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Let $\Gamma$ be a discrete group of isometries of the hyperbolic plane $\mathbb{H}^2$, and let $x_0\in \mathbb{H}^2$ have trivial stabiliser. Then the set $$D(\Gamma,x_0)=\textrm{Int}(\{x\in \mathbb{H}^2\mid d(x_0,x)\le d(x_0,\gamma x)\;\forall\gamma\in\Gamma\})$$ defined as the interior of a closed set is a Dirichlet fundamental domain for $\Gamma$ centred on $x_0$. See proposition 1.10 on p 149 of Geometry II edited by EB Vinberg.

I have been searching for a reference which treats the idea of a Dirichlet fundamental domain in its most general setting (or at least a reasonably general one). For example, if $(X,d)$ is a connected (and locally connected) metric space, and $\Gamma$ acts on $X$ discretely by isometries, then is this enough to guarantee that a set defined analogously to $D(\Gamma,x_0)$ above is a fundamental domain? More generally my question is:

Question What is the most general setting (ie the weakest hypotheses on a group $\Gamma$ acting on a metric space $(X,d)$), for $D(\Gamma,x_0)$ to be a fundamental domain?

This question touches on what I am asking, but falls short of being explicit of the level of generality possible.

Edit 1: the definition of a fundamental domain I'm working with is a set $\mathcal{F}\subset X$ which satisfies:

  • $\mathcal{F}$ is open and connected
  • translates of its closure cover $X$: ie $\bigcup_{\gamma\in\Gamma}\gamma\overline{\mathcal{F}}=X$
  • distinct translates are disjoint: ie if $\gamma\in \Gamma$ does not fix $X$ pointwise, then $\mathcal{F}\cap\gamma\mathcal{F}=\emptyset$

Edit 2: thanks to @MoisheKohan who referred me (in the comments on their answer below) to this very useful MO post along similar lines to my question. I wont go to the effort of editing my original question to become a facsimile of that post, but it points out some of the subtleties not mentioned in my question, for example

  • The need for $\mathcal{F}$ to be regular in some sense
  • The difficulty of defining a fundamental domain outside of the context of group actions on complete manifolds because it may be possible for a finite order group element to fix a nonempty open subset (this is illustrated in the example in the answer below)
David Sheard
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  • What is your definition of a fundamental domain? There are many inequivalent definitions in the literature... – Moishe Kohan Jun 18 '22 at 15:02
  • @MoisheKohan I have added my definition as an edit – David Sheard Jun 18 '22 at 15:11
  • Then, by your definition, Dirichlet domain is hardly ever fundamental since it is not open. Did you mean the strict inequality? – Moishe Kohan Jun 18 '22 at 15:32
  • Yes, I was being a little cavalier with open vs closed in the original version of the question because I don't think its the most important condition to worry about, but when I made my edit you'll see I also adjusted the definition of $D(\Gamma,x_0)$ from what appears in Geometry II to make it open. – David Sheard Jun 18 '22 at 15:41

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Here is an example which shows that "connected and locally connected" is not enough. Consider $X$ which is the union of the coordinate lines in the plane with the restriction of the Euclidean metric, $p=(1,0)$, $\Gamma$ generated by the involution $(x,y)\mapsto (-x,y)$. Then $D(p,\Gamma)$ is the union of the y-xis and the half-line $\{(x,0): x\ge 0\}$. Its interior equals $D=D(p,\Gamma)$ with the origin removed. However, this interior fails the 3rd axiom of the fundamental domain since the generator fixes all the point (except for the origin) in the intersection of the y-axis with $Int(D)$. As for the "weakest" conditions that guarantee the fundamental domain property, I do not know, it sounds like the generality level is too high to identify such. If you ask me about a specific class of spaces, I might be able to tell you, as I did for connected and locally connected spaces above...

Moishe Kohan
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  • Thanks, that is is a very helpful example, particularly as it's resistant to replacing the definition of $D$ with ${x\in X\mid d(x,x_0)<d(gx,x_0}$ In your example I think this would yield $D={(x,0)\mid x>0}$ which now satisfies the third condition, but fails the second. – David Sheard Jun 19 '22 at 10:00
  • If finding generality is a problem, could I play it safe as follows: if $\Gamma$ acts discretely by isometries on a connected Riemannian manifold, is this sufficient to guarantee that $D$ is a fundamental domain? – David Sheard Jun 19 '22 at 10:02
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    @DavidSheard Yes, assuming a complete manifold. I think, Smilga asked this on Mathoverflow and got an answer. – Moishe Kohan Jun 19 '22 at 12:24
  • Thanks, I'll add a link to post here for reference – David Sheard Jun 19 '22 at 12:32
  • @MoisheKohan Are there any resources for Dirichlet Fundamental domain (DFD for short) for a Lie group $G$ acting on a Riemannian manifold $M$ so that $G$ is not discrete, and say $dim(G)>0?$ I guess in this case, if such a DFD exists, then its interior (as a manifold) must be mapped by the projection by an open and dense subset of the quotient $M/G.$ But I'm having trouble finding any resources for any non-discrete groups, except for https://www.jstor.org/stable/2034968, but they don't quite mention anything about Dirichlet Fundamental Domain (DFD). – Learning Math Feb 27 '24 at 16:46
  • @LearningMath: I do not even understand a definition of such a domain. For Lie group actions people use slices instead of fundamental domains. – Moishe Kohan Feb 27 '24 at 16:48
  • @MoisheKohan It's the same one mentioned in the OP: $D(\Gamma,x_0)=\textrm{Int}({x\in \mathbb{H}^2\mid d(x_0,x)\le d(x_0,\gamma x);\forall\gamma\in\Gamma}),$ but this time $\Gamma$ need not be discrete. Will this be a fundamental domain? I'll consult the topic of slices in this case. – Learning Math Feb 27 '24 at 16:52
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    @LearningMath: this would be just ${x_0}$, hence, useless. In any case, if you have a new question, you should not be asking it in comments. – Moishe Kohan Feb 27 '24 at 16:54
  • @MoisheKohan Thanks, okay I'll ask a new question then. – Learning Math Feb 27 '24 at 16:55
  • @MoisheKohan I asked a question here: https://math.stackexchange.com/questions/4872077/why-dont-i-see-any-mention-of-fundamental-domain-in-the-context-of-non-discrete – Learning Math Feb 29 '24 at 10:08