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

In summary, the conversation discusses a previous inquiry about math structures, and the discovery of geometric group theory. The participants discuss the application of group theory to geometry, including concepts like group actions and Lie groups. They also mention the role of calculus of variations in the subject and suggest including Felix Klein's Erlangen program. The conversation ends with a question about the current state of lattice gauge theory.
  • #1
diegzumillo
173
18
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.
 
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  • #2
diegzumillo said:
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.
 
  • #3
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.
 
  • #4
WWGD said:
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.
 
  • #5
fresh_42 said:
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)##.
 
  • #6
WWGD said:
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.
 
  • #7
Conversation laws?
 
  • #8
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.
 
  • #9
fresh_42 said:
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?
 
  • #10
WWGD said:
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.
 
  • #11
I guess you meant call conversion? And, how about including your paysan Felix Klein's Erlangen program in geometric g.t?
 
  • #12
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?
 
  • #13
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.
..
 

What is geometric group theory?

Geometric group theory is a branch of mathematics that studies the connections between groups, which are algebraic structures, and geometric spaces, such as graphs and manifolds. It uses tools and concepts from both group theory and geometry to understand the properties and behaviors of groups and their actions on geometric spaces.

How does geometric group theory relate to different geometries?

Geometric group theory has applications in various areas of geometry, including hyperbolic geometry, Euclidean geometry, and topological geometry. For example, it has been used to study the symmetries of hyperbolic spaces, the fundamental groups of manifolds, and the isometries of Riemannian manifolds.

What are some key concepts in geometric group theory?

Some key concepts in geometric group theory include group actions, which describe how a group acts on a geometric space, and geometric invariants, which are properties of a geometric space that are preserved under group actions. Other important concepts include Cayley graphs, which represent groups as graphs, and the word problem, which asks if a given element can be expressed as a combination of other elements in a group.

What are some applications of geometric group theory?

Geometric group theory has applications in many areas of mathematics, including topology, algebra, and geometry. It has also been used in computer science and physics. For example, it has been used to study the structure of DNA molecules, to understand the symmetries of crystals, and to analyze the behavior of certain algorithms.

What are some open problems in geometric group theory?

There are many open problems in geometric group theory, including the study of the growth of groups, which measures the size of a group as it grows, and the classification of groups, which seeks to understand the different types of groups that exist. Other open problems include the existence of groups with certain properties, such as being non-amenable or having exotic group actions.

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