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Here's a book I've been reading lately:
1) Kenneth S. Brown, Cohomology of Groups, Graduate Texts in Mathematics 182, Springer, 1982.
I should have read this book a long time ago - but I probably wouldn't have enjoyed it as much as I do now. All sorts of things I struggled to learn for years are neatly laid out here. Best of all, he comes right out and admits from the start that the cohomology of groups is really a branch of topology, instead of hiding this fact like some people do.
This is something every mathematician should know: you can take any group and turn it into a space, thus "reducing" group theory to topology. In particular, if you have any trick for telling spaces apart, like "cohomology theory", you can apply it to groups as well.
Of course topology is harder than group theory in many ways - hence my quotes around "reducing". Indeed, algebraic topology was invented as a trick for reducing topology to group theory! But, the bridge turns out to go both ways, and there's a lot of profitable traffic in both directions.
Ultimately, as James Dolan likes to point out, it's all about the unity of mathematics. Topology is about our concept of space, while group theory is about our concept of symmetry... but the amazing fact is that they turn out to be two aspects of the same big thing! Mathematics is a source of endless surprises, but this is one of the biggest jaw-droppers of all.
The idea goes back at least to Evariste Galois, who noticed that you can classify the ways a little thing can sit in a bigger thing by keeping track of what we now call its "Galois group": the group of all symmetries of the big thing that map the little thing to itself. For example, you can pick out a point or line in the plane by keeping track of which symmetries of the plane map this point or line to itself.
However, the idea of using groups to classify how a little thing sits in a big one was really made explicit in Felix Klein's "Erlangen program", a plan for reducing geometry to group theory.
You may know Klein for his famous one-sided bottle:
A mathematician named KleinOr maybe you know that the symmetry group of a rectangle, including reflections, is called the "Klein 4-group":
Thought the Möbius strip was divine.
Said he: "If you glue
The edges of two
You'll get a weird bottle like mine!"
1 a b c
---------
1| 1 a b c
a| a 1 c b
b| b c 1 a
c| c b a 1
He is also known for some other groups called "Kleinian groups", which
act as symmetries of fractal patterns like this:
2) Jos Leys, Kleinian Pages, http://www.josleys.com/creatures42.htm
If you like cool pictures, check out this website! I've linked you to the page that most closely connects to Kleinian groups, but there are lots of other more fanciful pictures. And if you get interested in the math lurking behind these fractals, you've got to try this book:
3) David Mumford, Caroline Series, and David Wright, Indra's Pearls: The Vision of Felix Klein, Cambridge U. Press, Cambridge, 2002.
Mumford is a world-class mathematician, so this book is completely different from the superficial descriptions of fractals one often sees in math popularizations - but it's still readable, and it's packed with beautiful pictures. You can learn a lot about Kleinian groups from this!
The Kleinian groups arose from Klein's studies of complex functions, which he considered his best work. But, he was also a mathematical physicist. Among other things, he wrote a four-volume book on tops with one of the fathers of quantum mechanics, Arnold Sommerfeld:
4) Felix Klein and Arnold Sommerfeld, Über die Theorie des Kreisels, 4 vols, 1897-1910. Reprinted by Johnson, New York, 1965. Also available at http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV7354.0001.001
This came after a book he wrote on his own:
5) Felix Klein, The Mathematical Theory of the Top, Scribner's, New York, 1887.
This may seem like a lot of books about a kid's toy! But, tops are profoundly related to the rotation group, and the "exactly solvable" tops discovered by Euler, Lagrange/Poisson, and Sofia Kowalevskaya are solvable because of their symmetries - deeply hidden symmetries, in the case of the Kowalevskaya top. So, one can imagine why Klein liked this subject.
Klein also wrote a book on the icosahedron and the quintic equation:
6) Felix Klein, Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, 1888. Reprinted by Dover, New York, 2003. Also available at http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=03070001&seq=7
Galois had already noticed that the number field you get by taking the rationals and throwing in the roots of a typical quintic:
ax5 + bx4 + cx3 + dx2 + ex + f = 0has as its symmetry group all the permutations of the 5 roots. Indeed, he saw that the "unsolvability" of this group, in a technical sense, is what makes it impossible to solve the quintic by radicals. It must have been common knowledge that the symmetry group of the icosahedron is the group of all even permutations of 5 things. But, Klein took this much further! Alas, I've never really understood what he did. Perhaps if I read these and think hard, I'll understand:
7) Jerry Shurman, Geometry of the Quintic, John Wiley and Sons, New York, 1997.
Peter Doyle and Curt McMullen, Solving the quintic by iteration, Acta Math. 163 (1989), 151-180. Available at http://math.dartmouth.edu/~doyle/docs/icos/icos/icos.html
Anyway, it should be clear by now that Klein was a lover of symmetry. He was also a bit of a visionary, and his obituary by Grace Chisholm Young shows that this got him in some trouble:
One of Weierstrass' pupils, still alive, told me that at Berlin Klein was anathema: it was said that his work was not mathematics at all, but mere talk. This criticism shows a want of appreciation of his rare type of mind. It teemed with ideas and brilliant reflections, but it is true that his work lacks the stern aspects required by mathematical exactitude. It was in personal contact that this was corrected, at least in so far as his students were concerned. His favourite maxim was, "Never be dull".In a talk he wrote in 1872 when he was made professor at Erlangen University - a talk he didn't actually give! - Klein outlined what is now called his "Erlangen program". The idea here is that different kinds of geometry correspond to different symmetry groups. Taken to the extreme, this philosophy says that a geometry is just a group! In a given geometry, a "figure" of any kind - like a point or line - can be detected by the subgroup of symmetries that map that figure to itself. So, a figure is just a subgroup!
This program eventually led to a grand theory of groups and geometry based on "flag manifolds", which I tried to sketch in "week178", "week180", and "week181".
It's important to realize how similar the Erlangen program is to Galois theory. Galois had also used group theory to classify how a little thing can sit in a bigger thing, but in situations where the "things" in question are commutative algebras - for example, the rational numbers with some roots of polynomials thrown in.
Now, commutative algebra is like topology, only backwards. Any space has a commutative algebra consisting of functions on it, and if we're very clever we can think of any commutative algebra as functions on some space - though this was only achieved long after Galois, by Alexander Grothendieck.
What do I mean by "backwards"? Well, suppose you have a "covering space" - a big space sitting over a little one, like a spiral sitting over the circle. In this situation, any function on the little space downstairs defines a function on the big one upstairs. So, the algebra of functions on the little space sits inside the algebra of functions on the big space.
Notice how it's backwards. Classifying how a little commutative algebra can sit in a big one amounts to classifying how a big space can cover a little one! For more details on this analogy, try "week198", "week201" and especially "week205".
I should warn you: the Galois group has a different name when we apply it to the classification of covering spaces - we call it the group of "deck transformations". The idea is pretty simple. Suppose Y is a covering space of X, like this:
----------------
---------------- Y
↓ p
---------------- X
We've got a function p: Y → X, and sitting over each point of X are
the same number of points of Y, living on different "sheets" that look
locally just like X. You should imagine the sheets being able to
twist around from place to place, like the edges of a Moebius strip.
Anyway, a "deck transformation" is just a way of mapping Y to itself that permutes the different points sitting over each point of X.
The theory of this was worked out by Riemann, Poincare, and others. Poincare showed you could use this idea to turn any connected space X into a group - its "fundamental group". There are different ways to define this, but one is to form the most complicated possible covering space of X that's still connected - its "universal cover". Then, take the group of deck transformations of this! Following Galois' philosophy, all the other connected covering spaces of X correspond to subgroups of this group.
The theory of the fundamental group was just the beginning when it came to groups and topology. One of many later big steps, back in the late 1940s, was due to Sammy Eilenberg and Saunders Mac Lane. They saw how to reverse the "fundamental group" idea and turn any group back into a space!
More precisely: for any group G, there's a space whose fundamental group is G and whose higher homotopy groups vanish. It's sometimes called the "Eilenberg - Mac Lane space" and denoted K(G,1), but sometimes it's called the "classifying space" and denoted BG. It's pretty easy to build; I described how back in "week70".
You start with a point:
o
Then you stick on an edge looping from this point to itself for
each element a in G. Unrolled, it looks like this:
o--a->--o
where a is an element of our group. Then, whenever we have ab = c in
our group, we stick on a triangle like this:
o
/ \
a b
/ \
o---c---o
Then, whenever we have abc = d in our group, we stick on a tetrahedron
like this:
o
/|\
/ | \
/ b \
a | bc
/ _o_ \
/ / \_ \
/ _ab c_ \
/_/ \_\
o--------d--------o
And so on, forever! For each list of n group elements, we get an
n-dimensional simplex in our Eilenberg-Mac Lane space. The resulting
space knows everything about the group we started with. In particular,
the fundamental group of this space will be the group we started with!
Using this idea, we can do some fiendish things. For example, for each n we can form a set Cn(G,A) consisting of all functions that eat n-dimensional simplices in the Eilenberg-Mac Lane space of G and spit out elements of some abelian group A. There are maps
d: Cn(G,A) → Cn+1(G,A)
reflecting the fact that each (n+1)-simplex has a bunch of n-simplices as its faces. Since the boundary of a boundary is zero,d2 = 0
Guys who live in the kernel of
d: Cn(G,A) → Cn+1(G,A)
are called "n-cocycles", and guys who live in the image of
d: Cn-1(G,A) → Cn(G,A)
are called "n-coboundaries". Since d2 = 0, every coboundary is a cocycle, but not always vice versa. So, we can form the group of cocycles mod coboundaries. This is called the "nth cohomology group" of G with coefficients in A, and it's denoted
Hn(G,A)
This sounds unmotivated at first, but the nth cohomology group of a space is really just a clever way of keeping track of n-dimensional holes in that space. So, what we're doing here is cleverly defining a way to study "holes" in a group! There are deeper, more conceptual ways of understanding group cohomology, but this is not bad for starters.
For example, let's take the simplest group that's not utterly dull - the integers mod 2, or Z/2. Here we get
K(Z/2,1) = RP∞
where RP∞ is the space formed by taking an infinite-dimensional sphere and identifying opposite points. This space has holes of arbitrarily high dimension, so the cohomology groups of Z/2 go on being nontrivial for arbitrarily high n. I sketched a "picture proof" here:
8) John Baez, Fall 2004 Quantum Gravity Seminar, week 10, notes by Derek Wise, http://math.ucr.edu/home/baez/qg-fall2004/
and I showed that, for example
Z if n = 0
Hn(Z/2,Z) = 0 if n is odd
Z/2 if n is even and > 0
I also explained how this stuff is related to topological quantum field
theory.
Anyway, all this is just the very superficial beginnings of the subject of group cohomology. Read Brown's book to dig deeper!
Personally, what I find most exciting about this book now are the remarks on the "Euler characteristic" of a group. Let me explain this... though now I'll have to pull out the stops and assume you know some group cohomology.
We can try to define the "Euler characteristic" of a group G to be the Euler characteristic of K(G,1). This is the alternating sum of the dimensions of the rational cohomology groups
Hn(G,Q)Of course, this alternating sum only converges if the cohomology groups vanish for big enough n. Also, they all need to be finite-dimensional.
Unfortunately, not many groups have well-defined Euler characteristic with this naive definition!
For example, people have studied groups G whose nth cohomology vanishes for n > d, regardless of the coefficients. If we take the smallest d for which this holds, such a group G is said to have "cohomological dimension" d. Eilenberg and Ganea showed that for d ≥ 3, a group has cohomological dimension d whenever we can build K(G,1) as a simplicial complex (or CW complex) with no cells of dimension more than d.
This is a nice geometrical interpretation of the cohomological dimension. But, one can show that groups with torsion never have finite cohomological dimension! We've seen an example already: Z/2, whose Eilenberg-Mac Lane space is infinite-dimensional.
However, it turns out that there's a generalization of the Euler characteristic that makes sense for any group G that has a torsion-free subgroup H whose Euler characteristic is well-defined in the naive way, as long as H has finite index in G. We just define the Euler characteristic of G to be the Euler characteristic of H divided by the index of H in G. The answer doesn't depend on the choice of H!
Take my favorite example, SL(2,Z). This has torsion, so its cohomological dimension is infinite and its naive Euler characteristic is undefined! Indeed, I wrote a whole issue of This Week's Finds about some elements of orders 4 and 6 sitting inside SL(2,Z), related to the symmetries of square and hexagonal lattices - see "week125".
But, SL(2,Z) has a torsion-free subgroup of index 12, namely its commutator subgroup - the group you need to quotient by to make SL(2,Z) be abelian. This subgroup has finite cohomological dimension and its Euler characteristic is -1. I'm not sure why this is true, but Brown says so! This means the Euler characteristic of SL(2,Z) works out to be -1/12.
If you've read my stuff about Euler characteristics in "week147", you'll see why this gets me so excited - I can add this stuff to my list of weird ways of calculating the Euler characteristic. Plus, it's related to the magical role of the number "24" in string theory, and also the Riemann zeta function!
Indeed, the Riemann zeta function gives a way to make rigorous Euler's zany observation that
1 + 2 + 3 + .... = -1/12,as I explained here:
9) John Baez, Euler's Proof that 1+2+3+ ... = -1/12, Bernoulli Numbers and the Riemann Zeta Function, Winter 2004 Quantum Gravity Seminar, homework for weeks 5,6,7, available at http://math.ucr.edu/home/baez/qg-winter2004/
This suggests that there should be a version of the Eilenberg-Mac Lane space for SL(2,Z) which has 1 cell of dimension 0, 2 cells of dimension 2, 3 cells of dimension 4, and so on. Does anyone know if this is true?
More generally, G. Harder computed the (generalized) Euler characteristic for a large class of arithmetic groups:
10) G. Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. Ecole Norm. Sup. 4 (1971), 409-455.
For example, he looked at the symplectic group defined over the integers, Sp(n,Z), and showed that its Euler characteristic is this product of values of the Riemann zeta function:
ζ(-1) ζ(-3) ⋅⋅⋅ ζ(1-2n)
In the case n = 1 we get back SL(2,Z) and ζ(-1) = -1/12.
In fact, every Chevalley group over the integers has a well-defined Euler characteristic, and Harder was able to compute it in terms of Bernoulli numbers. A Chevalley group is sort of like a simple Lie group, but defined algebraically. Sp(n,Z) is one example. SL(n,Z) is another, but it's Euler characteristic turns out to vanish for n > 2, so it's not too interesting.
Harder worked them all out. For example, he showed the Euler characteristic of the integral form of the exceptional group E7 is some wacky number like
- 691 × 43867 / 221 × 39 × 52 × 73 × 11 × 13 × 19
Serre went even further, computing Euler characteristics of Chevalley groups defined over algebraic number fields. He also noticed that when you write the Euler characteristic of a group as a fraction, the primes in the denominator are precisely the primes p for which the group has p-torsion. He was thus able to conclude, for example, that E7 defined over the integers has p-torsion for p = 2, 3, 5, 7, 11, 13 and 19.
For more, see:
11) Jean-Pierre Serre, Cohomologie des groups discretes, Ann. Math. Studies 70 (1971), 77-169.
This only takes us up to 1971. I shudder to think what bizarre results along these lines are known by now! Probably they'd seem not bizarre but beautiful if I understood this stuff better: I don't really have a clue how the Riemann zeta function gets into this game, so everything after that seems like black magic to me - bewitching but bewildering.
But, it's clear that the study of groups and symmetry has not lost its ability to turn up surprises.
There is a group called the Klein group. It is denoted V4 = The Vierer-Gruppe (The fours group).This group of order 168 has made an appearance here before, in "week194": it's PSL(3,Z/2) - the group of symmetries of the projective plane over Z/2, or "Fano plane", whose points can also be thought of as imaginary unit octonions. It's also PSL(2,Z/7). I've long been mystified by its relation to Klein's quartic, mainly because I've never spent time trying to understand it! - it's just one of those things that's been gnawing at the edges of my consciousness, especially ever since I saw this book come out:Klein worked with the simple group of order 168 and found the "Klein quadric" which has it for symmetry group.
The suggestion is that friends decided to call the non-cyclic abelian group of order 4 the "Klein group" = the "little group" as a joke.
I have a question you may like to posit to your readers:
Is V4 the abstract group or a permutation group?
There are other points ... I presume you know that your -1/12 is ζ(-1). There is a paper by Lepowsky on the occurrence of such ζ(-n) involving vertex algebras.
I dearly wish I understood cohomology!
I am busy tethering moonshine!
Best, John
12) Silvio Levy, The Eightfold Way: the Beauty of Klein's Quartic Curve, MSRI Research Publications 35, Cambridge U. Press, Cambridge 1999. Available at http://www.msri.org/publications/books/Book35/
It has a translation of Klein's original paper on this subject. Someday I'll break down and study this.
Anyway....
James Dolan mentioned some other folklore saying that the "Kleinian groups" were also named after Klein as a joke:
by the way, i enjoyed the latest twf a lot (although i don't know why we seem to never get a chance to talk about all this stuff ourselves that much), but i noticed that you (apparently non-ironically) mentioned kleinian groups as a famous thing named after klein, without telling the story that i always hear about how poincare gave kleinian groups the name "kleinian groups" after klein complained to poincare about poincare's use of the terminology "fuchsian groups" for something that fuchs apparently didn't event.Noam Elkies suggested that the commutator subgroup of SL(2,Z) has Euler characteristic -1 because it's a a free group on 2 generators, so its classifying space is a figure 8, with Euler characteristic 1 - 2 = -1 since it has one vertex and two edges.i guess that the versions of the story that i'd heard seemed to suggest that klein was complaining because he thought that fuchs hadn't significantly contributed to the study of fuchsian groups, and that poincare may have been naively trying to placate klein and/or not-so-naively twitting him by then giving the name "kleinian groups" to something that klein hadn't significantly contributed to the study of.
however i did just look for the story on the web, and the tellings that i found there i guess don't really suggest that klein didn't "significantly contribute to the study of" kleinian groups (or at least not by my standards). it's still not clear though what sort of reaction poincare may have been trying to provoke in klein, and whether he succeeded in provoking it. it's claimed that poincare did come up with the name "kleinian function" later in the same day after klein complained about the name "fuchsian function", and also that klein was subsequently just as vociferous in complaining to poincare about the name "kleinian function" as he was in complaining about the name "fuchsian function". but apparently klein's complaints were based on _very_ exacting concerns about absolute priority, so that the names "fuchsian function" and "kleinian function" can be seen as inappropriate only by the standards of someone with similarly ridiculous concerns about absolute priority, rather than by a reasonable person such as myself.
i'd also heard that klein's nervous breakdown was provoked by the stress of trying to keep up with a genius like poincare, but maybe it was actually provoked by poincare's apparently casual attitude towards priority disputes and/or concept-naming.
i'd thought of asking you about this issue of whether klein really did have much to do with kleinian groups right after i read the advance copy of twf that you sent me, but i guess that i didn't notice that it was an advance copy. i guess that it doesn't matter though, since apparently there _is_ a case to be made that klein had lots to do with developing the theory of kleinian groups; just not by his own apparently ridiculous standards.
This sounds right. In particular, I already mentioned how Brown claims the commutator subgroup of SL(2,Z) is torsion-free. Further, Kevin Buzzard shows below that any torsion-free subgroup of SL(2,Z) is a free group. So, we just need to check that the commutator subgroup of SL(2,Z) can be generated by two elements but not by just one.
Laurent Bartholdi just made this job easier; he sent me an email saying these are free generators for the commutator subgroup of SL(2,Z):
( 2 -1) (-1 1)and
( 1 -1) (-1 2)In fact, Kevin Buzzard's email was packed with wisdom. He wrote:
I know one elementary argument which you don't appear to, so I thought I'd fill you in. The argument below is waffly but rather easy really.I replied:John Baez wrote:
> But, SL(2,Z) has a torsion-free subgroup of index 12, namely its > commutator subgroup - the group you need to quotient by to make SL(2,Z) > be abelian. This subgroup has finite cohomological dimension and its > Euler characteristic is -1. I'm not sure why this is true, but Brown says > so! This means the Euler characteristic of SL(2,Z) works out to be -1/12.One doesn't have to use such a "strange" subgroup as the commutator subgroup of SL(2,Z). People who do modular forms, like me, far prefer "congruence subgroups", as these are the ones that show up when you study automorphic forms for SL(2,Z). So here's an easy way to compute the Euler characteristic of SL(2,Z): take your favourite congruence subgroup which has no torsion, work out its Euler characteristic (this is easy, I'll show you how to do it in a second) and then deduce what the Euler characteristic of SL(2,Z) is.Here are some examples of congruence subgroups: for any integer N≥1, consider the subgroup Γ1(N) of SL(2,Z), defined as the matrices
(a b) (c d)in SL(2,Z) such that c=0 mod N and a=d=1 mod N. It's just the pre-image in SL(2,Z) of the upper triangular unipotent matrices in SL(2,Z/NZ) so it's a subgroup of SL(2,Z). Here's a neat fact that makes life easy:Lemma: if N≥5 then Γ1(N) has no torsion.
Proof: say g in SL(2,Z) has finite order d≥1. Then its min poly divides Xd-1 so over the complexes it has distinct linear factors so it's diagonalisable with roots of unity z and w on the diagonal. Now |z|=|w|=1 so |trace(g)| ≤ 2. But it's an integer, so it's -2,-1,0,1,2. And for N>=5 the only one of these congruent to 2 mod N is 2. So z=w=1 and so g is the identity.
Deeper, but also completely standard (and not logically necessary for what follows)---any torsion-free subgroup of SL(2,Z) is free! This is because SL(2,Z) acts very naturally on a certain tree in the upper half plane. This is a neat piece of mathematics. SL(2,Z) acts on the upper half plane {z=x+iy:y>0} via the rule:
(a b) (c d)sends z to (az+b)/(cz+d). Now draw dots at the points i=√-1 and ρ=exp(2πi/6), the primitive 6th root of unity in the upper half plane, and draw the obvious arc between them (the one that lies on the circle |z|=1), this is our first edge, and now look at the image of what you have under the SL(2,Z) action. It's a rather pretty tree, with two kinds of vertices---those in the i orbit have valency 2 and stabiliser of order 4, and those in the ρ orbit have valency 3 and stabiliser of order 6. Now a group is free iff it acts freely on a tree, and anything torsion-free in SL(2,Z) must be acting freely on this tree because the stabiliser of each vertex and edge under the SL(2,Z) action is finite.So Γ1(5) is, by this general theorem, free. In fact I don't really need this general nonsense, one can give a hands-on proof of this fact, which I'll do now. We've seen that SL(2,Z), and hence Γ1(5), acts on the upper half plane. There is no torsion in Γ1(5) so the action is very nice, one checks easily that the action is free in fact by a similar sort of argument to the lemma above, it's the sort of thing you can find in the first few pages of any book on modular forms. So we can quotient out the upper half plane by Γ1(5) and get a quotient Riemann surface. The point is that this computation is very manageable and can be done "in practice". There is a standard argument which shows how to quotient out the upper half plane by SL(2,Z)---the answer is a Riemann surface isomorphic to the complex plane (although you have to take care at the points where the action isn't free---this is exactly the vertices of the tree above), and the isomorphism can even be given "explicitly" via the j-function coming from the theory of elliptic curves---there is a standard fundamental domain even, the one with corners ρ, ρ2 and +i∞. I'm sure you'll have come across this sort of thing many times before. Now SL(2,Z) surjects onto SL(2,Z/5Z) so the index of Γ1(5) in SL(2,Z) is just the index of
(1 *) (0 1)in SL(2,Z/5Z) and by counting orders this comes out to be 24. Now it's not hard to find explicitly 24 translates of the standard fundamental domain and then glue them together to work out the quotient of the upper half plane by Γ1(5)---it turns out that it is isomorphic to the Riemann Sphere minus 4 points.In fact there is no need to do this sort of computation---the modular forms people have automated it long ago. The quotient of the upper half plane by Γ1(N) is a Riemann surface called Y1(N) and I can just ask my computer to compute the genus of its natural compactification (this exists and is called X1(N)) and also to compute how many cusps were added to compactify it. So in practice you just have to find a friendly modular forms person and then say "hey, what's the genus of X1(5) and how many cusps does it have?" and then you have a complete description of Γ1(5) because it's π1 of the answer.
OK, the upper half plane modulo Γ1(5) is the sphere minus 4 points, so Γ1(5) is π1 of this, i.e. it's free on three generators. That makes the Euler characteristic of Γ1(5) equal to 1-3=-2. And we already checked that the index was 24, so the Euler Characteristic of SL(2,Z) works out to be -1/12. Grothendieck wouldn't have chosen Γ1(5); he would have chosen something called Γ(2), the subgroup of SL(2,Z) consisting of the matrices which are the identity mod 2. There is another classical modular function λ inducing an isomorphism of Y(2), the quotient of the upper half plane by Γ(2), with the sphere minus three points---this is what gives the one-line proof of the fact that any analytic function C→C that misses two points must be constant, because it then lifts to a function from C to the upper half plane which is the same as the unit disc, so we're done by Liouville. There is a subtlety here though:
(-1 0) (0 -1)is in Γ(2). So you have to work with PSL(2,Z)=SL(2,Z)/{±1} instead. Let PΓ(2) denote the image of Γ(2) in PSL(2,Z). Note that -1 is kind of a pain in the theory of modular forms sometimes because it acts trivially on everything but isn't the identity. Grothendieck was very interested in the sphere minus three points but it's much older than this that PΓ(2) is its fundamental group, so PΓ(2) has Euler characteristic 2-3=-1 and index 6 in PSL(2,Z), so PSL(2,Z) has Euler characteristic -1/6, so SL(2,Z) has Euler characteristic -1/12 because that's how they work. :-)John Baez wrote:
> This only takes us up to 1971. I shudder to think what bizarre > results along these lines are known by now! Probably they'd seem not > bizarre but beautiful if I understood this stuff better: I don't really > have a clue how the Riemann zeta function gets into this game, so > everything after that seems like black magic to me - bewitching but > bewildering.Nowadays almost any analytic function that is involved in number theory, when evaluated at certain "natural" points, gives an answer which has a conjectural interpretation in terms of relations between cohomology theories---this is the subject of many conjectures (Deligne, Beilinson, Bloch-Kato,...). It is still absolutely black magic! Actually I'm being unfair, the relation between special values of ζ and Euler characteristics is somehow less profound than this stuff. I wish I knew more about it! It can actually be used to compute certain values of L-functions (things more general than the zeta function but along the same lines)...Kevin
Hi -Kevin replied:Thanks VERY much for this email. I was actually wondering why Brown used the commutator subgroup of SL(2,Z) as a kind of "warmup" for computing the Euler characteristic of SL(2,Z) instead of one of the congruence subgroups. It seems this subgroup is not any of the beloved congruence subgroups....
In fact, I've finally managed to turn up the thing I was looking for. How does this relate to the stuff you're saying? It involves Γ(3) rather than the Γ1(N) groups:
In "week 97", I wrote:
Where does the extra 24 come from? I don't know, but Stephan Stolz said it has something to do with the fact that while PSL(2,Z) doesn't act freely on the upper half-plane - hence these elliptic curves with extra symmetries - the subgroup "Γ(3)" does. This subgroup consists of integer matrices (a b) (c d) with determinant 1 such that each entry is congruent to the corresponding entry of (1 0) (0 1) modulo 3. So, if we form H/Γ(3) we get a nice space without any "points of greater symmetry". To get the moduli space of elliptic curves from this, we just need to mod out by the group SL(2,Z)/Γ(3) = SL(2,Z/3) But this group has 24 elements! In fact, I think this is just another way of explaining the period-24 pattern in the theory of modular forms, but I like it.Kevin wrote:>It's a rather pretty tree,Yes, there's a picture of it in Brown's book, drawn on top of an old picture by Klein of a triangulation of the hyperbolic plane.What Brown seems to be doing there is showing that this tree is a deformation retract of that triangulation (with its simplicial topology, where the points on the boundary of the hyperbolic plane form a discrete set), and thus proving that the cohomological dimension of SL(2,Z) is just 1.
Anyway, this is all great stuff. Do you mind if I attach a copy of your email to the copy of "week213" on my website? I think people will find it helpful, especially because of its friendly straight-to-the-point style, which books rarely seem to manage....
Best,
jb
John Baez wrote:
> I was actually wondering why > Brown used the commutator subgroup of SL(2,Z) as a kind of "warmup" > for computing the Euler characteristic of SL(2,Z) instead of one of > the congruence subgroups. It seems this subgroup is not any of the > beloved congruence subgroups....You're right, I don't think it is. For N≥1 define Γ(N) to be the kernel of the obvious map SL(2,Z)→SL(2,Z/NZ); a congruence subgroup is any subgroup of SL(2,Z) that contains one of these Γ(N)'s. Clearly such things have finite index in SL(2,Z). But unfortunately there exist subgroups of finite index in SL(2,Z) that are not congruence subgroups. This is a "low-dimensional" phenomenon---the moment you have a bit more freedom, e.g. you're working with SL(3,Z) or indeed SL(n,Z) for any n≥3, or even SL(2,Z[1/p]) for some prime p, then any subgroup of finite index is a congruence subgroup---these groups satisfy the "congruence subgroup property". But I've never understood the commutator of SL(2,Z) precisely for the reason that it's not a congruence subgroup (this is essentially because the commutator subgroup of SL(2,Z/NZ) never has index 12 in SL(2,Z/NZ)! The index is always smaller than 12 because SL(2,Z/pZ) is essentially a simple group.)John Baez wrote:
> In fact, I've finally managed to turn up the thing I was looking > for. How does this relate to the stuff you're saying? It involves > Γ(3) rather than the Γ1(N) groups.Anything will do. If you know about Γ(3) then great. The same key observation is true---Γ(3) contains no elements of finite order, because any finite order element(a b) (c d)of Γ(3) which isn't the identity must have trace in {-2,-1,0,1} congruent to 2 mod 3, so the trace must be -1, so d=-1-a, so the det is a(-1-a) mod 9, which is never 1 mod 9. Now the index of Γ(3) in SL(2,Z) is 24, and the modular curve X(3) has genus 0 (everyone knows this because Wiles needed it to prove Fermat's Last Theorem!) and four cusps (zero, 1, 1/2 and ∞) and hence the Euler Characteristic of Γ(3) is 2-4=-2, so we recover the result that the Euler Characteristic of SL(2,Z) is -1/12 again.John Baez wrote:
> Where does the extra 24 come from? I don't know, but Stephan Stolz > said it has something to do with the fact that while PSL(2,Z) doesn't > act freely on the upper half-plane - hence these elliptic curves with > extra symmetries - the subgroup "Γ(3)" does.One can see that any subgroup of SL(2,Z) which has finite index and is free, must have index a multiple of 12 (and hence at least 12). Because if it has index d and is free on g generators, when we know (1-g)/d=-1/12, so 12 divides the denominator of (1-g)/d in lowest terms. Geometrically what is going on is that perhaps the "correct" quotient of the upper half plane by SL(2,Z) is not just the complex numbers, it's something that looks a bit like the complex numbers except there is a little bit of extra magic going on at i and ρ, corresponding to the fact that one shouldn't really have attempted to quotient out there, one should just remember that really the quotient is kind of "crumpled up" near there. So for example the fundamental group of the quotient shouldn't be the trivial group---if you take a small loop around i then this should not be regarded as contractible---you have to go around i twice before you can hope to contract the loop. Similarly you have to go around ρ three times. Even worse---if you do this carefully enough then even going around i twice or ρ three times isn't enough to contract the loop---because the resulting loop somehow corresponds to the element -1 in SL(2,Z), which acts trivially but which isn't the identity! So you have to do everything again before you get to the element 1. Mumford thought hard about how to make all this sort of thing rigorous, and managed in the late 60s to prove that the Picard group of the quotient of the upper half plane by SL(2,Z) was in fact Z/12Z.John Baez wrote:
> Anyway, this is all great stuff. Do you mind if I attach a copy of > your email to the copy of "week213" on my website?Go ahead!Kevin
In point of fact, it has traditionally been the "continuous" aspect of things which has been the central focus of Geometry, while those properties associated with "discreteness", notably computational and combinatorial properties, have been passed over in silence or treated as an afterthought. It was therefore all the more astonishing to me when I made the discovery, about a dozen years ago, of the combinatorial theory of the icosahedron, even though this theory is barely scratched (and probably not even understood) in the classic treatise of Felix Klein on the icosahedron. I see in this another significant indicator of this indifference (of over 2000 years) of geometers vis-a-vis those discrete structures which present themselves naturally in Geometry: observe that the concept of the group (notably of symmetries) appeared only in the last century (introduced by Evariste Galois), in a context that was considered to have nothing to do with Geometry. Even in our own time it is true that there are lots of algebraists who still haven't understood that Galois theory is primarily, in essence, a geometrical vision, which was able to renew our understanding of so-called "arithmetical" phenomena. - Alexander Grothendieck
© 2005 John Baez
baez@math.removethis.ucr.andthis.edu
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