# Universal acylindrical actions

25 Jun

I’m at a fantastic summer graduate school at MSRI (the Mathematical Sciences Research Institute, a.k.a. “math heaven”) right now and re-met a friend I’d seen at a few earlier conferences.  I saw that she’d posted a preprint up on arXiv recently, so I thought I’d try to blog about it!

Remember that a group is a collection of elements paired with some kind of operation between them (the integers with addition, rational numbers with multiplication, symmetries of a square with composition).  For that operation, you put in two group elements and get another group element out.  You can imagine different functions with different inputs and outputs.  Like you might have a function where you put in Yen and late night, and it outputs pumpkin.  Or you could put one group element in, and a location, and get a different location [like if you put in the group element -2 to the location (3,3), maybe you get (1,1)].  More precisely, a group action on a space is a homomorphism* which takes in a group element and a point in the space and outputs a (possibly different) point on that space.  For instance, you can give an action of the integers on the circle by saying that rotates the circle by $n/2\pi$. Each integer rotates the circle by pi/2 times the integer. Looks like circle is getting a little sick of the action…

In the picture above, if you input the integer 2 and the original purple dot, you get the new location of the dot (180 degrees from its original location, aka pi away).  If you say the original purple dot is location and the new location is y, the notation is that 2.x=y.  A homomorphism is a function that respects this: f(xy)=f(x)f(y).

We say a space is hyperbolic if it locally “looks like” hyperbolic space (there’s a particularly nice function between it and hyperbolic space).  The title of Carolyn’s paper is “Not all acylindrically hyperbolic groups have universal acylindrical actions,” so we need to learn what “acylindrical” means (look, we’ve already learned what groups and actions are, and we know the words “not”,”all”,and “have”!  We’re doing great!)

Here’s the precise definition, and then I’ll break it down:

An action of a group on a hyperbolic space is called acylindrical if, for any $\epsilon >0,$ there exist numbers M,N>0 such that for every pair of points x,y with distance d(x,y)>M, the number of group elements that translate both x,y by less than epsilon is bounded by N: $|\{g: d(x,g.x)\leq \epsilon, d(y,g.y)\leq \epsilon\}| \leq N$.

Here’s the non math-y intuition for this: if you have a pool noodle and you spin one end around, the other one generally will fly away from where it used to be. Here’s the math-y intuition for this: choose two points that are M-far apart.  Make a little $\epsilon$-circle around each, then connect the two with a cylinder.  The condition says that only a few group elements preserve the cylinder (that means that when acts on all the points in the cylinder, it maps them back into other points in the cylinder).  So if you have a bunch (perhaps infinitely many) elements that preserve one circle, most of them send the other circle/rest of the cylinder away.

A group is called acylindrically hyperbolic if you can find a hyperbolic space on which the group acts acylindrically.  In practice, such groups actually act on a whole bunch of different spaces acylindrically.

Now suppose that you’ve got an element in G and you want to see how that particular element acts.  We say is loxodromic if you can find a space and a point in it so that the map $\mathbb{Z}\to X$ that sends an integer to the orbit of the point $n\mapsto g^n.s$ is a quasi-isometry– roughly, if you draw all the points that gets mapped to if you apply over and over again, you get something that looks like a line. The older tree is the same as the younger tree up to scaling (multiplication) and adding some constants (the leaves). This is an example of a quasi-isomeTREE.  [Also pretend both trees go on forever.]

Just for fun here’s a picture of something that’s not a quasi-isometry: The ribbon on the right goes on forever in both directions, so it’s not quasi-isometric to the tree

You might’ve noticed above that we say an element is loxodromic if we can find space on which it acts in this particular way.  But we also said that a group can act on several different spaces.  So even if an element acts loxodromically on one space, that doesn’t necessarily mean it acts loxodromically on another space (even if the group acts on that other space).  We actually call an element generalized loxodromic if there exists some space on which it acts loxodromically.  Then if you can find an action so that all generalized loxodromic actions are, in fact, loxodromic, you’ve found a universal acylindrical action.  So this paper gives an example of an acylindrically hyperbolic group that doesn’t have such an action.

Blog notes: For the summer I’m going to blog every Thursday (day was chosen arbitrarily).  Also, I went back and tagged all the gluten-free recipes as gluten-free.  And you should know that whenever I mention a person in this blog by name or link to them, that means that I admire them/am inspired by them.