Tag Archives: hyperbolic

Connecting hyperbolic and half-translation surfaces, part II (math)

1 Mar

Last week we saw the standard definition for a hyperbolic surface.  You can tweak this standard definition to define all sorts of surfaces, and we tweaked it for a definition of half-translation surfaces.  Here are the two definitions:

•  A hyperbolic surface is a topological space such that every point has a neighborhood chart from the hyperbolic plane and such that the transition maps are isometries.
• A half-translation surface is a topological space such that all but finitely many points* have a neighborhood chart from the Euclidean plane and such that the transition maps are combinations of translations and flips.  These finitely many points are called singularities.

Precision note: according to Wikipedia, we need to add the adjective “Hausdorff” to our topological space.  We won’t worry about this or give a precise definition of it; you can just know that Hausdorff has something to do with separating points in our space.

Half-translation spaces come with something nifty that occurs in Euclidean space.  You know how when you look at a piece of notebook paper, it has all these nice parallel lines on it for writing?  Or if you look at a big stack of paper, each sheet makes a line which is parallel to the hundreds of others?

This is from clip art panda.  How useful!

Mathematicians call that a foliation: each sheet of paper is called a leaf.  This is an intuitive definition; we’re not going to go into a technical definition for foliation.  Just know that Euclidean space comes with a foliation of all horizontal lines y=r, where is some real number.  Then since transition maps of half-translation spaces come from either straight translations or flips, the foliation carries over to the half-translation space (though orientation might have flipped, we don’t care about those in this application).

All the slides are generously shared by Aaron Fenyes

Notice in the picture in the lower left that there are a few points where the horizontal foliation doesn’t quite work.  Those are the singularities that show up in the definition of a half-translation surface (we need them if we want our surface to be anything besides an annulus).

At those singular points, we glue together patches of Euclidean space.  The orange color in this picture shows the path of the critical leaf as it winds all the way around the surface some number of times.

Last week we had those nify gifs to show us how to think about curvature as positive, negative, or zero.  Here’s the example of zero curvature, because the last arrow is the same as the first arrow:

We can actually get precise numbers instead of just signs for curvature.

Here the triangle encompasses -π/3 curvature.  Notice that it embeds straight down into the hyperbolic surface, so we see an actual triangle down in the lower left.  If we made this triangle bigger and bigger, eventually it’d wrap around the surface and we wouldn’t see a triangle, just a bunch of lines hinting at a triangle up in the hyperbolic plane.  That’s the next picture.

Curvature can range from -π to π.  Here’s an example of an extremely negatively curved triangle which has -π curvature:

Such a triangle in hyperbolic space has all three corners on the boundary/at infinity.  This is called an ideal triangle.  So all ideal triangles encompass -π curvature.  You can see also how in the surface, we have a collection of lines whose preimage is the ideal triangle.

We can also use the same process to find curvature in other places.  For instance, if we make a little hexagon around a singularity of a half-translation surface, we can go around it with the same parallel transport process to figure out how much curvature the singularity contains.  We’ll make use of that horizontal foliation we saw earlier.

This looks very similar to our ideal triangle: the arrow starts off pointing up, and ends up pointing exactly the opposite direction.  So this singularity has -π curvature too, just like the ideal triangles.

Now for the math part!  Here’s the question: given a hyperbolic surface, how can we construct an associated half-translation surface?

Answer: we’ll use those foliations that we had before, as well as something called a geodesic lamination: this is when you take a closed subset of your surface, and give it a foliation.  So it’s like a foliation, only there’ll be holes in your surface where you didn’t define how the pages stack.  The first example of a geodesic lamination is a plain ol’ geodesic curve in your surface: the curve itself is a closed subset, and the foliation has exactly one leaf, the curve itself.  After this example they get real funky.

You don’t even have to take me to funkytown; geodesic laminations are already there!

Given a book, we might want to know how many pages we’ve read once we stick our finger in somewhere.  Luckily there are page numbers, so we can subtract the page number we started at from the page number we’re standing at.  Similarly, given a foliation, we might want to have a measure on it, transverse to the leaves.  If we have one, it’s called a measured foliation.  These exist.

So let’s start with our hyperbolic surface, and choose a maximal measured geodesic lamination on it.  Maximal means that the holes are the smallest they could possibly be.  Turns out this means they’re the images of ideal triangles under the atlas.

Told you they were funky.

Also, there are only finitely many of these triangle-shaped holes down in the surface (we’re sweeping some math under the rug here).  Now we need to get from this surface to a half-translation surface.  We’ll keep that foliation given by the lamination, and we need to get rid of those complementary triangles somehow.  So the lamination’s foliation will become the horizontal foliation of the half-translation surface, and the ideal triangles will correspond to singular points.  We can’t just collapse the ideal triangles to singular points, because as we saw earlier, images of ideal triangles are really funky and wrap around the surface.  We need to find smaller triangles to turn into singular points.  Here’s the picture:

Upstairs, we made a new purple foliation (transverse to the lamination) of the complementary ideal triangles, by using arcs of circles perpendicular to the boundary circle (these circles are called horocycles).  So now we have teensier triangles in the middle of the ideal triangles, called orthic triangles.  To make a half-translation surface, we’ll quotient out the horocycles, which means that in each ideal triangle, we identify an entire purple arc with one point.

Quotienting out horocycles a.k.a. identifying the pink lines all as individual pink points.  That means each side of the orthic triangle is identified with a point, so the orthic triangle disappears.

In this way we get tripods from triangles.  The middles of these tripods are singular points of the half-translation surface.  The measure from the measured lamination gives a measure on the foliation of the half-translation surface.

But Euclidean space actually comes with horizontal and vertical distances defined (remember, half-translation surfaces locally look like Euclidean space).  So far we have a way to get one direction.  How do we get the transverse distance?  We use the fact that we chose a geodesic lamination of our hyperbolic surface.  Geodesics are curves of shortest length; in particular they have length.  So if I’m in my translation surface and moving along a leaf of the foliation, I can look back at where I was in the lamination of the hyperbolic surface and use that distance.  [There’s some rug math here too.]  So we’ve made neighborhoods in the half-translation surface look like Euclidean space.

So that’s that!  You can also go backwards from a half-translation surface to a hyperbolic surface by blowing up the singular points into ideal triangles.  [More math, especially when the singularities of the half-translation surface are messy or share critical leaves].  Aaron claims this is folklore, but a quick google search led me to this paper (in French) and this one by the same author who connects flat laminations (on half-translation surfaces) to the geodesic ones we see in hyperbolic surfaces in section 5.

*I lied about finitely many points.  You can have infinitely many singularities in a half-translation surface; they just have to be discrete (so you should be able to make a ball around each other disjoint from the others, even if the balls are different sizes).  Examples of discrete sets: integers, $2^x, x>0$. Examples of not-discrete sets: rational numbers, $2^x, x<0$.

Connecting hyperbolic and half-translation surfaces, part I (definitions)

24 Feb

I love talks that start with “I haven’t seen this written down explicitly anywhere, but…” because that means someone is about to explain some math folklore!  There are some statements floating around in mathland that specialists in those fields believe, so the rest of us believe them because the specialists said so, but no one knows a citation or a written proof for them.  That’s folklore.  Two weeks ago I gave a talk and someone asked a question (are RAAGS uniquely determined by their defining graphs?) and I said “probably, but I have no references for you.”  I found a reference a day later and emailed it to her, but the reference was way hard and had way more machinery than I was expecting.  The power of folklore!

Anyways, this series of posts will be based on a talk by a grad student at UT, Aaron Fenyes.  This was a great, great talk with lots of pretty slides, which Aaron has generously allowed me to put up here.  We’ll review curvature and surfaces, and then talk about how to go back and forth between two kinds of surfaces.

We’ve chatted about hyperbolic space v. Euclidean and spherical space before in terms of Euclid’s postulates, but let’s chat a bit about curvature. We say the Euclidean plane/real space has curvature 0, that hyperbolic space is negatively curved, and spherical space is positively curved.  There’s a nice way to see curvature: draw a triangle in your space (that old link also has some triangle conditions in it), and imagine standing at a point on that triangle and looking toward one corner of the triangle.  By “looking out” I mean your gaze should lie tangent to the triangle.  Remember:

Left: tangent; line hits circle at exactly one point.
Center: not tangent, line hits circle at two points
Right: not tangent, line doesn’t hit circle

Now walk toward the corner you’re facing, and then walk down the second side of the triangle still facing that direction (so you’re sidestepping), and walk around the next corner (so you’re now walking backwards) and keep going until you end up where you started.  This is called parallel transport.  If your triangle was in Euclidean space, then you’re facing the same way you were when you started.

Gif!  Note the ending blue vector is identical to the beginning purple vector.  So we have curvature 0.

If your triangle was slim, then you might find yourself facing the opposite way that you started!  Or if your triangle isn’t that curved, you’ll find yourself facing a direction counterclockwise from your original one.

Aaron Fenyes made the pictures; I gif’d it!  My first gif!

This is the summary of the gif: you can see how the ending light blue vector is pointed away from the original purple vector

From the original to the new vector: it’s a pi/3 counter-clockwise turn.

Similarly, if your triangle was fat, you’ll end up facing a direction clockwise from your original.

Here’s the picture of just the first and last arrows:

The green arrow is now clockwise from the red arrow, which means this has positive curvature.

So curvature is a way to measure how far clockwise you’ve turned after doing this parallel transport.

I love that description of curvature vs. the way I did it before, but they’re all good ways of seeing the same thing.  Next we need to review surfaces.  When we first met hyperbolic surfaces, we built them by gluing pairs of pants together, which themselves were stitched together from right angled hexagons which lived in hyperbolic space.  Redux:

Now if I take a little patch from my hyperbolic surface, I can trace back through one or two pairs of pants to find the original hexagon(s) in hyperbolic space where my patch came from.  So I have a map from hyperbolic space to my patch of hyperbolic surface, describing the metric and geometry around that patch.  This map is called a chart, and every point on a hyperbolic surface will have a chart associated with it, sending some part of hyperbolic space to a neighborhood of that point.

Here Aaron picked a blue patch and an orange patch in the surface, and the picture shows their charts from the hyperbolic plane to the patches.

This picture might make you leery: what happens when images of charts overlap, like they do here?  The preimages in the hyperbolic plane are disjoint, but they map to the same yellow area in the surface.  We want to say there’s some reasonable relationship between the yellow preimage patches in the hyperbolic plane.  That relationship is the only one we know, isometry:

There’s an isometry of the hyperbolic plane sending the orange patch to the new orange patch, so that the yellow parts overlap exactly.

If we look only at the yellow patch, we can find another way to describe the map in the picture: first, do the blue chart sending the blue patch to the surface.  Then, do the inverse of the orange chart, which sends the orange surface patch to its preimage.  Restricted to the yellow overlap patch, this is the definition of a transition map.

So here’s another way to think of hyperbolic surfaces, instead of gluing hexagons together like before.  A hyperbolic surface is a topological space such that every point has a neighborhood chart from the hyperbolic plane and such that the transition maps are isometries.

If you change where the chart is coming from, we can change the adjective before surface.  For instance, a flat surface is when the charts come from the Euclidean plane.  Now we’re going to define half translation surfaces, where the charts come from the Euclidean plane, but we have some more conditions on the transition maps.  The isometries of the Euclidean plane all come from a combination of translations and rotations.  Instead of allowing all isometries, we’ll only allow some of them:

In this picture you can see the orange and blue patches on the surface which come from the Euclidean plane.  Now we’re allowing translations and pi (180 degree) rotations only for our transition maps.  That’s why they’re called half-translation surfaces: charts from the Euclidean plane, and transition maps are translations plus half-rotations (flips).  As an aside, a translation surface is when we allow translations only, and no flips.

In the next post in this series, I’ll go through Aaron’s explanation of how we can go from hyperbolic surfaces to half-translation surfaces and back, and we’ll get to revisit our old friend the curve complex.  It’ll be fun!

Playtime with the hyperbolic plane

2 Feb

Update: Thanks to Anschel for noting that I messed up the statement of the last exercise.  It’s fixed now.  Thanks to Justin for noting that I messed up a square root.  Pythagorean theorem is hard, yo.

About a year and a half ago I explained what hyperbolic space is, specifically by contrasting it with Euclidean space and spherical space.  We’ve also run into hyperbolic groups a few times, which are groups whose Cayley graphs are somehow like hyperbolic space.  More precisely, a group is hyperbolic if, whenever you have a Cayley graph of that group, triangles are $\delta-$thin, which means the third side of any triangle is contained in a $\delta$ neighborhood of the other two sides.  It’s important that the same $\delta$ works for every triangle in the space.

Here the bottom side is contained in a neighborhood of the other two sides, and the triangle looks like it belongs in Star Trek

Here each side is contained in a small neighborhood of the other two sides, and it seems like the triangle is curving inward

Note that triangles in Euclidean space are way totally far from being $\delta-$hyperbolic.  For any big number n, you can make a triangle so that the third side is not contained in an n-neighborhood of the other two sides: just take a 2n horizontal segment and a 2n vertical segment to make an isoceles right triangle.  If is bigger than 2, then the midpoint of the hypotenuse is farther than away from the other two sides.  As usual, this long paragraph could be better done in a picture.

Soooo not hyperbolic: you can make arbitrarily fat triangles in Euclidean space.  Also, the purple line should have $\sqrt{2}}n as its length, not the square root of n.[/caption] I thought today we could just play around with hyperbolicity. I'm running a small reading group on geometric group theory with some grad students, and today we got sidetracked a few times by just basic thoughts about geodesics in the hyperbolic plane. We all thought they were interesting, so here I am trying to share it with you! There are lots of other definitions of hyperbolicity, but I like$latex \delta-thin triangles. Oh I forgot to mention that a nneighborhood of a point/line/shape consists of all the points within n of that point/line/shape. So, for instance, a 3-neighborhood of a point in Euclidean space is a circle. But with a taxicab metric, that 3-neighborhood is a squarey circle. [caption id="attachment_3081" align="alignnone" width="181"] Purple points are all distance three or less from red point Anyways, I just put in that definition because it’s the first thing you’ll hear or see in a colloquium talk that involves the word “hyperbolic.” Let’s play with the upper half plane model of hyperbolic space! Here’s a repeat picture from that October 2014 post (wow that’s when baby was born! He’s walking around and getting into trouble now, btw.). Straight lines are ones that go straight up to infinity, and segments of half-circles whose diameters lie on the bottom line The graph paper lines in this picture are misleading; they contrast hyperbolic geodesics with Euclidean ones. So the gray lines are Euclidean geodesics, and the colored ones are hyperbolic. All geodesics in this model are either straight lines perpendicular to the horizontal axis, or semicircles perpendicular to the horizontal axis. All of the horizontal axis and everything that the straight up and down geodesics end at (sort of like a horizontal axis infinitely far away) represent infinity. I’ll write down the metric in case you were wondering, but we won’t need it for what we’ll be doing: $ds^2=\frac{dx^2+dy^2}{y^2}$ [I took this formulation from wikipedia]. What this says is that the hyperbolic metric is a lot like the Euclidean one, except that the higher up you go on the y-axis, the less distance is covered (because of that 1/y factor). More precisely, if you’re just looking at the straight line geodesics, the distance between two points at heights a<b is $ln(\frac{b}{a})$. All the lines have the same length ln(2). Blue: ln (8/4), green: ln (16/8) The other fact we might want to know is that things that look like Euclidean dilations (stretching something like your pupil dilates from looking in a bright light to a dark room) are isometries in this model. You can see that in the picture above: the lines look like they’re stretching longer and longer in the Euclidean metric, but they’re actually all the same length. Speaking of isometries, if you have any two geodesics (like a vertical line and a big old semi-circle somewhere else), you can find an isometry that sends one to the other. First question: what do circles look like? Whenever you have a metric space, it’s nice to know what neighborhoods look like, and the first thing you might want to consider are neighborhoods of points. Turns out circles in this model look like circles in Euclidean space, but the centers aren’t where you think they are. For instance, here’s a picture of circles with radius ln(2), which we saw in the straight lines above. The center of each circle is at the top of its surprised mouth. The next highest line segment shows that each vertical diameter is actually a diameter (twice the radius). Notice that the centers of these circles hang a lot lower in the circle than they do in the Euclidean metric! Isn’t playtime fun?! Generally when I play with math I throw out a lot of garbage ideas, and then eventually one of them is somewhat right. Other people apparently think for awhile before they put out an idea. Anyways, here are some sketches of what I thought a 2-neighborhood of a vertical line might look like: This is the most subtle joke I have ever put in this blog Maybe you looked at these and were like “Yen that is nonsense what were you thinking?!” Maybe you are my advisor or a practiced mathematician. Let’s go through the nonsense-ness of each of these pictures: The rightmost picture is a 2-neighborhood of the vertical line in Euclidean space. We know hyperbolic space is pretty drastically different from Euclidean space, so we wouldn’t expect the neighborhoods to be so similar. The middle and left pictures have similar shapes but different curviness, and yes we’d expect a hyperbolic neighborhood to look different so those are guesses based in some more intution. However, let’s try to figure out the actual size of a neighborhood of a vertical line. We can use our previous pictures, and switch to a ln(2) neighborhood. Changed my mind this is the most subtle joke I’ve put in this blog please someone get it and appreciate it please please Here I moved all our ln(2) circles so that their centers laid on the same line. A neighborhood of a line is just the union of the neighborhoods of all of the points on that line, so if we just keep making ln(2) circles along the line we’ll end up with a neighborhood of the whole line. So you can see that our actual neighborhood ended up being upside down from my middle picture above. If this explanation didn’t make sense, here’s [half] a 2-neighborhood of a Euclidean line: Note how the denser the circles, the closer their boundaries on the left get to becoming that straight line we see on the right. Actually using Euclidean intuitions and then mixing them up a bit is a great way to play with the hyperbolic plane. This next exercise was an actual exercise in the book but it is just so crazy I have to share it with you. It’s just dramatically different from Euclidean space, just like the triangles were. If you have a circle in the hyperbolic plane and project it to a geodesic segment that it doesn’t intersect (which means for any point on the circle, you find the closest point to it on the geodesic and draw a dot on the geodesic there), the projection is shorter than $ln(\frac{\sqrt{2}+1}{\sqrt{2}-1})$. Here’s the picture in Euclidean space where this makes no sense: Third place likes getting on the podium. I meant, the vertical lines show the projections from the faces to the horizontal line, and you can see they can be as big as you want if you just make bigger and bigger circles. And here’s a picture in hyperbolic space that might make you think this could possibly just maybe be true. Any circle will eventually fit inside a big huge circle that looks like the blue one in the picture, so its projection would be shorter than the projection of the blue one. That means you only have to worry about big huge circles in that particular position. And by “big huge,” I mean “of (Euclidean) radius n“. Remember, if we’re just looking at vertical lines, we know how to measure distance: it’s $ln(\frac{a}{b})$. So if you can show that the small orange circle hits the vertical line at $\sqrt{2}n-n$ and the big orange circle hits it at $\sqrt{2}n+n$, you’ll have proved the contraction property. Try using Euclidean geometry, and think about how we did the triangles case. That was fun for me I hope it was fun for you! Quick post: research updates of friends 18 Aug I noticed a few papers up on arXiv last week that correspond to some old posts, so I thought I’d make a quick note that these people are still doing math research and maybe you are curious about it! We last saw Federica Fanoni and Hugo Parlier when they explored kissing numbers, and they gave an upper bound on the number of systoles (shortest closed curves) that a surface with cusps can have. This time they give a lower bound on the number of curves that fill such a surface. Remember, filling means that if you cut up all the curves, you end up with a pile of disks (and disks with holes in them). So you can check out that paper here. Last time we saw Bill Menasco, he was working with Joan Birman and Dan Margalit to show that efficient geodesics exist in the curve complex. This new paper up on arxiv was actually cited in that previous paper- it explains the software that a bunch of now-grad students put together with Menasco when they were undergrads in Buffalo, NY (UB and Buffalo State) during this incredible sounding undergrad research opportunity– looks like the grant is over, but how amazing was that- years of undergrads working for an entire year on real research with a seminar and a semester of preparation, and then getting to TA a differential equations class at the end of your undergraduate career. Wow. I’m so impressed. I got sidetracked: the software they made calculates distances in the curve complex and the paper explains the math behind it and includes lots of pretty pictures. My friend Jeremy did a guest post about baklava and torus knots a long time ago, and of course he’s got his own wildly popular blog. He also has a bunch of publications up on arXiv, including one from this summer. They’re all listed in computer science but have a bunch of (not-pure) math in them. The paper I worked on over that summer at Tufts with Moon Duchin, her student Andrew Sánchez (note to self: I need a good looking website I should text Andrew), my old friend Matt Cordes, and graduate student superstar Turbo Ho is up on arXiv and has been submitted: it’s on random nilpotent quotients. Moon and Andrew and others from that summer have another paper which has been accepted to a journal, it’s also about random groups and is here. It was super cool, I saw a talk at MSRI during my graduate summer school there and John Mackay (also a coauthor on that paper) was in the audience and this result came up organically during the talk. Pretty great! There’s another secret project from that summer which isn’t out yet, but I just checked two of the three co-authors webpages and they had three and four papers out in 2015 (!!!) That’s so many papers! So I don’t know when secret project will be out but I’ll post about it when it is. I really enjoy posting about current research in mathematics and trying to translate it into undergrad-readability, so I’ll try to continue doing so. But this Thursday you’ll read about cinnamon buns instead. Yum. 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 pointn\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.

Kissing numbers, current research in hyperbolic surfaces

30 Mar

I just got back from the fantastic Graduate Student Topology & Geometry Conference, where I gave a talk and also brought my baby.  I tried to google “bringing baby to academic conference” as I’ve seen one baby at a conference before (with his dad), and I knew this kid would be the only baby at ours.  But it was cold enough/uncomfortable enough that I just had him stay in the hotel with my mom, and I ran back during breaks to feed him.  Also, it was my first time being “heckled” by both of these two brother professors famous for “attacking” speakers- they happen to know just about everything and are also suckers for precision, which I am not (and should be).  But I got a lot of good feedback on my talk, and I’m generally a very capable speaker (though I was not as prepared as I would’ve liked, thanks to somebody who likes to interrupt me every five minutes…)  Anyways, this is not about me, this is about my friend who gave one of the best talks of the conference and more importantly, her research.  This post is based on notes I took during her talk + skimming her paper (joint with her advisor) on which it is based.

Remember that we had our introduction to hyperbolic space.  This research is focused on hyperbolic surfaces, which are shapes that locally look like hyperbolic space- this means that if you look at one point on the surface and just a little area around it, you think you’re in hyperbolic space.  A good analogy is our world- we live on a sphere, but locally it looks like flat space.  If you didn’t know better, you’d think the earth is flat, based on your local data.  So how can we build a hyperbolic surface?

While hexagons in flat space always have angles that sum to 720 degrees, that’s not true in hyperbolic space.  In fact, you can make right angled hexagons, which means that every single corner has 90 degrees.  If you pick three lengths a,b,c>0 and assign these lengths to three sides of the hexagon like the picture, you’ll fully determine the hexagon- hyperbolic space is wacky!

Now glue two copies of a hexagon together along those matching a,b,c sides.  You’ll have a funny shape with three holes in it, and those holes will have circumference 2a, 2b, 2c.  This is called a pair of pants in topology.

You can glue together a bunch of pants to form a hyperbolic surface, by gluing them together along holes with the same length.  Any hyperbolic surface, conversely, can be cut up into pairs of pants (this pants decomposition is not unique, as you can see below).

You could also set one of those lengths equal to 0, so you’d get a right angled pentagon as one of the hexagon’s sides would collapse.  You can still do the pants thing here by gluing together copies of the pentagon, but instead of having a hole with circumference 2a like we had before, you’ll have a cusp that goes off to infinity- it’s like an infinite cone with finite volume.

Now we’ve built every hyperbolic surface (there are some more details, like how you glue together pants, but let’s just stick with this broad schematic for now). As long as the expression 2-2*(number of holes)-(number of cusps)<0, your surface is hyperbolic.  So, for instance, a sphere isn’t hyperbolic, because it has no holes and no cusps, so you get 2 which is not smaller than 0.  And a torus isn’t hyperbolic, because it only has one hole, so you get 2-1=1.  But all the surfaces in the pictures in this post are hyperbolic- try the formula out yourself!

One thing you can ask about a hyperbolic surface is: how long is its shortest essential curve?  By “essential,” we mean that it isn’t homotopic (this is a link to a previous post defining homotopy) to a cusp or a point.  This shortest curve is called the systole of the surface.  Systolic geometry is a whole area of study, as a side note.  But we’re interested in the question: how many systoles can a surface have?  This is called the kissing number of the surface.

A few notes: a “generic” surface has Kiss(S)=1, that is, there’s only one shortest curve if you happen to pick one “random” surface (scare quotes because no precise definitions).  And it’s relatively “easy” to make a surface with Kiss(S)=3*(number of holes)-3+(number of cusps).  Check for yourself that this number is exactly the number of curves in a pants decomposition of a surface.  Using some hyperbolic geometry you can prove that there won’t be any shorter curves if you make all of the pants curves very “short.”

So what Fanoni and Parlier do in their paper is come up with an upper bound on the kissing number of surfaces with cusps.  I won’t go into that, but I will try to explain part of a lemma they use on the way.

If your surface doesn’t have any cusps, then systoles can pairwise intersect at most once.  But if you do have cusps, then Fanoni & Parlier prove that your systoles can intersect at most twice (and they build examples of surfaces with cusps that have systoles that pairwise intersect twice).

First they show that two systoles which intersect at least twice can only intersect in the way pictured to the left below, and not as in the right:

This picture from the Fanoni-Parlier paper I did not make this!

This matters because it implies that two systoles which intersect at least twice must intersect an even number of times.  In particular, if two systoles intersect more than twice, then they intersect at least four times.

So assume for contradiction that two systoles and intersect more than twice.  So they intersect at least four times.  That means that there’s some intersection point somewhere such that the b-arcs coming out of it make up no more than half of the systole length (see picture below)

If the green arc is more than half the length of the circle, then the blue one is less than half the length of the circle.

So if you look at these short b-arcs, plus the path, and wiggle things around, you’ll see a four-holed sphere (two holes above” the curve, and two holes “below,” one of each inside a arc).

Left: a schematic of how a and b intersect. Black dots represent holes or cusps.
Center: the short b arcs plus the full a path
Right: the short b arcs plus the a path, after moving four dots to be holes of a 4-holed sphere

This four-holed sphere has a curve on it, determined by part of and the arcs, which is shorter than the original systoles.  This contradicts the definition of systole, so our premise must be wrong- two systoles can intersect at most twice.

This was proposition 3.2 in their paper- tomorrow I’m going to share propositions 3.1-3.3 with my advisor’s small seminar.  Hopefully I don’t get heckled too badly this time!

OOPS I ALMOST FORGOT: life update.  We bought a house and are moving to Austin, TX.  I’m still planning on finishing my Ph.D., just virtually.  [Up to a finite-index subgroup, obviously.  Bad math joke].  I’ll probably be flying up to Chicago every so often to meet with my advisor/eventually defend my thesis.  But yes, we’re driving in our minivan to Texas on Thursday.  So… we’ll see when we get the internet set up in the new house.  I’ll try not to make too long a break until my next post.

What is hyperbolic space? (updated)

2 Oct

Thanks very much to reader “ilikemathyoudont” for pointing out that I, yet again, messed up the triangle inequality.  Corrected it below.  If you are like me and always forget it, you should draw triangles like I do below, and probably you’ll get it right.

I spend a lot of my time (most/all of my time) thinking about shapes/structures/arguments in hyperbolic space, so I thought I’d take a post and explain what it is.  Maybe I’ll be able to put up some research-y posts one day about it.  NOTE: in this post, we’re only going to talk about two dimensions.  In the future we might talk about higher dimensions (like three, the one we live in).

First, let’s intuitively figure out what I mean by “space.”  For us, a space is somewhere that a point can walk around and measure how far it’s moved.  The way we measure distance is called a metric (I’ve written about metrics before)- here are a few examples.  Let’s take as our space the plane a.k.a. a flat land.

The orange path (taxicab metric) takes 14 steps between the two dots. The green only takes 10 (Euclidean metric). I just made up the purple one but it’s a weird length.

Each of these metrics measures a different distance between where a point starts and where it ends.  There aren’t too many rules to be a metric:

1. you need to have distance zero if and only if start point = end point,
2. distances need to be positive (or zero, see 1)
3. we need to satisfy the triangle inequality: for any three points x,y,z in your space, this should be true: $d(x,y)+d(y,z)\geq d(x,z)$

I actually use the triangle inequality ALL THE TIME and I always forget what it is and need to draw a triangle.  Essentially, you need to be able to draw a triangle with the distances, so the sum of the two sides can’t   must be longer than the third side.

The orange one is fine. The yellow guy on the left isn’t quite a triangle, but his sides are long enough that we can swing around and make a triangle. But poor blue- there’s no way to use those little stumps to close up into a triangle.

So one way you can have a different metric space is by taking your space (like the plane) and putting a different metric on it.  But what if we change the space?  Like, what if instead of walking around the plane, we walk around a sphere (like the Earth)?  Our space will have different properties.  For instance, if you walk in a straight line on the plane, you’ll never get back to yourself.  But you can walk around the equator (with, yknow, walk-on-water shoes and infinite endurance) and end up right back where you started.  Somehow the sphere is fundamentally different from the plane.

Let’s have a super short history lesson.  Once upon a time (around 300 B.C.), a Greek named Euclid wrote a super cool book called the Elements.  In it he wrote a bunch of definitions and five axioms (things that we assume are true without proof), which laid the groundwork for the study of geometry.  These are the axioms:

1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. Given a line and a point not on it, there exists exactly one line that passes through that point and never intersects the line.

And here are the pictures that illustrate them:

For a really long time, people were happy with the first four and unhappy with the fifth one.  Euclid’s fifth postulate really irked mathematicians for a millenium or so, and in the mid-1800s  we finally got non-Euclidean geometry, which accepts the first four postulates and rejects the fifth one.

There are a couple of non-Euclidean geometries (hyperbolic geometry is one of them), but I think spherical geometry is a little bit easier to get your head around first, because the Earth is a sphere (thank you, history).  Let’s get back to the metric on the sphere.  We want distance to measure the shortest path between any two points, so instead of drawing a straight line on a map we do those funny arcs that they have in airline maps.

These arcs look funny, but if you had Venice and Toronto on a globe and taped a piece of floss to the globe between the two, it would map out the arc exactly as it looks on the map above.  Turns out all lines in spherical geometry can be extended to great circles on the sphere, a.k.a. the longest possible circle you can draw.  The equator is an example of a great circle.  Or any circle that includes the north and south poles.  And that brings us to Euclid’s postulate no. 5- given a line on the sphere, and a point not on that line, there’s no way to draw a line through the point which is parallel to the first.

No matter how we try to draw a line through the mole, it’ll hit the equator.

Another fun thing about spherical geometry: triangles don’t add up to 180 degrees like they do in Euclidean geometry.  This picture from Wikipedia proves it better than I can:

On the sphere, you can have a triangle with way more than 180 degrees: this one has 230 degrees! But in flat space, all triangles add up to 180.

So a few differences we’ve noticed between flat and spherical geometry so far:

• In Euclidean geometry, there’s exactly one line parallel to an original line that goes through some other point.  In spherical geometry, there are none.
• In Euclidean geometry, all triangles add up to 180 degrees.  In spherical geometry, they add up to more than 180 degrees.

(Side note: this number-of-degrees-in-a-triangle fact is equivalent to the parallel postulate, so these two facts are basically the same).

There’s a natural question that comes up from these two differences:

• Is there a geometry with more than one line parallel to an original that goes through some other point?
• Is there a geometry where all triangles add up to less than 180 degrees?

The answer is yes!  This is called hyperbolic geometry and is where lots of research lives nowadays.  In this land, if you draw a line there are infinitely many lines parallel to it that go through some other point.  And all triangles add up to less than 180 degrees.  There are many models of hyperbolic space, but we’ll just look at two.  The first one is the Poincare disk model.  Escher does a really good picture for this:

All the fish are the same size!

In this model, the outside circle represents the end of space a.k.a. infinity (we call it the circle at infinity or the boundary of hyperbolic space).  One way I’ve explained this picture is imagine that there’s an infinitely large bowl printed with all these fish, which are all the same size.  If you stick your head into the bowl, the fish at the “bottom” of the bowl will be pretty big, while the fish in your peripheral vision will get smaller and smaller the further away they are from the bottom of the bowl.  This is a nice way to start to wrap your head around hyperbolic space, which is fundamentally different from flat space in the opposite way that spherical space is.  We say that spheres are positively curved, while hyperbolic space is negatively curved (and flat space isn’t curved or has curvature 0).

The metric is a little harder to see in this model, so mathematicians often use the upper half-space model instead.  It’s sort of like using a map to think about the Earth instead of a globe.  When we use maps, they’re finite, because the Earth has finite surface area.  But in hyperbolic space, since we go off to infinity, we’re going to have to use something that is infinite.  So we use the top half of the Euclidean flat grid.  While straight lines on a map of Earth are arcs, as we saw above, straight lines on this model of hyperbolic space look a little different.

Straight lines are ones that go straight up to infinity, and segments of half-circles whose diameters lie on the bottom line

This model includes the boundary at infinity too, but it’s infinitely far away up (just like infinity in Euclidean space is infinitely far out).  If you have two points (x,y) and (a,b) in the Euclidean plane/flat space, the distance formula (which measures the metric) is $\sqrt{(x-a)^2 + (y-b)^2}$.  To write this in terms of differentials (nope, not defining that now), we can say $ds^2 = dx^2 + dy^2$ for the Euclidean plane.  In the upper half plane model of hyperbolic space, the metric is $ds^2 = \frac{ dx^2 + dy^2}{y^2}$.  Roughly, this means that the further up you go, the shorter horizontal distances are.  That’s why the fastest way between two points on the bottom line is using those half circles we drew above.

OK that’s our introduction to hyperbolic geometry.  I really wanted to put in a math post before my life derails for a bit.  So I apologize if we don’t have a post for awhile after this one- I’ll be dealing with a newborn.  Here’s a picture of me now: