Examples for sequence defintion of proper group action

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Sorry for the (edited) duplicate from overflow but I did not receive any answer there.

For an action by a Lie group $G$ on a second countable and Hausdorff differentiable manifold $M$ to be proper, it suffices to show that the (extended) map $G times M rightarrow M times M$ is closed with respect to prescribed topologies on these spaces and that the fibers are compact. No local compactness is assumed here for either $G$ or $M$ though. As mentioned in many texts, one way to proceed is to verify that given a sequence $g_n$ in $G$ and a converging sequence $x_n$ in $M$, if the sequence $g_n cdot x_n$ converges in $M$, then $g_n$ contains a converging subsequence.

That this formulation is equivalent, under appropriate assumptions, to some other forms of definition for proper group action is clear to me. However, I have searched places I could think of including posts at this site but could not seem to find any specific example that actually uses this sequence characterization to prove properness of a group action.

I am just beginning to learn this material, and understand that different techniques are called for in different scenarios to extract a converging subsequence. But could someone kindly provide or point me to some less trivial and concrete example(s) illustrating necessary procedures to arrive at a converging subsequence in the differentiable context?

Thanks a lot!

differential-geometry lie-groups group-actions

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asked Jul 31 at 18:46

X-Naut PhD

61

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Local compactness need not be assumed because it is true in the context of your question: every manifold is locally compact; and every Lie group is a manifold and hence is locally compact.
  – Lee Mosher
  Jul 31 at 18:54
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I simply did not need to exclude infinite-dimensional groups and manifolds modeled, for example, on hilbert spaces.
  – X-Naut PhD
  Jul 31 at 19:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Alright. It's a good idea to be explicit about those matters, usually.
  – Lee Mosher
  Jul 31 at 20:45
  

  
  
  
  

add a comment | 

up vote
0
down vote

favorite

Sorry for the (edited) duplicate from overflow but I did not receive any answer there.

For an action by a Lie group $G$ on a second countable and Hausdorff differentiable manifold $M$ to be proper, it suffices to show that the (extended) map $G times M rightarrow M times M$ is closed with respect to prescribed topologies on these spaces and that the fibers are compact. No local compactness is assumed here for either $G$ or $M$ though. As mentioned in many texts, one way to proceed is to verify that given a sequence $g_n$ in $G$ and a converging sequence $x_n$ in $M$, if the sequence $g_n cdot x_n$ converges in $M$, then $g_n$ contains a converging subsequence.

That this formulation is equivalent, under appropriate assumptions, to some other forms of definition for proper group action is clear to me. However, I have searched places I could think of including posts at this site but could not seem to find any specific example that actually uses this sequence characterization to prove properness of a group action.

I am just beginning to learn this material, and understand that different techniques are called for in different scenarios to extract a converging subsequence. But could someone kindly provide or point me to some less trivial and concrete example(s) illustrating necessary procedures to arrive at a converging subsequence in the differentiable context?

Thanks a lot!

differential-geometry lie-groups group-actions

share|cite|improve this question

asked Jul 31 at 18:46

X-Naut PhD

61

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Local compactness need not be assumed because it is true in the context of your question: every manifold is locally compact; and every Lie group is a manifold and hence is locally compact.
  – Lee Mosher
  Jul 31 at 18:54
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I simply did not need to exclude infinite-dimensional groups and manifolds modeled, for example, on hilbert spaces.
  – X-Naut PhD
  Jul 31 at 19:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Alright. It's a good idea to be explicit about those matters, usually.
  – Lee Mosher
  Jul 31 at 20:45
  

  
  
  
  

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Sorry for the (edited) duplicate from overflow but I did not receive any answer there.

For an action by a Lie group $G$ on a second countable and Hausdorff differentiable manifold $M$ to be proper, it suffices to show that the (extended) map $G times M rightarrow M times M$ is closed with respect to prescribed topologies on these spaces and that the fibers are compact. No local compactness is assumed here for either $G$ or $M$ though. As mentioned in many texts, one way to proceed is to verify that given a sequence $g_n$ in $G$ and a converging sequence $x_n$ in $M$, if the sequence $g_n cdot x_n$ converges in $M$, then $g_n$ contains a converging subsequence.

That this formulation is equivalent, under appropriate assumptions, to some other forms of definition for proper group action is clear to me. However, I have searched places I could think of including posts at this site but could not seem to find any specific example that actually uses this sequence characterization to prove properness of a group action.

I am just beginning to learn this material, and understand that different techniques are called for in different scenarios to extract a converging subsequence. But could someone kindly provide or point me to some less trivial and concrete example(s) illustrating necessary procedures to arrive at a converging subsequence in the differentiable context?

Thanks a lot!

differential-geometry lie-groups group-actions

share|cite|improve this question

asked Jul 31 at 18:46

X-Naut PhD

61

Sorry for the (edited) duplicate from overflow but I did not receive any answer there.

For an action by a Lie group $G$ on a second countable and Hausdorff differentiable manifold $M$ to be proper, it suffices to show that the (extended) map $G times M rightarrow M times M$ is closed with respect to prescribed topologies on these spaces and that the fibers are compact. No local compactness is assumed here for either $G$ or $M$ though. As mentioned in many texts, one way to proceed is to verify that given a sequence $g_n$ in $G$ and a converging sequence $x_n$ in $M$, if the sequence $g_n cdot x_n$ converges in $M$, then $g_n$ contains a converging subsequence.

That this formulation is equivalent, under appropriate assumptions, to some other forms of definition for proper group action is clear to me. However, I have searched places I could think of including posts at this site but could not seem to find any specific example that actually uses this sequence characterization to prove properness of a group action.

I am just beginning to learn this material, and understand that different techniques are called for in different scenarios to extract a converging subsequence. But could someone kindly provide or point me to some less trivial and concrete example(s) illustrating necessary procedures to arrive at a converging subsequence in the differentiable context?

Thanks a lot!

differential-geometry lie-groups group-actions

share|cite|improve this question

asked Jul 31 at 18:46

X-Naut PhD

61

share|cite|improve this question

share|cite|improve this question

asked Jul 31 at 18:46

X-Naut PhD

61

asked Jul 31 at 18:46

X-Naut PhD

61

asked Jul 31 at 18:46

X-Naut PhD

61

61

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Local compactness need not be assumed because it is true in the context of your question: every manifold is locally compact; and every Lie group is a manifold and hence is locally compact.
  – Lee Mosher
  Jul 31 at 18:54
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I simply did not need to exclude infinite-dimensional groups and manifolds modeled, for example, on hilbert spaces.
  – X-Naut PhD
  Jul 31 at 19:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Alright. It's a good idea to be explicit about those matters, usually.
  – Lee Mosher
  Jul 31 at 20:45
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Local compactness need not be assumed because it is true in the context of your question: every manifold is locally compact; and every Lie group is a manifold and hence is locally compact.
  – Lee Mosher
  Jul 31 at 18:54
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I simply did not need to exclude infinite-dimensional groups and manifolds modeled, for example, on hilbert spaces.
  – X-Naut PhD
  Jul 31 at 19:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Alright. It's a good idea to be explicit about those matters, usually.
  – Lee Mosher
  Jul 31 at 20:45
  

  
  
  
  

Local compactness need not be assumed because it is true in the context of your question: every manifold is locally compact; and every Lie group is a manifold and hence is locally compact.
– Lee Mosher
Jul 31 at 18:54

Local compactness need not be assumed because it is true in the context of your question: every manifold is locally compact; and every Lie group is a manifold and hence is locally compact.
– Lee Mosher
Jul 31 at 18:54

I simply did not need to exclude infinite-dimensional groups and manifolds modeled, for example, on hilbert spaces.
– X-Naut PhD
Jul 31 at 19:36

I simply did not need to exclude infinite-dimensional groups and manifolds modeled, for example, on hilbert spaces.
– X-Naut PhD
Jul 31 at 19:36

Alright. It's a good idea to be explicit about those matters, usually.
– Lee Mosher
Jul 31 at 20:45

Alright. It's a good idea to be explicit about those matters, usually.
– Lee Mosher
Jul 31 at 20:45

add a comment | 

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