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Suppose $G$ is a subgroup of $O_n(\mathbb{R})$ generated by finitely many reflections, where a reflection is defined to be a linear transformation fixing a hyperplane and sending a normal vector to the hyperplane to its negative. Suppose the normal vectors corresponding to these hyperplanes are linearly independent. Is G necessarily isomorphic to a Coxeter group?

This problem arose upon considering the fact that every Coxeter group is isomorphic to its Tits reflection representation, and I was wondering if there was a converse statement. Looking at the literature, a common statement is that this is true for finite reflection groups, but I could not find a statement for finitely generated reflection groups.

I might be missing some easy counterexample here as it seems like it would be a standard result if it were true.

Thanks!

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    It depends on what you mean by a Coxeter group. Take two lines in the Euclidean plane passing through the origin whose intersection angle is an irrational multiple of $2\pi$. The group generated by reflection in these two lines is an infinite dihedral group, and the restriction of its action to the unit circle has dense orbits in the circle. This is not a Coxeter group in the classical sense of a discrete reflection group. However it is an abstract Coxeter group. Which of these two senses are you interested in?2012-05-22

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