What is actually the geometry or analysis behind the fact that $Mob(hat{Bbb C})$ is simple?
$begingroup$
Let, $Mob(hat{Bbb C})$ be the group of all Mobius transformations from the extended complex plane to itself i.e. from $hat{Bbb C} to hat{Bbb C}$ .
I have been able to prove that (i) $Mob(hat{Bbb C}) cong PSL_{2}(Bbb C)$ where $PSL_{2}(Bbb C)$ is defined to be , $PSL_{2}(Bbb C) := SL_{2}(Bbb C)/ {pm I}$ and that,
(ii) $SL_{2}(Bbb C)$ does not have a proper normal subgroup containing its center i.e. ${pm I}$ and thus by correspondence theorem, $PSL_{2}(Bbb C) = SL_{2}(Bbb C)/ {pm I}$ is simple.
Combining (i) and (ii) we get that $Mob(hat{Bbb C})$ is simple!
My question precisely is,
What is actually the geometry or analysis behind the fact that $Mob(hat{Bbb C})$ is simple ?
complex-analysis geometry mobius-transformation
asked Jan 15 at 16:12
CoherentCoherent
1,176523
$endgroup$
add a comment |
$begingroup$
Let, $Mob(hat{Bbb C})$ be the group of all Mobius transformations from the extended complex plane to itself i.e. from $hat{Bbb C} to hat{Bbb C}$ .
I have been able to prove that (i) $Mob(hat{Bbb C}) cong PSL_{2}(Bbb C)$ where $PSL_{2}(Bbb C)$ is defined to be , $PSL_{2}(Bbb C) := SL_{2}(Bbb C)/ {pm I}$ and that,
(ii) $SL_{2}(Bbb C)$ does not have a proper normal subgroup containing its center i.e. ${pm I}$ and thus by correspondence theorem, $PSL_{2}(Bbb C) = SL_{2}(Bbb C)/ {pm I}$ is simple.
Combining (i) and (ii) we get that $Mob(hat{Bbb C})$ is simple!
My question precisely is,
What is actually the geometry or analysis behind the fact that $Mob(hat{Bbb C})$ is simple ?
complex-analysis geometry mobius-transformation
asked Jan 15 at 16:12
CoherentCoherent
1,176523
$endgroup$
$begingroup$
Two thoughts - first, I suggest looking at the proof that $SL_2(mathbb{C})$ is simple and use the isomorphism you constructed to pull the argument back to $Mob(hat{mathbb{C}})$. Second, you might find insight in the geometric interpretation of $Mob(hat{mathbb{C}})$ as isometries of $mathbb{H}^3$.
$endgroup$
– Neal
Jan 21 at 14:10
$begingroup$
@Neal I have studied the elements Mobius group as isomteries of $Bbb H^3$, I want to picturize how this phenomena (being simple) affects the theory of fixed points under Mobius transformations etc.
$endgroup$
– Coherent
Jan 21 at 14:13
add a comment |
$begingroup$
Let, $Mob(hat{Bbb C})$ be the group of all Mobius transformations from the extended complex plane to itself i.e. from $hat{Bbb C} to hat{Bbb C}$ .
I have been able to prove that (i) $Mob(hat{Bbb C}) cong PSL_{2}(Bbb C)$ where $PSL_{2}(Bbb C)$ is defined to be , $PSL_{2}(Bbb C) := SL_{2}(Bbb C)/ {pm I}$ and that,
(ii) $SL_{2}(Bbb C)$ does not have a proper normal subgroup containing its center i.e. ${pm I}$ and thus by correspondence theorem, $PSL_{2}(Bbb C) = SL_{2}(Bbb C)/ {pm I}$ is simple.
Combining (i) and (ii) we get that $Mob(hat{Bbb C})$ is simple!
My question precisely is,
What is actually the geometry or analysis behind the fact that $Mob(hat{Bbb C})$ is simple ?
complex-analysis geometry mobius-transformation
asked Jan 15 at 16:12
CoherentCoherent
1,176523
$endgroup$
Let, $Mob(hat{Bbb C})$ be the group of all Mobius transformations from the extended complex plane to itself i.e. from $hat{Bbb C} to hat{Bbb C}$ .
I have been able to prove that (i) $Mob(hat{Bbb C}) cong PSL_{2}(Bbb C)$ where $PSL_{2}(Bbb C)$ is defined to be , $PSL_{2}(Bbb C) := SL_{2}(Bbb C)/ {pm I}$ and that,
(ii) $SL_{2}(Bbb C)$ does not have a proper normal subgroup containing its center i.e. ${pm I}$ and thus by correspondence theorem, $PSL_{2}(Bbb C) = SL_{2}(Bbb C)/ {pm I}$ is simple.
Combining (i) and (ii) we get that $Mob(hat{Bbb C})$ is simple!
My question precisely is,
What is actually the geometry or analysis behind the fact that $Mob(hat{Bbb C})$ is simple ?
complex-analysis geometry mobius-transformation
complex-analysis geometry mobius-transformation
asked Jan 15 at 16:12
CoherentCoherent
1,176523
asked Jan 15 at 16:12
CoherentCoherent
1,176523
asked Jan 15 at 16:12
CoherentCoherent
1,176523
asked Jan 15 at 16:12
CoherentCoherent
1,176523
asked Jan 15 at 16:12
CoherentCoherent
1,176523
1,176523
$begingroup$
Two thoughts - first, I suggest looking at the proof that $SL_2(mathbb{C})$ is simple and use the isomorphism you constructed to pull the argument back to $Mob(hat{mathbb{C}})$. Second, you might find insight in the geometric interpretation of $Mob(hat{mathbb{C}})$ as isometries of $mathbb{H}^3$.
$endgroup$
– Neal
Jan 21 at 14:10
$begingroup$
@Neal I have studied the elements Mobius group as isomteries of $Bbb H^3$, I want to picturize how this phenomena (being simple) affects the theory of fixed points under Mobius transformations etc.
$endgroup$
– Coherent
Jan 21 at 14:13
add a comment |
$begingroup$
Two thoughts - first, I suggest looking at the proof that $SL_2(mathbb{C})$ is simple and use the isomorphism you constructed to pull the argument back to $Mob(hat{mathbb{C}})$. Second, you might find insight in the geometric interpretation of $Mob(hat{mathbb{C}})$ as isometries of $mathbb{H}^3$.
$endgroup$
– Neal
Jan 21 at 14:10
$begingroup$
@Neal I have studied the elements Mobius group as isomteries of $Bbb H^3$, I want to picturize how this phenomena (being simple) affects the theory of fixed points under Mobius transformations etc.
$endgroup$
– Coherent
Jan 21 at 14:13
$begingroup$
Two thoughts - first, I suggest looking at the proof that $SL_2(mathbb{C})$ is simple and use the isomorphism you constructed to pull the argument back to $Mob(hat{mathbb{C}})$. Second, you might find insight in the geometric interpretation of $Mob(hat{mathbb{C}})$ as isometries of $mathbb{H}^3$.
$endgroup$
– Neal
Jan 21 at 14:10
$begingroup$
Two thoughts - first, I suggest looking at the proof that $SL_2(mathbb{C})$ is simple and use the isomorphism you constructed to pull the argument back to $Mob(hat{mathbb{C}})$. Second, you might find insight in the geometric interpretation of $Mob(hat{mathbb{C}})$ as isometries of $mathbb{H}^3$.
$endgroup$
– Neal
Jan 21 at 14:10
$begingroup$
@Neal I have studied the elements Mobius group as isomteries of $Bbb H^3$, I want to picturize how this phenomena (being simple) affects the theory of fixed points under Mobius transformations etc.
$endgroup$
– Coherent
Jan 21 at 14:13
$begingroup$
@Neal I have studied the elements Mobius group as isomteries of $Bbb H^3$, I want to picturize how this phenomena (being simple) affects the theory of fixed points under Mobius transformations etc.
$endgroup$
– Coherent
Jan 21 at 14:13
add a comment |
1 Answer
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Assume that $N$ is a normal subgroup of the Mobius group which is not the trivial subgroup. Then $N$ contains some non-identity element $f$. If $f$ is hyperbolic/loxodromic, i.e., if it has two fixed points $z_0$ and $z_1$, then you can pick a third point $z_2$ different from $z_0$ and $z_1$, and conjugate $f$ by a Mobius transformation $h$ which fixes $z_0$, and which maps $z_1$ to $z_2$, to obtain $g = hfh^{-1} in N$ with fixed points $z_0$ and $z_2$. It is then easy to show that the commutator $k = [f,g] = f^{-1}g^{-1}fg$ fixes $z_0$ with $k'(z_0)=1$, but that it is not the identity, so it must be a parabolic Mobius transformation. This establishes that $N$ contains a parabolic Mobius transformation, and since any two parabolic Mobius transformations are conjugate, $N$ contains all of them. In particular $N$ contains $F(z) = z+1$ and $G(z) = frac{z}{1+az}$ for any $a ne 0$. Explicit calculations of the commutator of $F$ and $G$ shows that you get a hyperbolic/loxodromic element with trace $2+a^2$. (This calculation is probably easier with matrices than with explicit Mobius transformations.) This shows that $N$ contains hyperbolic/loxodromic elements with any given trace, so $N$ is the whole Mobius group.
answered Jan 21 at 23:08
Lukas GeyerLukas Geyer
13.5k1454
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$begingroup$
Assume that $N$ is a normal subgroup of the Mobius group which is not the trivial subgroup. Then $N$ contains some non-identity element $f$. If $f$ is hyperbolic/loxodromic, i.e., if it has two fixed points $z_0$ and $z_1$, then you can pick a third point $z_2$ different from $z_0$ and $z_1$, and conjugate $f$ by a Mobius transformation $h$ which fixes $z_0$, and which maps $z_1$ to $z_2$, to obtain $g = hfh^{-1} in N$ with fixed points $z_0$ and $z_2$. It is then easy to show that the commutator $k = [f,g] = f^{-1}g^{-1}fg$ fixes $z_0$ with $k'(z_0)=1$, but that it is not the identity, so it must be a parabolic Mobius transformation. This establishes that $N$ contains a parabolic Mobius transformation, and since any two parabolic Mobius transformations are conjugate, $N$ contains all of them. In particular $N$ contains $F(z) = z+1$ and $G(z) = frac{z}{1+az}$ for any $a ne 0$. Explicit calculations of the commutator of $F$ and $G$ shows that you get a hyperbolic/loxodromic element with trace $2+a^2$. (This calculation is probably easier with matrices than with explicit Mobius transformations.) This shows that $N$ contains hyperbolic/loxodromic elements with any given trace, so $N$ is the whole Mobius group.
answered Jan 21 at 23:08
Lukas GeyerLukas Geyer
13.5k1454
$endgroup$
add a comment |
$begingroup$
Assume that $N$ is a normal subgroup of the Mobius group which is not the trivial subgroup. Then $N$ contains some non-identity element $f$. If $f$ is hyperbolic/loxodromic, i.e., if it has two fixed points $z_0$ and $z_1$, then you can pick a third point $z_2$ different from $z_0$ and $z_1$, and conjugate $f$ by a Mobius transformation $h$ which fixes $z_0$, and which maps $z_1$ to $z_2$, to obtain $g = hfh^{-1} in N$ with fixed points $z_0$ and $z_2$. It is then easy to show that the commutator $k = [f,g] = f^{-1}g^{-1}fg$ fixes $z_0$ with $k'(z_0)=1$, but that it is not the identity, so it must be a parabolic Mobius transformation. This establishes that $N$ contains a parabolic Mobius transformation, and since any two parabolic Mobius transformations are conjugate, $N$ contains all of them. In particular $N$ contains $F(z) = z+1$ and $G(z) = frac{z}{1+az}$ for any $a ne 0$. Explicit calculations of the commutator of $F$ and $G$ shows that you get a hyperbolic/loxodromic element with trace $2+a^2$. (This calculation is probably easier with matrices than with explicit Mobius transformations.) This shows that $N$ contains hyperbolic/loxodromic elements with any given trace, so $N$ is the whole Mobius group.
answered Jan 21 at 23:08
Lukas GeyerLukas Geyer
13.5k1454
$endgroup$
add a comment |
$begingroup$
Assume that $N$ is a normal subgroup of the Mobius group which is not the trivial subgroup. Then $N$ contains some non-identity element $f$. If $f$ is hyperbolic/loxodromic, i.e., if it has two fixed points $z_0$ and $z_1$, then you can pick a third point $z_2$ different from $z_0$ and $z_1$, and conjugate $f$ by a Mobius transformation $h$ which fixes $z_0$, and which maps $z_1$ to $z_2$, to obtain $g = hfh^{-1} in N$ with fixed points $z_0$ and $z_2$. It is then easy to show that the commutator $k = [f,g] = f^{-1}g^{-1}fg$ fixes $z_0$ with $k'(z_0)=1$, but that it is not the identity, so it must be a parabolic Mobius transformation. This establishes that $N$ contains a parabolic Mobius transformation, and since any two parabolic Mobius transformations are conjugate, $N$ contains all of them. In particular $N$ contains $F(z) = z+1$ and $G(z) = frac{z}{1+az}$ for any $a ne 0$. Explicit calculations of the commutator of $F$ and $G$ shows that you get a hyperbolic/loxodromic element with trace $2+a^2$. (This calculation is probably easier with matrices than with explicit Mobius transformations.) This shows that $N$ contains hyperbolic/loxodromic elements with any given trace, so $N$ is the whole Mobius group.
answered Jan 21 at 23:08
Lukas GeyerLukas Geyer
13.5k1454
$endgroup$
Assume that $N$ is a normal subgroup of the Mobius group which is not the trivial subgroup. Then $N$ contains some non-identity element $f$. If $f$ is hyperbolic/loxodromic, i.e., if it has two fixed points $z_0$ and $z_1$, then you can pick a third point $z_2$ different from $z_0$ and $z_1$, and conjugate $f$ by a Mobius transformation $h$ which fixes $z_0$, and which maps $z_1$ to $z_2$, to obtain $g = hfh^{-1} in N$ with fixed points $z_0$ and $z_2$. It is then easy to show that the commutator $k = [f,g] = f^{-1}g^{-1}fg$ fixes $z_0$ with $k'(z_0)=1$, but that it is not the identity, so it must be a parabolic Mobius transformation. This establishes that $N$ contains a parabolic Mobius transformation, and since any two parabolic Mobius transformations are conjugate, $N$ contains all of them. In particular $N$ contains $F(z) = z+1$ and $G(z) = frac{z}{1+az}$ for any $a ne 0$. Explicit calculations of the commutator of $F$ and $G$ shows that you get a hyperbolic/loxodromic element with trace $2+a^2$. (This calculation is probably easier with matrices than with explicit Mobius transformations.) This shows that $N$ contains hyperbolic/loxodromic elements with any given trace, so $N$ is the whole Mobius group.
answered Jan 21 at 23:08
Lukas GeyerLukas Geyer
13.5k1454
answered Jan 21 at 23:08
Lukas GeyerLukas Geyer
13.5k1454
answered Jan 21 at 23:08
Lukas GeyerLukas Geyer
13.5k1454
answered Jan 21 at 23:08
Lukas GeyerLukas Geyer
13.5k1454
13.5k1454
add a comment |
add a comment |
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$begingroup$
Two thoughts - first, I suggest looking at the proof that $SL_2(mathbb{C})$ is simple and use the isomorphism you constructed to pull the argument back to $Mob(hat{mathbb{C}})$. Second, you might find insight in the geometric interpretation of $Mob(hat{mathbb{C}})$ as isometries of $mathbb{H}^3$.
$endgroup$
– Neal
Jan 21 at 14:10
$begingroup$
@Neal I have studied the elements Mobius group as isomteries of $Bbb H^3$, I want to picturize how this phenomena (being simple) affects the theory of fixed points under Mobius transformations etc.
$endgroup$
– Coherent
Jan 21 at 14:13