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.
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 $D^{n-1}$.
