What Maps Belong to Aut(S2)/SO(3)?

[lugita15]

Consider Aut(S2), the automorphism group of the Riemann sphere, i.e. the group of bijective holomorphic maps from C2 U {∞} to itself. Clearly some automorphisms of a sphere are the rotations of the sphere, SO(3). But what other maps are in Aut(S2)? To put it another way, what bijective holomorphic maps from C2 U {∞} to itself belong to the quotient group Aut(S2)/SO(3)?

Any help would be greatly appreciated.

Thank You in Advance.

 

[morphism]

The automorphisms of the Riemann sphere are the mobius transformations, so Aut=PGL(2,C). Those which are rotations have a particularly special form - they come from unitary matrices, i.e. they form the subgroup PSU(2) (which is isomorphic to SO(3)). However, PSU(2) isn't normal in PGL(2,C), so it makes no sense to talk about the "quotient group" PGL(2,C)/PSU(2).

 

[lugita15]

OK, let me put it this way. Any möbius transformation is a composition of an element of PSU(2) and an element of ... what?

 

[lugita15]

Any more thoughts on this?

 

[morphism]

OK, let me put it this way. Any möbius transformation is a composition of an element of PSU(2) and an element of ... what?

I don't think there's a nice way of describing "what?" here, in the sense that there is no subgroup complement to PSU(2) in PGL(2,C). This is because you can express each rotation as a composition of automorphisms that aren't rotations (so the "complement" of PSU(2) in PGL(2,C) won't be closed under multiplication); in fact, you can express every automorphism in PGL(2,C) as the composition of dilations, translations and inversions (see here).

 

[lugita15]

I don't think there's a nice way of describing "what?" here, in the sense that there is no subgroup complement to PSU(2) in PGL(2,C). This is because you can express each rotation as a composition of automorphisms that aren't rotations (so the "complement" of PSU(2) in PGL(2,C) won't be closed under multiplication); in fact, you can express every automorphism in PGL(2,C) as the composition of dilations, translations and inversions (see here).

First of all, you didn't actually put a link in your post. Second of all, is it possible to describe the automorphism group PGL(2,C) of the Riemann sphere in geometric terms, i.e. describing what you're physically doing to the sphere?

 

[morphism]

Oops - sorry about that. Here's the link again: http://books.google.ca/books?id=61BO2UY9QN4C&lpg=PA65&ots=5cjSkzzf1n&pg=PA65#v=onepage. It's to Gamelin's complex analysis book, Chap. II, Sec. 7.

As for your second question: check out this video.

 

[lugita15]

As for your second question: check out this video.

OK, so the video says that every Mobius transformation can be written as a composition of rotations of the complex plane, inversions, translations of the complex plane, and dilations. And according to the video, a rotation of the complex plane is a vertical rotation of the Riemann sphere, an inversion is a horizontal rotation of the Riemann sphere, a translation of the complex plane is a horizontal translation of the Riemann sphere, and a dilation is a vertical translation of the Riemann sphere. So is it safe to say that any automorphism of the Riemann sphere is a composition of rotations of the Riemann sphere and translations of the Riemann sphere?

Also, just like the rotation group of the Riemann sphere can be written in concrete terms as PSU(2), how would you describe the translation group of the Riemann sphere in concrete terms? To put it another way, what is the name of the group of all transformations that can be written as compositions of translations of the complex plane and dilations?

 

[morphism]

OK, so the video says that every Mobius transformation can be written as a composition of rotations of the complex plane, inversions, translations of the complex plane, and dilations. And according to the video, a rotation of the complex plane is a vertical rotation of the Riemann sphere, an inversion is a horizontal rotation of the Riemann sphere, a translation of the complex plane is a horizontal translation of the Riemann sphere, and a dilation is a vertical translation of the Riemann sphere. So is it safe to say that any automorphism of the Riemann sphere is a composition of rotations of the Riemann sphere and translations of the Riemann sphere?

No. In the video they're keeping track of what happens to the stereographic projection.

Also, just like the rotation group of the Riemann sphere can be written in concrete terms as PSU(2), how would you describe the translation group of the Riemann sphere in concrete terms? To put it another way, what is the name of the group of all transformations that can be written as compositions of translations of the complex plane and dilations?

Translation doesn't move the sphere around. Don't confuse what the video was showing (i.e. the interaction of mobius transformations on the plane and stereographic projection) with the the automorphisms of S^2. When you talk about Aut(S^2), you're thinking of intrinsic automorphisms. A translation in this sense translates points on there sphere to other points. If you view the Riemann sphere as $\mathbb C \cup \{\infty\}$ then a translation is just a map of the form $z\mapsto z+b$ (for some $b \in \mathbb C$). This is fairly concrete. Similarly, a dilation looks like $z \mapsto az$ ($a \in \mathbb C\backslash \{0\}$).

The subgroup generated by these is simply the group of all automorphisms of the form $az+b$ ($a \in \mathbb C\backslash \{0\}, b\in \mathbb C$). Note that every such automorphism fixes the point $\infty$, so it is simply an automorphism of $\mathbb C$. In fact, this is precisely the group of automorphisms of the plane. So you can call it $\text{Aut}(\mathbb C)$.

 

[lugita15]

No. In the video they're keeping track of what happens to the stereographic projection.

But I thought the stereographic projection IS how we understand the Riemann sphere as a sphere.

Translation doesn't move the sphere around. Don't confuse what the video was showing (i.e. the interaction of mobius transformations on the plane and stereographic projection) with the the automorphisms of S^2. When you talk about Aut(S^2), you're thinking of intrinsic automorphisms. A translation in this sense translates points on there sphere to other points. If you view the Riemann sphere as $\mathbb C \cup \{\infty\}$ then a translation is just a map of the form $z\mapsto z+b$ (for some $b \in \mathbb C$). This is fairly concrete. Similarly, a dilation looks like $z \mapsto az$ ($a \in \mathbb C\backslash \{0\}$).

The subgroup generated by these is simply the group of all automorphisms of the form $az+b$ ($a \in \mathbb C\backslash \{0\}, b\in \mathbb C$). Note that every such automorphism fixes the point $\infty$, so it is simply an automorphism of $\mathbb C$. In fact, this is precisely the group of automorphisms of the plane. So you can call it $\text{Aut}(\mathbb C)$.

Would it be correct to say that the transformations in PSU(2) correspond to rotations of the stereographic projection? Also, would it be correct to say that transfomations in the automorphism group PGL(2,C) of the Riemann sphere can be written as compositions of elements of PSU(2) and elements of Aut(C), the automorphism group of the complex plane?

 

[morphism]

But I thought the stereographic projection IS how we understand the Riemann sphere as a sphere.

Stereographic projection, in its traditional form, tells you that if you delete the north pole from S^2 (a.k.a. the point at infinity) you get the plane. That is, stereographic projection is a map $S^2 \backslash \{\infty\} \to \mathbb C$. An automorphism of the sphere doesn't necessarily respect this map: it could move the point $\infty$ around, i.e. the point at infinity could go to a point in the finite plane and vice versa.

Would it be correct to say that the transformations in PSU(2) correspond to rotations of the stereographic projection?

There seems to be some confusion in this thread. When you say rotation, what do you mean? Note that the map $z\mapsto 1/z$ is a dilation of the plane. But applied to the sphere, this map is a rotation.

 

[lugita15]

Stereographic projection, in its traditional form, tells you that if you delete the north pole from S^2 (a.k.a. the point at infinity) you get the plane. That is, stereographic projection is a map $S^2 \backslash \{\infty\} \to \mathbb C$. An automorphism of the sphere doesn't necessarily respect this map: it could move the point $\infty$ around, i.e. the point at infinity could go to a point in the finite plane and vice versa.

OK, then I mean stereographic projection in an extended sense, i.e. including the point at infinity.

There seems to be some confusion in this thread. When you say rotation, what do you mean? Note that the map $z\mapsto 1/z$ is a dilation of the plane. But applied to the sphere, this map is a rotation.

I mean rotations of the stereographic projection of the Riemann sphere, which is a sphere S2. I want to know what transformations correspond to this, for which you seem to be saying that the answer is PSU(2). And the video seems to be saying that Aut(C) corresponds to translations of the stereographic projection. So I want to know whether the automorphisms of the Riemann sphere can be written a compositions of elements of PSU(2) and elements of AUT(C), i.e. rotations of the stereographic projection S2 and translations of the stereographic projection.

 

[morphism]

I think you're missing my point about stereographic projection...but anyway, yes every automorphism of the Riemann sphere is a composition of elements of PSU(2) and elements of Aut(C). (In fact, you only need one element in PSU(2): namely the inversion z->1/z.) This is just a reformulation of the fact that everything is a composition of inversions, translations and dilations.

 

[lugita15]

I think you're missing my point about stereographic projection...but anyway, yes every automorphism of the Riemann sphere is a composition of elements of PSU(2) and elements of Aut(C). (In fact, you only need one element in PSU(2): namely the inversion z->1/z.) This is just a reformulation of the fact that everything is a composition of inversions, translations and dilations.

OK, then can you give me an example of a transformation in the automorphism group of the Riemann sphere that cannot be written as a composition of just inversions and translations i.e. a transformation that requires dilations as well? Or equivalently, why can't dilations be written as compositions of inversions and translations?

 

[morphism]

OK, then can you give me an example of a transformation in the automorphism group of the Riemann sphere that cannot be written as a composition of just inversions and translations i.e. a transformation that requires dilations as well? Or equivalently, why can't dilations be written as compositions of inversions and translations?

You really should be able to figure this out yourself.

There are probably faster and more sophisticated proofs, but one low-level approach would be to write down the matrices corresponding to z->az, z->1/z and z->z+b, and show that no matter how you multiply out matrices of the latter two types, you'll never get a matrix of the type to z->az (unless a=1).

 

[lugita15]

You really should be able to figure this out yourself.

One approach would be to write down the matrices corresponding to z->az, z->1/z and z->z+b, and show that no matter how you multiply out matrices of the latter two types, you'll never get a matrix corresponding to z->az (unless a=1).

OK, first of all how in general do I find the transformation matrix corresponding to a complex function f(z)?

Second of all, based on the video it looks like all automorphisms of the Riemann sphere correspond to rigid body motions of the stereographic projection. But any rigid body motion can be written as a horizontal rotation plus a horizontal translation, which correspond respectively to inversions and translations in the complex plane. So how is it that a dilation, corresponding to a vertical translation of the sphere, cannot be written in that way?