Tag Archives: Cayley graph

The (2,3,7) Triangle Group

8 Nov

Hi!  Today we’re going to use some stuff we learned about a long time ago (non-Euclidean geometry, manifolds, and groups) and put it together to explore a particular group.  This is based on a talk given by my dear friend and co-organizer Michelle Chu.  “Co-organizer?  Of what?” you ask.  Well thanks for asking!  Last weekend we did held the first Texas Women in Mathematics Symposium – we had over 60 people come, lots of talks, lots of networking, and lots of food.  By the end of the day I got to add “annual” to that description, and it seems like a lot of schools were interested in hosting it in future years.  Maybe some time I’ll write a post about how to found an annual conference (this is my second!).

Anyways, let’s first talk about tilings by triangles.  We first choose some integers p, q, r and set the three angles of a base triangle equal to $\frac{\pi}{p}, \frac{\pi}{q}, \frac{\pi}{r}$.  Now reflect over each of the three sides to start tiling your space.  Turns out this tiling will lead to a group.  Here’s an example with p=q=4 and r=2 (so we have a right isosceles triangle):

Start with the pink triangle, and reflect it over each of the three sides to make the colored triangles as shown.

Now do the reflections again.  I kept the pink base triangle and grayed out the first images.  Note that I colored the bottom left image yellow, for reflecting over the vertical side of the bottom orange triangle, but I also could color it orange, for reflecting over the horizontal side of the left yellow triangle.  This means that yellow+orange = orange+yellow in the group.

A third iteration of the same process; there are more relations here (that I didn’t color)

I picked a particularly good example, so that my triangles could tile the Euclidean plain.  We learned some time ago about non-Euclidean geometries: the space is spherical if the sum of triangle angles is more than $\pi$, and hyperbolic if triangles are thin and their sum of angles is less than $\pi$.  So based on how I choose my p, q, and r, I’ll find myself tiling different spaces.  Here’s an example of one iteration on a sphere for p=q=2 and r=5:

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It’s pretty easy to find the integer solutions for p, q, r to tile each space.  The only triangles that tile the flat plane are when (p,q,r) = (2,3,6), (2,4,4), and (3,3,3).  We already saw (2,4,4), and I’ll just tell you that (3,3,3) is when you use equilateral triangles (so there are 6 around each vertex), and (2,3,6) are those 30-60-90 triangles we learned about back in trigonometry class: here’s the picture from wikipedia:

Similarly there are only a few (relatively) that tile the sphere: (2,3,3), (2,3,4), (2,3,5), and (2,2, n), where is some number bigger than 1.  Of course this forms an infinite family of tilings, since you can choose n.  In the example above I chose n=5, and if is bigger the base pink triangle just gets skinnier.

But I say there’s only a few that tile the sphere because everything else tiles the hyperbolic plane.  That’s a lot of choices!  It might also make you think, “hm, maybe the hyperbolic plane is interesting.”

Let’s bring us back to groups.  How does a tiling of a space lead to a group?  Well, let the reflections over the (extended) sides of the base triangle be the generators of our group.  If I name these a, b, and c, I immediately get the relators $a^2=b^2=c^2=1$.  Next we have to figure out the rest of the relators.  I hinted at them in the pictures above.  They are $(ab)^p=(bc)^r=(ca)^q$.  Now we have a group presentation [link for definition] $R\triangle(p,q,r)\cong \langle a, b, c \ | a^2=b^2=c^2=(ab)^p=(bc)^r=(ca)^q=1\rangle$.

Also, fun coincidence: if you create the dual tree to the tiling by putting a vertex inside each triangle and connecting two vertices by a line if one triangle is the image of another under one of the reflections, you get something that looks a lot like the Cayley graph of the reflection triangle group.  The only difference is that each edge needs to be two edges (like a little loop) to reflect that each generator has order 2.

So what’s special about the (2,3,7) triangle group?  We know from above that it tiles the hyperbolic plane.  Check out this great picture from wikipedia of the tiling:

Maybe we’ll take a second here to point out that you can see the p, q, r values in the tilings, both in this picture and the other wikipedia picture above for (2,3,6): pick your favorite triangle, and look at its three vertices.  Each vertex is adjacent to other triangles, and since there are $2\pi$ angle around any vertex, we can figure out that p,q,r are just $\frac{n}{2}$.

Also, of all the integers you can pick for p, q, r, it turns out that 2, 3, and 7 maximize the sum $\frac{\pi}{p}+\frac{\pi}{q}+\frac{\pi}{r}$ while staying less than $\pi$.  [It ends up giving you $\frac{41\pi}{42}$ for those of you keeping track at home.]

So we maximize something with the numbers 2, 3, 7.  Well it turns out we do more than that- we also minimize the volume of the resulting quotient-we touched on and defined those in this post about manifolds.  And this is unique (up to conjugation/fiddling), and makes the smallest possible quotient.  Huzzah!

On a personal note, I’ve had a demonic cold keeping me in bed for the past two weeks, so forgive me if I messed up (pretty sure I did on the reflections; I’ll try to fix those soon).  Also, hope you voted today!  I voted a week and a half ago.

What is a covering space?

14 Nov

We’ve briefly covered fundamental groups before, and also I’ve talked about what geometric group theory is (using spaces to explore groups and vice versa). One way to connect a group to a space is to look at a covering space associated to that group. So in this post, we’ll come up with some covering spaces and talk about their properties. This is in preparation for talking about separability (we already have an advanced post about that).

Aside: you might catch me slipping into the royal we during my math posts.  This is standard practice in math papers and posts, even if a paper is written by a single author.  Instead of saying “I will show” and proving stuff to you the reader, we say “we will show” and we go on a journey together.  I’m sure that’s not why mathematicians do this, but I like to think of it that way.

Also, sometimes I say “group” when I’m obviously referring to a space, and then I mean the Cayley graph of that group (which changes depending on generating set, but if it’s a finite generating set then all Cayley graphs are quasi-isometric).

Let’s start with an example, and then we’ll go on to the definition.  Here’s an old picture to get us in the mood:

This blue curve goes around the circle three times.

This picture was from the short fundamental groups post: you’re supposed to see that the blue spiral up above represents a curve going three times around the circle below.  Now consider this next picture:

Blue line covering the happy circle below

Here the blue spiral goes on forever in both directions.  If you unwound it, you’d get a line stretching on forever in both directions, which we’ll call the real line (the same number line you’re used to, with real numbers along it).  This picture sums up the intuition that the real line covers the circle: for any point on the circle, there are a bunch of points on the real line directly above it that project down to that point.  In fact, it does more than that:

Pink parts of blue line cover the pink part of the circle

For any point on the circle, there’s a neighborhood (the pink part) so that up in the real line, there are a bunch of neighborhoods that map down to that pink part.  And those neighborhoods aren’t next to each other nor all up in each other’s business: they’re disjoint.  So here’s the definition:

A covering space X of a space Y is a space with a map p: X->Y such that any point in Y has a neighborhood N whose preimage in $p^{-1}(N)\subset$ X is a collection of disjoint sets which are homeomorphic to N.

So why is this helpful?  Well, in our example we can say that the real line covers the circle, from the pink pictures.  We could also say that the circle wound around itself three times covers the circle, from the first picture in this post:

The three highlighted parts up above are homeomorphic to the the pink part on the bottom circle’s chin

The picture I just drew might not convince you, because every point on the bottom space needs to have a neighborhood that lifts up to the top space, and what about the left most point of the circle?  Well, up above that neighborhood just winds around between the top and bottom copies:

Still a cover: each of those pink things up above are homeomorphic to the bottom cheek

The fundamental group of the circle is the integers, so maybe using geometric group theory (or algebraic topology, really) we can come up with conclusions about the integers using facts about the circle or the line, and vice versa.  In fact, there’s a correspondence between group structures and covering spaces!  With some conditions, covering spaces correspond to subgroups of fundamental group.

Let’s see how this correspondence works in our example with the integers.  We know that the even integers are a subgroup of the integers, and so are $3\mathbb{Z}, 4\mathbb{Z}$, etc.  In fact, these are all of the subgroups (and the trivial subgroup, which is just the element {0}).  Above, we drew two covering spaces of the circle: the real line, where each neighborhood of the circle has infinitely many homeomorphic copies hanging out in the real line, and the circle wound around itself three times, where each neighborhood has three copies.  The number of copies is called the degree of the cover, and sometimes one says the cover is an n-fold covering.  You can wind the circle around itself times for any n, which will correspond to the $n\mathbb{Z}$ subgroup.  How does this correspondence work?  Well, looking at the degree three/3-fold picture again, if you go around the covering circle once, you’ll project down to going around the base circle three times.  So if you go around the covering circle and count, you’ll get 0, 3, 6, 9… In contrast, the real line corresponds to the trivial subgroup (and is an infinite degree cover), and it’s called the universal cover of the circle.  Every space has a unique universal cover, which is a covering space with trivial fundamental group.

Now a preview of why we’ll like this.  Sometimes spaces are tricky and not fun and it’s easier to look upstairs at a cover, and then go back downstairs.  Let’s let the downstairs space be two circles pinched together at a point.

First, you should get convinced that the picture above is a cover; I colored the homeomorphic copies in order to highlight what’s happening.  Also, pretend the branching part goes on forever, a la the Cayley graph of the free group on 2 generators:

from wikipedia

So here’s an example: let’s say we have a path downstairs that goes around the green circle several times.  And maybe we don’t want this path to hit itself over and over again, so we look at a cover upstairs so it turns into a line instead.  So instead of just being immersed (locally injective), the path is embedded (injective) in the cover upstairs.

The orange scribble downstairs goes around the green loop over and over again, hitting itself. Upstairs, it’s a line and doesn’t hit itself

Next time I write about current math research, I’ll be using covering spaces a lot!  In fact, one of the main questions is this: if you have a path downstairs that hits itself (is immersed), what’s the minimum degree cover you need to ensure that the path is embedded in the cover?  This question isn’t explicitly answered yet for loops on surfaces, but the research I’ll blog about gives some bounds on the degree.

Cayley graphs, trees, Schreier’s theorem Part I (actually just groups)

6 Mar

So that’s a big agenda, but hey, dream big, right?  I’m giving a talk in our geometry, topology, and dynamics graduate seminar tomorrow on Schreier’s theorem, so I thought I’d try to write a blog post or three about it.   And then maybe blog about some cranberry cookies I made tonight after preparing my talk.

First, we’ve got to talk about the fundamental object of abstract algebra, pretty much the first thing you learn about as a math major: groups.  Abstractly, a group is a collection of objects that have some type of operation between them, where the operation satisfies a good number of properties: doing the operation to any two objects (or elements) results in another element in the group (this is called closure); there’s an identity element so that if you operate it with any other element, you get that other element back; the operation should be associative which means you can move parentheses around; and each element should have an inverse that takes you back to the identity.  Concretely, here are some groups:

• The integers, with addition as your group operation.  Adding any two integers gives you an integer (closure); adding 0 to any number gives you back, so 0 is the identity; addition is associative; and x+ (-x) = 0, so every element has an inverse.
• Symmetries of a square.  This means all the ways you can turn, flip, and change a square.  The group operation is doing one symmetry, and then another (like flipping, and then turning).  What’s interesting about this group is that it can be generated by just two different symmetries: rotating to the right by 90 degrees, and flipping vertically.   You can convince yourself of this by taking a piece of paper and numbering the corners 1, 2, 3, 4 and figuring out how many ways there are total to get the square back again.  Or you can try drawing a non-symmetric smiley face in Paint!

Symmetries of a square. Red guy is totally random

• Another is $\mathbb{Z}/ n\mathbb{Z}$, which is also written as $\mathbb{Z}_n$.  This can be read as the integers modulo n, which means every time you hit n, you jump back to 0.  It’s exactly how we count the hours up to 11, and then 12, and then we’re back at 1 again.  So our way of telling time $\mathbb{Z}_{12}$.  If I tell you that it’s 10 o’clock now, and ask you what time it’ll be in 70 hours, and you answer 8 o’clock, you just did modular arithmetic.  So you figured out that the closest multiple of 12 to 70 was 60, and then you knew we had to add 10 to our time, but since you’re not in not-the-US you knew the answer couldn’t be 20 o’clock, and you got 8 o’clock.  Good for you, you just did some group theory!

Flava Flav gets group theory

Okay, but I’m interested in geometric objects and pictures we can draw besides these little smiley face squares.  So how can I encode the data in our examples (the integers, the integers mod n, and the reflections of a square) into a geometric object?  I’ll use a diagram that computer scientists use a lot, called a graph.

Building our graph is like when we built our curve complex last time we did math.  We’ve got vertices, each of which represents a group element.  To connect two group elements with edges, we’ve got to figure out a generating set for our group.  This is a subset of the group elements, where if you operate with them over and over again, you end up with all of the group.  For instance, I can use {1,-1} as my generating set for the integers: I can add 1 to itself n times to get any positive integer n, and similarly for the negative integers.  And we used the rotation r and the vertical flip s in our smiley face picture above to generate the symmetries of a square.

So to connect two vertices, we draw an edge if we can operate by one of our generating set to get from one vertex to another.  So the picture of the integers looks like:

Here, we use little arrows for the vertices, and lines for the edges.  I very obviously didn’t make this picture.

This picture is from wikipedia!  It’s the picture for those rotations of the square.  Wikipedia used a letter ‘F’ instead of a smiley face.

So we can make these pictures for any group given some generating set (aside: generating sets aren’t unique.  For instance, you can generate the integers with $\pm 2, \pm 3$ instead of using 1, -1.  Different generating sets give you different pictures).  This picture is called the Cayley graph of the group with respect to the generating set S

A graph without any loops/cycles/ways to get from one vertex back to itself is called a tree.  Boom, we’ve got two vocabulary words from the title down!  In Part II, I’ll actually explain what Schreier’s theorem is and maybe give some idea as to its proof.