Salvetti complex, which is a local isometry on the 2-skeleton.

Proof in one direction: Suppose I’ve got a local isometry from the 2-skeleton of my cube complex X to a Salvetti complex.  Since the Salvetti complex is special from how we built it, and local isometries keep things tidy (you can’t uncross intersecting hyperplanes, for instance), that means my 2-skeleton is also special.  So from our remark above, X is special.

Proof in the other direction: Say my cube complex is special.  Then make a graph with vertices being the hyperplanes of X, and edges connecting intersecting hyperplanes.  Now make the Salvetti complex of this graph.  We can map X into the complex by sending an edge to the vertex of the hyperplane that crosses it, and then extend the rest of the map.  It’s a hop and skip (no jumping allowed) that this map is, in fact, a local isometry on the level of 2-skeleta.

Right angled Artin groups have really nice properties and are fun stuff, so this little theorem can lead to a whole bunch of conclusions.  Phew, first math blog post.  I’ll get better at these, I promise.  I have to figure out what audience I’m writing for.

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