Talk:Fundamental domain

message: This page should probably be merged with free regular set.linas 17:17, 15 Feb 2005 (UTC)
replies: text: Nahh. linas 05:19, 9 November 2006 (UTC)
thread_title: Suggestions

message: Some of the hyperbolic ones are related to knot complements. e.g. the classic number theory one is the complement of the trefoil knot mod the stabilizer SO(2). Or something like that. Note the presentation y^2=x^3 for the trefoil is the same presentation as for the modular group. Or something like that. Or so I overheard at a party. What I'd like to know is ... are all homogeneous spaces knot complements? if not, why not? linas 05:19, 9 November 2006 (UTC)
thread_title: Knots

message: Hello, I am wondering if the definition given in the article is general enough (a fundamental domain is a set D of representatives for the cosets of Γ in G, Γ being a lattice of the Lie group G). The article on Mathworld doesn't make any reference to Lie groups. I must also say that as a newcomer to these theories, I find the definition from Mathworld clearer. Is it legal to just copy it? (just the definition, which should be public domain). --Mathieu Perrin (talk) 06:07, 2 March 2008 (UTC)
thread_title:  General definition ? 

message: "The standard fundamental domain of the modular group probably deserves a page of its own.  Katzmik (talk) 14:11, 22 September 2008 (UTC)
thread_title:  modular group 

message: Let  be a group and  be a topological G-set. Then a closed subset  of  is called a fundamental domain of  in  if  is the union of conjugates of , i.e., 
and the intersection of any two conjugates has no interior. 
For example, a fundamental domain of the group of rotations by multiples of  in  is the upper half-plane  and a fundamental domain of rotations by multiples of  is the first quadrant . 
The concept of a fundamental domain is a generalization of a minimal group block, since while the intersection of fundamental domains has empty interior, the intersection of minimal blocks is the empty set."
*I hope this is right, since the  last sentence is due to me. I would say if it is right , that would be nice, because it is concise and seems easy to understand and use. The current wikipedia article, like many wikipedia math articles (such as how affine space USED to be)go on and on without giving an immediate, precise definition. Maybe that's necessary in the nature of these things and the clear, concise definition on MathWorld is wrong? I am genuinely concerned that I have contributed a plausible, userfriendly but wrong explanation to MathWorld.Rich (talk) 03:05, 10 February 2010 (UTC)
thread_title: definition from MathWorld--Correct?

message: In almost all instances of a fundamental domain occurring in the mathematical literature — including that of the modular group as in the article's illustration — the fundamental domain is taken to include its entire boundary.  In other words, it is not necessary that along the boundary, the fundamental domain be required to contain only one point in each orbit of the associated group action.65.157.43.50 (talk) 14:19, 14 August 2020 (UTC)
thread_title:  Normally, the fundamental domain is taken to include its entire boundary 

message: I think it's worth having a section titled "Canonical Fundamental Domain". Canonical is referenced in one of the image captions, but I'm actually not sure what is the correct definition of a canonical Fundamental Domain/Zone (this is something I'd like to know). Is it the one that's most accepted/used by the scientific community, or is it one where the "borders" are all high-symmetry points of the group?
: There is no notion of a canonical fundamental domain in general. For some groups some domains can be nicer than others (for example if your quotient has nontrivial isometries you can ask that the domain be stable under a lift of some subgroup) but i'm not sure this has been formalised/studied in any notable way. Also reflection groups have a "canonical fundamental domain" from their very definition. As for your last remark, note that the "high symmetry" points will always be on the border of the domain since a small neighborhood cannot be contained in the domain if a nontrivial element stabilises the point. jraimbau (talk) 06:29, 20 March 2021 (UTC)
thread_title:  Canonical Fundamental Domain