Conditions for finiteness of a reflection group Given a subgroup $G$ of $O(V)$ that is generated by reflections ($V$ a Euclidean space), under what conditions can we determine that $G$ is finite?
I realize that if the angles between any two roots of the generating reflections is an irrational multiple of $\pi$ then these reflections will generate an infinite group. Is it enough then to require that the angles between all pairs of roots of the generating reflections be rational multiples of $\pi$ for the group to be finite? Or is more still required?
 A: Expanding my comment to an answer:
Consider the case $n=3$. It is a classical result that for three numbers $\alpha, \beta, \gamma\in (0,\pi)$ there exists a  spherical triangle with these angles if and only if the following conditions are met:

*

*$\alpha+\beta+\gamma> \pi$.


*The set of complementary angles $$\alpha'=\pi-\alpha, \beta'=\pi-\beta, \gamma'=\pi-\gamma$$ satisfies the three triangle inequalities.
You can find a proof, for instance, in M.Berger's book "Geometry" (chapter on the spherical geometry).
For instance, if all angles are equal, then the only condition is that $\pi/3<\alpha<\pi$.
Now, take a triangle like that with angles that are rational multiples of $\pi$ and form the group generated by reflections in its sides. Then each pair of generating reflections generates a finite subgroup. However, only few of these reflection groups will be finite. For instance, unless two of the angles are $\pi/2$, there are only finitely many triples of angles (they correspond to symmetry groups of Platonic solids).
To be very concrete, you can take an equilateral triangle with the angles equal to $2\pi/5$ to obtain a counter-example.
On the other hand, there is a theorem (probably due to Coxeter) close to your question:
Theorem. Suppose that $D$ is a convex spherical polyhedron (in $S^n$) whose dihedral angles are all of the form $\pi/k_i$, $i=1,...,m$, where each $k_i$ is a natural number. Then the group $G< O(n+1)$ generated by reflections in the faces of $D$ is finite and, moreover, $D$ is a fundamental domain for the action of $G$ on the sphere. (The latter means that $g(int D)\cap int(D)\ne \emptyset \iff g=1$ and $\cup_{g\in G} gD=S^n$.)
You should be able to find a proof in any textbook on Coxeter groups.  Note that if $D$ contains no antipodal points, i.e. is strictly convex, if it satisfies the assumptions of this theorem, then it has to be a spherical simplex.
Remark. There is a related question that I donot know to to answer:
Suppose that $\Gamma$ is a finitely-generated subgroup of $O(n)$, which is generated by reflections, such that  product of every pair of reflections in $\Gamma$ has finite order. Does it follow that $\Gamma$ is finite?
The answer is positive (the proof a bit tricky) in the case $n=3$, but that's very special.
