What is a hierarchically hyperbolic space?

Hey everybody, it’s been a while…
This post is about the notion of hierarchically hyperbolic space we defined with Jason Behrstock and Mark Hagen in this paper, which is a generalisation of the notion of (Gromov-)hyperbolic space. The main examples of hierarchically hyperbolic spaces (HHSs) are mapping class groups and CAT(0) cube complexes admitting a proper cocompact action by isometries, and in fact what we wanted to do was to export the machinery developed by Masur-Minsky and other people from the world of mapping class groups to that of cube complexes. There are also other interesting examples including fundamental groups of non-geometric 3-manifolds, Teichmuller space with either of the standard metrics, and more. In the last couple of years or so we were able to prove quite a few things using the HHS machinery, the latest being a theorem about quasi-flats that I have to say I’m quite proud of. But this post is not about any of the applications…

“Too many axioms…”
The complaint we get most often about our definition is that it’s too long and abstract. However, the geometry that the definition captures is actually very reasonable, and it doesn’t take so long to explain this… just about a blog post 😉
I hope that this post will give you a good heuristic picture of what’s going on in an HHS. Here we go…

Standard product regions
The heuristic picture of an HHS that I want to discuss is the one provided by standard product regions.
If an HHS $\mathcal{X}$ is not hyperbolic, then the obstruction to its hyperbolicity is encoded by the collection of its standard product regions. These are quasi-isometrically embedded subspaces that split as direct products, and the crucial fact is that each standard product region, as well as each of its factors, is an HHS itself, and in fact an HHS of lower “complexity”. It is not very important at this point, but the complexity is roughly speaking the length of a longest chain of standard product regions $P_1\nsubseteq P_2\nsubseteq \dots\nsubseteq P_n$ contained in the HHS; what is important right now is that factors of standard product regions are “simpler” HHSs, and the “simplest” HHSs are hyperbolic spaces. This is what allows for induction arguments, where the base case is that of hyperbolic spaces.

Standard product regions encode entirely the non-hyperbolicity of the HHS $\mathcal{X}$ in the following sense. Given a, say, length metric space $(Z,d)$ and a collection of subspaces $\mathcal{P}$ , one can define the cone-off of $Z$ with respect to the collection of subspaces (in several different ways that coincide up to quasi-isometry, for example) by setting $d'(x,y):=1$ for all $x,y$ contained in the same $P\in\mathcal{P}$ and $d'(x,y)=d(x,y)$ otherwise, and declaring the cone-off distance between two points $x,y$ to be $\inf_{x=x_0,\dots,x_n=y}\sum d'(x_i,x_{i+1})$ . This has the effect of collapsing all $P\in\mathcal{P}$ to bounded sets, and the reason why this is a sensible thing to do is that one might want to consider the geometry of $Z$ “up to” the geometry of the $P\in\mathcal{P}$ . When $Z$ is a graph, as is most often the case for us, coning-off amounts to adding edges connecting pairs of vertices contained in the same $P\in\mathcal{P}$ .

Back to HHSs, when coning-off all standard product regions of an HHS one obtains a hyperbolic space, that we denote $\mathcal{C}(S)$ (this notation is taken from the mapping class group context, even though it’s admittedly not the best notation in other examples). In other words, an HHS is weakly hyperbolic relative to the standard product regions. Roughly speaking, when moving around $\mathcal{X}$ , one is either moving in the hyperbolic space $\mathcal{C}(S)$ or in one of the standard product regions. The philosophy behind many induction arguments for HHSs is that when studying a certain “phenomenon”, either it leaves a visible trace in $\mathcal{C}(S)$ , or it is “confined” in a standard product region, and can hence be studied there. For example, if the HHS is in fact a group, one can consider the subgroup generated by an element $g$ , and it turns out that either the orbit maps of $g$ in $\mathcal{C}(S)$ are quasi-isometric embeddings, or $g$ virtually fixes a standard product region, see this paper.

So far we discussed the “top-down” point of view on standard product regions, but there is also a “bottom-up” approach. In fact, one can regard HHSs as built up inductively starting from hyperbolic spaces, in the following way:

hyperbolic spaces are HHSs,
direct products of HHSs are HHSs,
“hyperbolic-like” arrangements of HHSs are HHSs.

The third bullet refers to $\mathcal{C}(S)$ being hyperbolic, and the fact that $\mathcal{C}(S)$ can also be thought of as encoding the intersection pattern of standard product regions. Incidentally, I believe that there should be a characterisation of HHSs that looks like the list above, i.e. that by suitably formalising the third bullet one can obtain a characterisation of HHSs. This has not been done yet, though. There is, however, a combination theorem for trees of HHSs in this spirit in this paper.

One final thing to mention is that standard product regions have well-behaved coarse intersections, meaning that the coarse intersection of two standard product regions is well-defined and coarsely coincides with some standard product region. In other words, $\mathcal{X}$ is obtained gluing together standard product regions along sub-HHSs, so a better version of the third bullet above would be “hyperbolic-like arrangements of HHS glued along sub-HHSs are HHSs”.

In the examples

We now discuss standard product regions in motivating examples of HHSs.

RAAGs
Consider a simplicial graph $\Gamma$ . Whenever one has a (full) subgraph $\Lambda$ of $\Gamma$ which is the join of two (full, non-empty) subgraphs $\Gamma_1,\Gamma_2$ , then the RAAG $A_\Gamma$ contains an undistorted copy of the RAAG $A_\Lambda\approx A_{\Lambda_1}\times A_{\Lambda_2}$ . Such subgroups and their cosets are the standard product regions of $A_\Gamma$ . In this case, $\mathcal{C}(S)$ is a Cayley graph of $A_\Gamma$ with respect to an infinite generating set (unless $\Gamma$ consists of a single vertex), namely $V\Gamma\cup\{A_\Lambda<A_\Gamma:\Lambda=join(\Lambda_1,\Lambda_2)\}$ . A given HHS can be given different HHS structures (which turns out to allow for more flexibility when performing various constructions, rather than being a drawback), and one instance of this is that one can regard as standard product regions all $A_\Lambda$ where $\Lambda$ is any proper subgraph of $\Gamma$ , one of the factors being trivial. In this case $\mathcal{C}(S)$ is the Cayley graph of $A_\Gamma$ with respect to the generating set $V\Gamma\cup \{A_\Lambda<A_\Gamma:\Lambda{\rm\ proper\ subgraph\ of\ }\Gamma\}$ , which is perhaps more natural.
For both HHS structures described above, $\mathcal{C}(S)$ is not only hyperbolic, but in fact quasi-isometric to a tree.

Mapping class groups
Given a surface $S$ , there are some “obvious” subgroups of $MCG(S)$ that are direct products. In fact, consider two disjoint (essential) subsurfaces $Y,Z$ of $S$ . Any two self-homeomorphisms of $S$ supported respectively on $Y$ and $Z$ commute. This yields (up to ignoring issues related to the difference between boundary components and punctures that I do not want to get into) a subgroup of $MCG(S)$ isomorphic to $MCG(Y)\times MCG(Z)$ . Such subgroups are in fact undistorted. One can similarly consider finitely many disjoint subsurfaces instead, and this yields the standard product regions in $MCG(S)$ . More precisely, one should fix representatives of the (finitely many) topological types of collections of disjoint subsurfaces, and consider the cosets of the subgroups as above. In terms of the marking graph, product regions are given by all markings containing a given sub-marking.
In this case it shouldn’t be too hard to convince oneself that $\mathcal{C}(S)$ as defined above is quasi-isometric to the curve complex, there’s something similar in Section 7 of Masur-Minsky I. To re-iterate the philosophy explained above, if some behaviour within $MCG(S)$ is not confined to a proper subsurface $Y$ , then the geometry of $\mathcal{C}(S)$ probably comes into play when studying it, and otherwise it is most convenient to study the problem on the simpler subsurface $Y$ .