The finite subgroups of SL(2,C)

Books can be written about the finite subgroups of $\mathrm{SL}(2,\mathbb C)$ (and their immediate family, like the polyhedral groups...) I am about to start writing notes for a short course about them and I would like to include references to as much useful and interesting information about them as possible. Since they show up in quite different contexts, and can be looked upon from many different points of view, I am sure the very varied MO audience knows lots of things about them I don't.

So, despite this being more or less canonically too broad/vague a question for MO according to the FAQ:

Can you tell me (or at least point me to) all about the finite subgroups of $\mathrm{SL}(2,\mathbb C)$?

LATER: Thanks to everyone who answered. So far, the information is essentially of algebraic and geometric nature. I wonder now about combinatorics and such beasts.

For example, it is a theorem of Whitney (or maybe it just follows easily from a theorem of Whitney) that a 3-connected simple planar graph with $e$ edges has an automorphism group of order at most $4e$, and that the order is $4e$ precisely when the graph comes from a polyhedron, so that the group is a polyhedral group.

Do you know of similar results?

• I happen to like the treatment in Maskit's "Kleinian Groups". – Steve D Feb 22 '10 at 4:29
• @Steve (and everyone else): please answer with an answer, so I can vote you up! – Mariano Suárez-Álvarez Feb 22 '10 at 4:30
• (That is why I did not make this a community wiki, by the way) – Mariano Suárez-Álvarez Feb 22 '10 at 4:40

This question is heavily related to one of my favorite relations between geometry and representation theory. Consider simple Lie algebras of the following types:

• $A_n$
• $D_n$
• $E_6$
• $E_7$
• $E_8$

Then these Dynkin diagrams correspond to all possible finite subgroups of $SL_2.$ The relation is given by special classes of isolated surface singularities known as Kleinian or Du Val singularities. These arise as follows. Given a finite subgroup $G \subset SL_2,$ we have an action of $G$ on $\mathbb{C}^2$ with no fixed points other than the origin. If we then look at the geometric quotient $\mathbb{C}^2/G$ corresponding to $G$-invariant polynomials in $\mathbb{C}[x,y],$ it is generated by three homogeneous polynomials $f_1, f_2, f_3$ which are related by a weighted homogeneous polynomial $g$ of degree 3 such that $g(f_1, f_2, f_3) = 0.$ We can then identify $\mathbb{C}^2/G$ with the hypersurface $\{ g = 0 \} \subset \mathbb{C}^3.$

The resulting hypersurfaces have the following equations (with corresponding subgroup):

• $A_n: x^{n+1} + y^2 + z^2$ (cyclic)
• $D_n: x^{n-1} + xy^2 + z^2$ (dihedral)
• $E_6: x^4 + y^3 + z^2$ (tetrahedral)
• $E_7: x^3y + y^3 + z^2$ (octahedral)
• $E_8: x^5 + y^3 + z^2$ (icosahedral)

The Dynkin diagram enter as follows. Each of these surfaces can be resolved through a finite number of blow-ups, and the exceptional fiber in the resolution will consist of a copy of $\mathbb{P}^1$ for each node of the Dynkin diagram, each of which is joined to the point of another $\mathbb{P}^1$ if there is a corresponding edge in the Dynkin diagram connecting the two nodes (so in the cyclic case, it's just a chain of $\mathbb{P}^1$'s).

Lastly, there is a neat connection between this and Springer theory that goes as follows. Let $\mathcal{N}$ denote the nilpotent cone of a Lie algebra of one of the types listed above, and let $\mathcal{O}$ denote the subregular orbit. Then $\mathcal{O}$ has codimension two in $\mathcal{N}$ and hence the corresponding Kostant/Slodowy slice is a surface in $\mathcal{N}.$ It then turns out that this surface is one of the surface singularities listed above, and that the corresponding Springer fiber of a subregular element is isomorphic to the exceptional fiber in the resolution of the surface mentioned above. So the Springer resolution encodes the information of the successive blow-ups of these surfaces.

A few good references:

Milnor, Singular Points of Complex Hypersurfaces

Dimca, Singularities and Topology of Hypersurfaces

Slodowy, Simple Singularities and Simple Algebraic Groups

• That really is an amazing relation. Thanks for the enlightenment. – Q.Q.J. Feb 22 '10 at 12:58
• Mike, you say that the generators of the ring of invariants are related by a homogeneous polynomial of degree 3, yet the explicit equations you give for the different cases are not homogeneous nor degree 3. Can you clarify? Thanks. – Alberto García-Raboso Feb 22 '10 at 14:11
• Alberto, thanks for the correction, I don't know why I said a homogeneous polynomial. The correct statement is that the polynomials $p_1, p_2, p_3$ are homogeneous of different degrees and that $g$ is simply weighted homogeneous. I'll correct it now. – Mike Skirvin Feb 22 '10 at 14:56
• Sorry, that should say $f_1, f_2, f_3$ in my comment above instead of $p_1, p_2, p_3.$ – Mike Skirvin Feb 22 '10 at 14:57

It seems only QQJ alluded to this, but it's worth remembering that any finite subgroup $G$ of $SL(2,C)$ can be made to preserve an Hermitian inner product on $C^2$ by averaging, hence is also a finite subgroup of $SU(2)$, which then makes it double cover a finite subgroup of rotations of $R^3$ via $SU(2)\to SO(3)$. Thus you get the "binary" versions of the finite subgroups of $SO(3)$, (eg the binary icosahedral group, binary tetrahedral group, the other platonic groups, binary dihedral groups,...) and since $SU(2)$ is the 3-sphere the translation action exhibits these as fundamental groups of 3-manifolds, i.e. $S^3/G$, universally covered by $S^3$. These 3 manifolds are the links of the singularities described in Mike Skirvin's answer, and the corresponding Dynkin diagrams give plumbing diagrams=Kirby diagrams for the smooth 4-manifolds you get by resolving the singularities with boundary these 3-manifolds.

Dolgachev has a note on the McKay correspondence in dimension $2$. It has a lot of cool stuff on subgroups of $SL(2,\mathbb C)$, mostly from the algebraic geometry point of view.

• That's a really great reference, thanks for posting it. – Mike Skirvin Feb 22 '10 at 5:44
• I agree - it really is a great reference – David Lehavi Feb 22 '10 at 6:09

I like Thurston's treatment in his book. The idea is that any finite subgroup $G< SU(2) \to SO(3)$ gives rise to an orbifold $S^2/G$. First, one classifies the possible quotient orbifolds, then one figures out the possible preimage subgroups in $SU(2)$. Exercise 4.4.6 gives a direct argument (at least for $SO(3)$). The lengthier, but more conceptual argument using orbifolds does not appear in the published book, but is in section 5.5 of a preliminary draft (presumably this would be part of the material to appear in volume 2), and also appears in Theorem 13.3.6 of Thurston's notes. Classifying spherical and euclidean 2-dimensional orbifolds is a satisfying exercise, that may be undertaken by undergraduates with very little mathematical background: see the notes from the course "Geometry and the Imagination".

The McKay correspondence mentioned in Hailong's and Mike's answers extends to maximal Cohen-Macaulay modules over the invariant rings $R=k[x,y]^G$, where $G$ is a finite subgroup of $GL(2,k)$ (with $|G|$ invertible in $k$). In particular (Herzog) all such subrings have only finitely many non-isomorphic indecomposable MCM modules, that is, they have finite CM type. The converse is true in characteristic zero -- that a two-dimensional complete normal domain over $\mathbb{C}$ of finite CM type is a ring of invariants -- by a result of Auslander.

The case $G \subset SL(2,k)$ corresponds to $R$ being Gorenstein, and in particular a hypersurface -- namely the ADE hypersurfaces listed in Mike's answer. The correspondence between the irreducible representations of $G$ and the components of the exceptional fiber extends to include the irreducible MCM modules, and the McKay quiver (aka Dynkin diagram) is the same as the stable Auslander-Reiten quiver. The modules are directly connected to the irreps by Auslander, and directly connected to the components of the fiber by Gonzales-Sprinberg--Verdier and Artin--Verdier, extended to the non-Gorenstein case by Esnault and Wunram.

Most of this is in the current draft of my notes with Roger Wiegand on MCM modules, chapters 4, 5, and 6. (Ignore the geometry in Chapter 5 -- it's riddled with errors and I'm currently rewriting it. Or you're welcome to point out errors I might not have noticed yet.) The question of what happens to the Auslander-Reiten-McKay correspondence for $G \not\subset SL(2)$ is addressed in some recent papers of Iyama and Wemyss. (You only get some of the indecomposable MCMs, the so-called special ones.)

Don't forget Euclid book 13.

Here are a few references for arithmetic Kleinian groups. One good reference is Chapter 12 of The Arithmetic of Hyperbolic 3-Manifolds (GTM 219) by Maclachlan and Reid, which is partially based on Chinburg and Friedman, The finite subgroups of maximal arithmetic Kleinian groups, Ann. Inst. Fourier (Grenoble) 50 no. 6 (2000), 1765--1798. Also, there's Vignéras, Arithmétique des Algébres de Quaternions, Lecture Notes in Math. 800. According to the notes at the end of Ch. 12 of Maclachlan--Reid, there's also a paper by V. Schneider in Math. Z. from `77.

If you are interested in the compact real form then "On Non-Linear Realizations of the group $SU(2)$" by Mickelsson and Niederle lists the conjugacy classes of closed proper subgroups of SU(2) as a recap before going on the nonlinear cases. They are

i) The unitary subgroup $U(1)$

ii) The subgroup $N[U(1)]$ (normalizer of $U(1)$)

iii) $C_n$, the cyclic subgroups of order $n$

iv) The subgroups $\tilde{D_{2n}}$ where $\tilde{D_{2n}}/Z_2$ is isomorphic to the dihedral group $D_n$ of order $2n$.

v) The subgroup $\tilde{T}$, where $\tilde{T}/Z_2$ is isomorphic to the tetrahedral group T of order 12.

vi) The subgroup $\tilde{O}$, where $\tilde{O}/Z_2$ is isomorphic to the octahedral group O of order 24.

vii) The subgroup $\tilde{Y}$, where $\tilde{Y}/Z_2$ is isomorphic to the icosahedral group Y of order 60.

They attribute this result to a 'method of Murnaghan' whose book is "The Theory of group representations" and from memory it is in the back as an appendix.

They go on to say which of these lead to homogeneous spaces that are 3-manifolds. An interesting read and possibly of some relevance to your notes.

Check out Curtis' "Construction of a family of Moufang Loops" in Math Proc Camb Phil Soc for a rather interesting extension of the finite subgroups in realtion to the octonions.

Springer, Invariant theory, esp Chapter 4.