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)
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...