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 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 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 in the following sense. Given a, say, length metric space and a collection of subspaces , one can define the cone-off of with respect to the collection of subspaces (in several different ways that coincide up to quasi-isometry, for example) by setting for all contained in the same and otherwise, and declaring the cone-off distance between two points to be . This has the effect of collapsing all to bounded sets, and the reason why this is a sensible thing to do is that one might want to consider the geometry of “up to” the geometry of the . When 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 .

Back to HHSs, when coning-off all standard product regions of an HHS one obtains a hyperbolic space, that we denote (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 , one is either moving in the hyperbolic space 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 , 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 , and it turns out that either the orbit maps of in are quasi-isometric embeddings, or 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 being hyperbolic, and the fact that 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, 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 . Whenever one has a (full) subgraph of which is the join of two (full, non-empty) subgraphs , then the RAAG contains an undistorted copy of the RAAG . Such subgroups and their cosets are the standard product regions of . In this case, is a Cayley graph of with respect to an infinite generating set (unless consists of a single vertex), namely . 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 where is any proper subgraph of , one of the factors being trivial. In this case is the Cayley graph of with respect to the generating set , which is perhaps more natural.

For both HHS structures described above, is not only hyperbolic, but in fact quasi-isometric to a tree.

*Mapping class groups*

Given a surface , there are some “obvious” subgroups of that are direct products. In fact, consider two disjoint (essential) subsurfaces of . Any two self-homeomorphisms of supported respectively on and 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 isomorphic to . Such subgroups are in fact undistorted. One can similarly consider finitely many disjoint subsurfaces instead, and this yields the standard product regions in . 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 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 is not confined to a proper subsurface , then the geometry of probably comes into play when studying it, and otherwise it is most convenient to study the problem on the simpler subsurface .

The tl;dr description is: The input of the construction is a family of geodesic metric spaces and “projection maps” between them, and the output is a metric space that contains all the metric spaces you started with. Such space is hyperbolic if you started with (uniformly) hyperbolic spaces. More in general, it is relatively hyperbolic, and actually even quasi-isometric to a tree-graded space with pieces your original metric spaces (this is due to David Hume).

Let me start by describing a simple example that you might wish to keep in mind. Consider this curve on the surface :

Now consider all its lifts to , the universal cover of . Schematically, the picture looks like this:

Now, consider the closest-point projections onto the lines in picture. As it turns out, the projection of one line onto another one has uniformly bounded diameter. Also, if two lines project far away onto another line, then any geodesic connecting them behaves like the blue curve in this picture:

That is to say, such geodesics pass close to the projection sets. What the BBF construction does in this case is roughly the following. You define a line to be projection-between two other lines if the projections of and onto are far away, as in the picture above. (This is highly *un*related to being “topologically” between.) Then, you form a path metric space in the following way: You consider the disjoint union of all lines, then you add paths of length 1 connecting all points in to all points in whenever there is no line projection-between and . (This is not entirely correct, but close enough.) The construction is natural in the sense that the fundamental group of the surface we started with acts on the metric space we constructed. Also, as it turns out, such space is a quasi-tree, i.e. it is quasi-isometric to a tree.

As an aside, if we start from a loop in a (say, closed) hyperbolic 3-manifold instead, then we again end up with an action of the fundamental group of on a quasi-tree . It is easy to see that there are elements of acting loxodromically on : The element of corresponding to the original geodesic has an invariant line, i.e. the same invariant line it has in . This is quite interesting, because there are ‘s so that every action of on a genuine tree fixes a point…

Ok, time to state the BBF axioms. I’ll mostly keep the same notation they use in their paper, which is motivated by curve complexes and subsurface projections. Let be a collection of geodesic metric spaces and for each let be a map assigning to each a bounded subset of . The axioms are as follows (all axioms start with “There exists ” and assume that all projections involved are well-defined).

**1.** The diameter of is bounded by for each .

Ok, that was easy.

Let’s set for convenience (i.e. “the distance between and from the point of view of “), where is the diameter.

**2.** Given , there exist only finitely many ‘s so that .

Another way of saying this: There are finitely many ‘s that are projection-between and . In the example of the lines in , this axiom is true because any geodesic from one line to another line fellow-travels for a bit all lines projection-between and . The length of any such geodesic is finite, so…

Ok, last axiom!

**3.** At most one of can be larger than .

A more compact way of formulating it is: . I formulated it in the other way because it makes it clearer what this axiom is useful for, i.e., ensuring that “being projection-between” behaves like the relation “being between” in a partial order. What I mean is that if is between and then, say, is not between and .

Exercise: Show that the third axiom holds in the example, using the picture about far-away projections.

Finally, as mentioned earlier, the construction spits out a metric space constructed from the union of the ‘s and some paths of length 1. The “right” statement about this metric space is that it is quasi-isometric to a tree-graded space whose pieces are copies of the ‘s (proven by David Hume). In particular, if the ‘s are (uniformly) hyperbolic, then is hyperbolic, and if the ‘s are (uniformly) quasi-trees, then is a quasi-tree (proven by BBF). Also, is hyperbolic relative to the copies of the ‘s that it contains. Just to wrap this all up, the BBF construction allows to construct interesting actions on hyperbolic and relatively hyperbolic spaces whenever projections with good properties can be defined. And if you’re not convinced that good actions on hyperbolic metric spaces are useful you can take a look at the paper by Dahmani-Guirardel-Osin, or wait for a future post about it.

]]>After discussing with Piotr Przytycki, we came up with a proof that takes place directly in the curve graph. To be more precise, the proof works in the closed case, i.e. exactly when the HPW argument requires the additional trick.

Here is the statement that we are going to prove.

**Theorem: **There exists so that for any closed surface of genus at least 2 the curve graph of is -hyperbolic.

The proof relies on a “guessing geodesics lemma” similar, but simpler, to the one described here. It is Proposition 3.1 in this paper by Bowditch. (I plan to report on it in a later post.)

[EDIT: it has been pointed out to me that the criterion was actually discovered by Masur-Schleimer, see Theorem 3.15 here.]

Informally speaking, it says that if a space contains a family of paths forming thin triangles, then it is hyperbolic.

**Proposition (Masur-Schleimer): **Let be a metric graph and suppose that we assigned to each pair of points a connected set Suppose that there exists so that

1) if are at distance at most 1 then the diameter of is at most and

2) the ‘s form thin triangles, i.e. is contained in the -neighborhood of for each

Then is hyperbolic, with hyperbolicity constant depending on only.

For mere technical convenience, we will work with the “augmented curve graph” where the edges connect curves that (in minimal position) intersect at most once. It is easy to see that is quasi-isometric to (with constants not depending on ) because if two curves intersect at most once then their distance in is at most 2.

**Bicorn curves. **So, we need to guess, given a family of curves forming a path from to And the way to do this is “interbreeding” with : we put them in minimal position and define to be the collection of all (simple closed) curves consisting of the union of an arc of and an arc of which we call the -arc and -arc of We also add to and we call the elements of bicorn curves.

**Connectedness. **First of all, we show that (the full subgraph of spanned by) is connected. There is a natural *partial* order on : we write if the -arc of contains the -arc of like in the following picture:

In other words we measure progress towards by looking at the how close the -arc is to being the whole of

So, the reason why is connected is that for every there is adjacent to and so that (unless ). In particular, every point in can be connected to

In order to see that such exists start with and prolong one side of the -arc until it hits the -arc. The prolonged -arc is part of a bicorn curve that (when put in minimal position) intersects at most once, and clearly

As an aside, I like to think of endowed with the partial order, as a “directed multi-path”.

**Thinness. **Property 1) from the proposition is straightforward: if are adjacent in then Property 2) is also rather easy. Fix curves and a bicorn curve Given another curve we have to find some close to

The procedure is as follows. Put all curves in minimal position and consider three consecutive intersections of with (if there are fewer than three intersections then is close to and we are done). The picture might look like this, for example:

We can assume that two of those intersections are on, say, the -arc. The subarc of from to is part of a bicorn curve in that (when put in minimal position) intersects at most twice. Therefore, and are within distance two of each other.

And this is it. Yep, really.

We welcome suggestions about further applications of bicorn curves…

The proofs available before HPW were quite complicated and relied on Teichmüller theory, so I was quite amazed to see that this could be proven in 6 self-contained pages. And, as the authors pointed out to me: “Well, there’s one page of introduction and one page of references.”

This post is an illustrated guide to the proof. I won’t define arc/curve complexes here, but I plan to write a BI soon. The more loudly you complain, the sooner I will actually do.

Ok, let’s start. First of all, as it turns out it’s more convenient to deal with arc graphs instead of curve graphs.

**Reduction to the arc graph.** (Really, just skip this ) Here’s a very quick sketch of how to deduce hyperbolicity of curve graphs once we know it for arc graphs, which for the moment we assume we do.

If we are dealing with a (complicated) surface with boundary then the idea we can use is that one can construct a path in the curve complex starting from a path in the arc complex by considering curves disjoint from the arcs appearing along (it will be clearer later how to exploit this, if you don’t know how to do it already). I’ll be honest, just this one time: There is a subtle issue I’m sweeping under the carpet here, but I would like not to make the post too technical…

Anyway, suppose instead that your surface, , is closed. If you call the surface obtained by adding a boundary component, then you have a boundary-forgetting map. This map is Lipschitz, this is immediate, and has a section, constructed as follows. Choose a hyperbolic metric on and realise all (homotopy classes of) curves as geodesics. Now you can take a puncture outside the union of all such (countably many) geodesics. Using the two properties it’s not hard to see that the curve graph of must be hyperbolic as well. The details are in section 5 of the paper.

**Unicorns!** Ok, let’s now take an arc graph. The strategy to show it’s hyperbolic is to prove that, as David puts it, “it smells hyperbolic”**, i.e. that there exists a family of preferred paths with nice properties (you may wish to take a look at this post ). The most important property that this family should satisfy is that triangles of preferred paths should be slim. The other properties you need (in simplified form) are that a preferred path is a geodesic if the endpoints have distance at most 1 and that a subpath of a preferred path is a preferred path.

HPW called the nice paths they constructed unicorn paths (!). They were initially called one-corner paths, but then Piotr realised that one-corner and unicorn are almost the same word in Polish, and that unicorn would be a much more awesome name.

And now, here are the paths. You have to start with two arcs in minimal position, and , and it’s convenient to fix preferred endpoints, and . We can assume that intersect minimally. The arcs appearing along the path from to , the unicorn arcs, have a rather simple form: You travel along until you reach some intersection point, , and then you go to .

Not all intersection points work, because the procedure described above might not give an embedded arc (look at , for example).

Let us also include and in the collection of unicorn arcs. So, given you have finitely many arcs, and there is a natural order on them: You write if the common subpath of and is longer than the common subpath of and . The idea is that you are describing a path from to , so the paths more closely resembling should appear first.

Now, we have to check that we actually described a path, i.e. that if is the successor of then can be realised disjointly.

The recipe to construct the successor is simple. Let be the intersection point defining , and let be the subpath of from to . Just travel along after until you hit in some point . Then it’s not hard to convince yourself that defines .

The picture also illustrates how to realise disjointly.

**Slim unicorns!** And here is the cool part. Unicorn triangles are 1-slim, i.e. given and a unicorn arc , defined by , on the unicorn path from to there is a point on at distance at most 1 from . The proof is very easy, just travel along until you hit , then go to or , whichever you can reach avoiding .

The picture hopefully clearly illustrates that the unicorn arc we just constructed can be realised disjointly from .

The authors told me that the initial argument for 1-slimness of triangles was based on their dismantlability, you may wish to take a look at this paper.

**The technical bit.** You need one more property to show that arc graphs are hyperbolic and unicorn paths are close to geodesics, that is to say that a subpath of a unicorn path is a unicorn path. Well, this is not actually true, sometimes you can have a subpath of a unicorn path of length 2 connecting points at distance 1 in the arc graph. But that’s the only bad thing that can happen, and this is good enough for us.

Stare at the recipe I told you about earlier to construct the successor. Done? It might now seem to you that if you have two unicorn arcs constructed from , the recipe will give you exactly the same successor of regardless of whether you work “between and ” or “between and .” So… what’s the problem? The problem is that might not be in minimal position!

So you have to understand when they don’t intersect minimally (Sublemma 3.6 in the paper). You can reduce to the case and the predecessor of , once again because of the recipe for the successor. The predecessor of is the unicorn arc travelling as little as possible along , so you just have to find the first intersection point of along .

(Yes, there’s a reason why I’ve drawn the picture in that weird way, see below )

Let me recall that two arcs are in minimal position if there are no bigons or half bigons:

Now, there cannot be a bigon between and , for otherwise you can easily convince yourself that there would be one between and . Similarly, you can only have a half-bigon if appears on it:

So, if there is a half-bigon, then it must look like this:

So, you have a very good control on what can go wrong. I won’t go into details but I hope that you’ll find it reasonable to believe that it’s possible to deal with this problem.

And that’s the end

Let me add the comment that Lemma 3.4, which I didn’t go into, is there just to get a better constant at the end, but without it you can prove, say, 19-hyperbolicity.

The lemma says that given 3 unicorn paths forming a triangle you can find a triple of points on them at pairwise distance at most 1. If you allow yourself to substitute 1 with 3, this follows from what we already know. In fact, suppose you start at and move towards . At first, you’ll be 1-close to , and at some point you’ll be 1-close to . When the switch happens, you have the 3 points you’re looking for. Btw, analysing this idea gives you a way of finding your candidate points at pairwise distance 1.

*David Hume was next to me when I wrote this and said “I have a paper that relies on that, so I’m not going to argue.”

** David suggested I add “I would like to thank David for being so quotable.”

**Theorem:** Let be a simple random walk on a non-virtually cyclic hyperbolic group . Then the expected Hausdorff distance between and a geodesic connecting to is .

Surprisingly, I couldn’t find this result in the literature, except for free groups (Ledrappier) [EDIT: I’ve been pointed out a reference (Corollary 3.9). The proof is quite different, but the overall aims are deeper.]. This post contains an essentially complete proof of the theorem. It’ll be a bit more technical than usual, but I just couldn’t resist blogging the full proof as it’s so short. I’m almost done writing up the same result for relatively hyperbolic groups (as well as a related result for mapping class groups). [EDIT: here it is.]

First, we show that any point on is close to a point in . This has very little to do with random walks, it’s just a fact about paths connecting the endpoints of a geodesic in a hyperbolic space.

*Lemma 1:* There exists so that if the path (say of length at leat ) connects the endpoints of a geodesic in a hyperbolic space and , then .

Another way of saying this is: hyperbolic spaces have (at least) exponential divergence.

The proof is not hard, you can find all the details in Bridson-Haefliger’s book, Proposition III.H.1.6.

The procedure is the following: you split into two subpaths of equal length and consider the triangle as in the picture:

is close to some on one of the sides. Then you repeat using instead of , and if you keep going in logarithmically many steps you get to .

And now we have to exclude the existence of large detours, something like in the following picture:

The idea is that the first and last point of the detour will be close to some point in , and hence close to each other, the details are below. Anyway, this motivates showing that if is at least logarithmically large, then so is .

*Lemma 2:* There exists so that

.

The “” part is there just to make the proof work, but we’re happy if for as in the picture above, because this gives a logarithmic bound on the length of the detour.

The proof is based on the following classical inequality due (afaik) to Kesten, which holds for nonamenable groups: There exists so that

.

In words, the probability that does not make linear progress decays exponentially.

In order to prove the lemma, we just have to sum these inequalities up (using the fact that has the same distribution as ).

Fixing first ( possibilities) and letting the difference between and vary, we see that the probability in the statement of Lemma 2 is at most

,

and this is it, for large.

Ok, the endgame. If there are no points on at distance more than from we are happy. Otherwise, let be so that . Then each point on is at distance at most from either or . As you move from to along the geodesic you’ll be at first close to and then you’ll be close to , so at some point you’ll be close to both. In particular, there are so that . By Lemma 2 (we can assume that) we have . Hence, is also logarithmically bounded, as well as . The end.

]]>Yeah, too much light, I know… Clearer one:

Explanation on the blackboard:

The cone point. Behold the negative curvature!

]]>First, the motivation. Here is a very cool conjecture.

**Borel Conjecture:** If two closed aspherical manifolds have isomorphic fundamental groups, then they are homeomorphic.

A manifold is aspherical if its universal cover is contractible, so that determines the homotopy type of . In case you don’t see the point, let me tell you that the Borel Conjecture for implies the Poincaré Conjecture.

The conjecture has been established for high dimensional manifolds of non-positive curvature by Farrell and Jones, and several people have worked on it since then, most notably showing that the conjecture holds if the fundamental groups involved are hyperbolic or CAT(0).

There are several conjectures related to the Borel Conjecture, for example the Novikov Conjecture, and here is another one.

**Stable Borel Conjecture (SBC): **If the closed aspherical manifolds have isomorphic fundamental groups, then there exists so that is homeomorphic to .

Stabilisation procedures appear all over topology. For example, there are many contractible 3-manifolds (without boundary) that are not homeomorphic to . However, when taking the product of such a manifold with one actually gets . Just to say that the SBC is a natural weaker version of the full Borel Conjecture. Of course, is said to satisfy the SBC if the statement holds for the given .

And here’s the motivation for being interested in FDC.

**Theorem (Guentner, Tessera, Yu): **If has finite decomposition complexity, then satisfies the Stable Borel Conjecture.

(Guoliang Yu told me that one can take in statement of the SBC.)

Time for the definition of FDC. We say that is an -decomposition of the metric space if for each . Let us consider the following decomposition game. Fix a metric space . Bob’s aim is to find a nice decomposition of , and Alice’s aim is to prevent this.

Step 0: Alice gives Bob a huge real number . Bob finds an -decomposition of .

Inductive step: A family of subspaces , coming from the previous step, is given. Alice gives Bob a real number and Bob finds an -decomposition of each . is the union of all subspaces appearing in each decomposition.

**Definition:** has FDC if Bob has a strategy to end up with a collection of uniformly bounded subspaces of .

Having FDC is a quasi-isometry invariant, and actually a coarse invariant.

If you know what asymptotic dimension is, it’s a nice exercise to show that it implies FDC. Otherwise, try to do it for . Hint: find a nice coloured covering with finitely many subsets of diameter , for each .

In particular, say, hyperbolic groups have FDC.

Not all groups with FDC have finite asymptotic dimension. Indeed, GTY showed that linear groups have FDC, but some of them contain an infinite sum of copies of and hence have infinite asymptotic dimension.

Another nice fact is that the class of groups with FDC is closed under extensions. Indeed, GTY observed that the following holds in the context of metric spaces. In order to state it, notice that it makes sense to play the decomposition game starting from a family of metric spaces, rather than one metric space.

*Fibering Theorem:* Suppose that is Lipschitz and that as well as have FDC, for each family of uniformly bounded subsets of . Then has FDC.

A similar statement for asymptotic dimension is usually called Hurewicz Theorem (as it is inspired by an actual theorem of Hurewicz about topological dimension).

The proof is simple. Bob first pretends he’s playing the decomposition game on , and when he as won in he’s left with preimages of bounded sets of in , where he also knows how to win.

Let me state another permanence result. It is not true that if a metric space is an infinite union of subspaces with FDC then the ambient space has FDC. However, the following result tells us that this is true if the subspaces can be “separated”. You may think of as obtained gluing metric spaces to some , and then you can let be the -neighborhood of .

*Infinite Union Theorem: *Suppose that and that has FDC. Also, suppose that for every there exists with FDC so that are pairwise -disjoint. Then has FDC.

Finally, as promised, I’ll tell you how one can show the following.

**Theorem: **A relatively hyperbolic group has FDC if and only if the peripherals do.

The only if part follows from the fact that having FDC is stable under taking subgroups.

For the if part, we can use arguments used by Denis Osin to show the analogue statement for finite asymptotic dimension.

What he did is considering the (Lipschitz) map from the relatively hyperbolic group to its coned off graph, , and showing that the hypothesis of Fibering/Hurewicz Theorem for asymptotic dimension apply to such map. In particular, he showed that has finite asymptotic dimension, and so in particular it has FDC. To show that the Fibering Theorem applies we now have to look at preimages of balls in , and proceed by induction. Let be the preimage of the ball of radius in . Then is obtained as a union of and some cosets of peripheral subgroups (almost true). This is the perfect setting for using the Infinite Union Theorem, with being suitable neighborhoods of , and indeed Osin showed that the result applies. It shouldn’t be too hard to convince yourself of this fact in the case of a free product.

This was the proof outline, and one can actually check that Denis only used the Fibering Theorem and the Infinite Union Theorem (and properties of relatively hyperbolic groups), and hence his proof also gives permanence of FDC.

]]>The most direct way of showing that is hyperbolic is trying to figure out how geodesics in look like, show that you guessed the geodesics correctly and then show that triangles are thin, right? Well, Bowditch tells you that you can skip the second step…

This might not seem much, but showing that a given path is (close to) a geodesic might be quite annoying… and instead you get this for free!

More formally, the statement is the following (in this form it’s due to Ursula Hamenstädt, see Proposition 3.5 here). We denote the -neighborhood by .

**Guessing Geodesics Lemma:** Suppose that we’re given, for each pair of points in the geodesic metric space , a path connecting them so that, for each ,

for some constant . Then is -hyperbolic and each is -Hausdorff-close to a geodesic, where .

(The statement is almost true: You also have to require two “coherence conditions”, i.e. that if and that for each the Hausdorff distance between and the subpath of between and is at most .)

This trick has been used many times in the literature: Bowditch used it to show that curve complexes are hyperbolic, Hamenstädt to study convex-cocompact subgroups of mapping class groups, Druţu and Sapir to show that asymptotic tree-graded is the same as relatively hyperbolic, Ilya Kapovich and Rafi to show that the free-factor complex is hyperbolic, etc.

The proof is quite neat. First, one shows a weaker estimate, i.e. that if is any path connecting, say, to , then the distance from any from is bounded, roughly, by .

To show this, you just split in two halves of equal length, draw the corresponding ‘s and notice that is close to one of them.

Then repeat the procedure logarithmically many times or, more formally, use induction (starting from when ).

Ok, now that we have the weaker estimate (for any rectifiable path) we have to improve it (for geodesics).

Let now be a *geodesic* from to , let be the furthest point from , and set . Pick before and after at distance from (let’s just ignore the case when or , I hope you trust it’s not hard to handle).

As is the worst point, we have , so that we can draw a red path of length at most like so:

*And now the magic happens*. In view of the weak estimate for the red path , we get

,

which gives a bound on . (You might have noticed that here we need the second “coherence condition”). That’s it!

Exercise: complete the proof showing that each is close to .

A relatively hyperbolic version of this trick will be available soon…

]]>As a consequence, for an orientable closed surface of genus at least 2 fixed from now on, we have a natural bijection

{complex structures on } {hyperbolic metrics on }

The bijection is given by the fact that a complex structure on is a way of seeing as a quotient of by a group of biholomophisms, and similarly for hyperbolic metrics.

There’s a more explicit way of comparing a complex structure with its corresponding hyperbolic metric. Look at the tangent space at some point of . The hyperbolic metric (as well as any other Riemannian metric) naturally gives a collection of circles covering . The complex structure does as well (consider multiplication by for ). As it turns out, a complex structure and its corresponding hyperbolic metric give the same collection of circles in every tangent space.

(In other words, they induce the same *conformal* structure.)

*Teichmüller space* is a space that parametrizes the complex structures/hyperbolic metrics on . It does not parameterize them up to biholorphism/isometry, but up to biholorphisms/isometries isotopic to the identity. The formal definition is below the informal discussion in the next paragraph.

Suppose you have a homeomorphism of supported in a disk and a hyperbolic metric on . Then one can take the push-forward of through and get another hyperbolic metric on . Teichmüller space does not distinguish between these metrics (if it did it would be huge!).

On the other hand, suppose you have a hyperbolic metric on with some non-trivial isometry . Then and actually give distinct points of .

And, now, if you really want to see it, here is the formal definition.

where “hyperbolic surface” just means surface endowed with a hyperbolic metric and the equivalence relation given by if and only if there exists an isometry so that the diagram

commutes up to isotopy, i.e. is isotopic to the identity. (One can safely substitute isotopies with homotopy equivalences.)

There is a topology on , but I will not discuss it. Here is probably the most important theorem about .

**Theorem: ** if homeomorphic to , where is the genus of .

More precisely, there is a system of coordinates for , the so-called *Fenchel-Nielsen coordinates*. Danny Calegari has a post on this which contains way more details than the discussion below.

Digression: The very nice structure of given by this theorem explains why it is reasonable to consider hyperbolic metrics up to isometry isotopic to the identity, rather than up to isometry, at least as a first step. Then, if one is interested in hyperbolic metrics up to isometry, one can then take the quotient of by the natural action of the mapping class group , and hence form the so-called moduli space of . This action is very nice as well, e.g. it is properly discontinuous, so that moduli space has the structure of an orbifold.

Back to the theorem. The proof is quite interesting, and is based on a pants decomposition of , that is to say describing as a union of pairs of pants, i.e. the following object:

Any maximal collection of disjoint simple closed curves on gives a pants decomposition, here is an example:

One can endow a pair of paints with a hyperbolic metric so that the boundary is a geodesic in the following way. In there exist right-angled hexagons, and one can also freely choose the lengths of three sides as in the picture.

Gluing two copies of such a hexagon along the blue sides yields a hyperbolic metric on a pair of pants. What is more, one can glue pairs of pants and still obtain a hyperbolic metric as long as corresponding boundary components have the same length. So, given a pair of pants decomposition of , one can assign a length to all the curves appearing in it, consider the corresponding metrics on pairs of pants and glue them all together to get a hyperbolic metric on .

Any maximal collection of disjoint simple closed curves in contains curves, not , which means that there are other parameters to consider.

The hyperbolic metrics we constructed on pairs of pants have the feature that for any given pair of boundary components there is a unique geodesic connecting them and orthogonal to both of them, namely the suitable blue side of one of the hexagons. Hence you see that gluing two pairs of pants in this way (the blue lines represent the geodesics we just described, in adjacent pairs of pants):

or this way:

gives different metrics. So, in order to specify a metric on we also have to assign a “twist parameter” to each simple closed curve in the decomposition. You might think that these parameters are defined modulo , but it is not the case: If we change one of the twist parameters by we obtain a new hyperbolic metric which is indeed isometric to the first one, but there’s no isometry *isotopic to the identity* between them.

To sum up, we choose a maximal collection of disjoint simple closed curves in , and in order to specify a hyperbolic metric on we assign to each of them a “length parameter” and a “twist parameter”. Some routine and some not-so-routine checks and the theorem is proven.

In another post I’ll talk about the tangent space of and the metrics one can put on using it.

]]>But it’s not! The icing is in the shape of a trefoil knot, so that the cake itself has the shape of a trefoil knot complement (in ). Here, take a look at the layers before they were put on top of each other:

Random facts about the cake:

-its fundamental groups is the braid group on 3 strands and has presentations ,

-it contains approximately 500 grams of butter, 500 grams of sugar and 7 eggs,

-it admits finite volume metrics with universal covers and ,

-the icing was very buttery,

-it has a Seifert fibration with base orbifold a once-punctured sphere with two cone-points,

-it took an afternoon minus a (spectacular!) rugby match to bake it.

Credits:

Designer: Rob

Manager: Rob

Cook: Rob

Sculptor: Rob

Icing artist: Rob

Dish washing: Alex