What is geometric group theory and how does it relate to different geometries?

[diegzumillo]

Hey all
I previously asked about some math structure fulfilling some requirements and didn't get much out of it ( Graph or lattice topology discretization ). It was a vague question, granted.

Anyway, I seem to have stumbled upon something interesting called geometric group theory. It looks relatively obscure, at least for someone with a physics background, and I would like to know more about it. Anyone has any idea where to start? The wikipedia page has many references, but if anyone wants to add some more to the pile or some personal comments I would really appreciate it.

 

[fresh_42]

Hey all
I previously asked about some math structure fulfilling some requirements and didn't get much out of it ( Graph or lattice topology discretization ). It was a vague question, granted.

Anyway, I seem to have stumbled upon something interesting called geometric group theory. It looks relatively obscure, at least for someone with a physics background, and I would like to know more about it. Anyone has any idea where to start? The wikipedia page has many references, but if anyone wants to add some more to the pile or some personal comments I would really appreciate it.

Sounds like graph theory or crystallography which should not sound obscure to someone with a physics background. I've recently had a short glimpse in it and found out, that it is far more than a few symmetry groups of crystals. And graph theory is an entire branch of mathematics, too.

 

[WWGD]

Lie groups , in my experience, are part of it. It is, as the name suggests, the application of group theory to geometry. In many cases you also include concepts like group actions, general linear groups.

 

[fresh_42]

Lie groups , in my experience, are part of it. It is, as the name suggests, the application of group theory to geometry. In many cases you also include concepts like group actions, general linear groups.

I think it's more the application of analysis to linear transformations. We always have the simple standard groups in mind when we say Lie group. However, they cover a far wider range. And they are not automatically subgroups of some general linear group, it's a theorem.

 

[WWGD]

I think it's more the application of analysis to linear transformations. We always have the simple standard groups in mind when we say Lie group. However, they cover a far wider range. And they are not automatically subgroups of some general linear group, it's a theorem.

Could be. I was describing an approx. to the syllabus I saw for this class. And you're right, they are not always subgroups of $GL(n,\mathbb F)$.

 

[fresh_42]

Could be. I was describing an approx. to the syllabus I saw for this class. And you're right, they are not always subgroups of $GL(n,\mathbb F)$.

I think Ado says they are for some $n$ but I haven't checked the details. If we go back to Lie and Noether, then it was actually the calculus of variations which originated the subject. In the end it turned out that conversation laws are encoded in Lie groups. A typical case of Zen: Everything is connected with everything. Although I think it was Humboldt who said it.

 

[WWGD]

Conversation laws?

 

[fresh_42]

Oops. Yes, that is what I meant by english is terribly bad at error correcting. And I haven't even the excuse of dyslexia. It would fit so well in this case.

 

[WWGD]

Oops. Yes, that is what I meant by english is terribly bad at error correcting. And I haven't even the excuse of dyslexia. It would fit so well in this case.

I guess it was conversion?

 

[fresh_42]

I guess it was conversion?

My suspicion is that it is due to a deep dislike of the word conservare, i.e. the associations of it.

 

[WWGD]

I guess you meant call conversion? And, how about including your paysan Felix Klein's Erlangen program in geometric g.t?

 

[fresh_42]

Sounds good, although I'm not so sure that lattice theory fits in as mentioned in the OP.

What is state of the art in lattice gauge theory?

 

[WWGD]

Not sure what you mean. Here geometries are classified by the groups acting on objects, that preserve basic structure . Translation for Euclidean geometry I remember, but not the others. My internet is down now, so I can't check.
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