# Configurations of $n$ points modulo isometries of the ambient space

Let $$M$$ be a Riemannian manifold and let $$n$$ a positive integer. Denote by $$F_n(M) \subset M^n$$ the space of all $$n$$-tuples of pairwise distinct points from $$M$$. The isometries of $$M$$ act co-ordinate wise on $$M^n$$, and this action restricts to an action on $$F_n(M)$$. What is known about the quotient of $$F_n(M)$$ by this action?

I am particularly interested in the case where $$M$$ is the hyperbolic plane and $$n > 3$$, but any starting points or references would be a great help.

• From a physicist's point of view, it should give you information on the symmetries of your configuration of points. – Sylvain JULIEN Dec 23 '18 at 19:27

Let us consider the case when $$M$$ is a hyperbolic plane, $$M=\mathbb H^2$$ and restrict to orientation preserving isometries of $$\mathbb H^2$$. Let's identify $$\mathbb H^2$$ with the open radius $$1$$ disk $$D\subset\mathbb C$$, centred at $$0$$. Then we have
$$D^n/PSL(2,\mathbb R)\cong (D)^{n-1}/S^1,\,\,\,\bf *$$ where $$S^1$$ acts on $$D^{n-1}$$ diagonally by rotating each $$D$$-factor around its centre.
To answer the original question, note that the quotient of $$\mathbb C^{n-1}$$ by the standard diagonal $$S^1$$-action is a cone over $$\mathbb CP^{n-2}$$. So, the quotient space $$(D)^{n-1}/S^1$$ is an open star-shape subset of the cone over $$\mathbb CP^{n-2}$$.
To see, that $$\bf *$$ holds, recall that one can choose in $$PSL(2,\mathbb R)$$ a parabolic subgroup $$P$$ acting transitively on $$\mathbb H^2$$ ($$P$$ preserves a point at infinity of $$\mathbb H^2$$). Now, we can obtain the quotient $$D^n/PSL(2,\mathbb R)$$ in two steps. First, for each $$n$$-tuple $$(x_1,\ldots, x_n)\subset D^n$$ we find a unique isometry $$p\in P$$ that sends $$x_1$$ to the centre of the disk $$D$$. This reduces the action of $$PSL(2,\mathbb R)$$ on $$D^n$$ to that of $$S^1$$ on $$D^{n-1}$$.
• Indeed, the question was rather vague. I think that quite a lot can be said about the quotient metric. Basically, we take a quotient of $(\mathbb H^2)^{n-2}$ by a diagonal $S^1$-action. If it were not $\mathbb H^2$ but $\mathbb R^2$, we would get a standard metric cone over $\mathbb CP^{n-2}$ with the standard Fubini-Studi metric. In the case of $\mathbb H^2$ one can take Kahler reduction. It should be possible to write down an explicit metric (since the Kahler metric on $\mathbb H^2$ is explicit). Though I am not sure it this what you are looking for – Dmitri Panov Dec 27 '18 at 20:33