Let $n$ be a large integer.

I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order.

Or equivalently, I need a cocompact properly discontinuous isometric action $\Gamma\curvearrowright\mathbb{H}^n$ such that $\mathbb{H}^n/\Gamma$ is simply connected, here $\mathbb{H}^n/\Gamma$ stays for quotient space (NOT for orbifold).

**Comments.**

- I am aware that for large $n$ these is no cocompact action generated by reflections (Vinberg, 1984).
- Consider the group of matrices with integer coefficients from $\mathbb{Q}[\sqrt{5}]$ which preserve the quadratic form $$\tfrac{1+\sqrt{5}}{2}\cdot x_0^2-x_1^2-\cdots-x_n^2.$$
This gives cocompact properly discontinuous isometric action, say $\Gamma\curvearrowright\mathbb{H}^n$. (For $n=2$, the group $\Gamma$ contains the Coxeter group of regular right-angled pentagon.)

*I would be very happy if the subgroup generated by the elements of finite order in $\Gamma$ would have finite index. Or, equivalently, if*$$|\pi_1(\mathbb{H}^n/\Gamma)|<\infty.$$

**P.S.**
It seems that examples of such actions are not known. (In addition to Agol's comment, I've got a letter from Vinberg stating this.)