When are root hyperplanes locally finite? For an even integral lattice $L$ (of arbitrary signature, non-degenerate but not necessarily unimodular), consider roots (vectors of square $2$) and their orthogonal hyperplanes. 
First, I'd like to know if the collection of root hyperplanes is always locally finite away from the origin. (I'm not actually sure what this means, but I have come across this condition in many papers. At the very least, it should mean that every non-zero point in $L \otimes \mathbb{R}$
 has a neighborhood meeting a finite number of the hyperplanes.) 
Second, if the above is true, I can take the complement of all of these hyperplanes to be left with a bunch of connected components and I'd like to know if choosing one of them always gives a fundamental domain for the group generated by reflections in the root hyperplanes. I know that this is the case for finite and affine root systems, and am wondering if this is a general fact.
 A: Any affine root system in a rank $n$ Lorentzian lattice gives rise to a system of hyperplanes in $\mathbb{R}^{n-1,1}$ that is not locally finite, since all of the root hyperplanes intersect the lightlike line (that is, the line of norm-zero vectors spanned by $\delta$).  Perhaps the easiest example is $\widehat{\mathfrak{sl}}_2$, where you have a singular 2-dimensional subspace of $\mathbb{R}^{2,1}$ that is is tangent to the light-cone in a distinguished lightlike line.  The real roots of the affine root system make up two chains of vertices in the singular subspace that are parallel to the lightlike line, and that line is perpendicular to everything in the singular subspace, including the real roots.  If you have a hyperbolic root system that contains an affine root system, you will get a similar failure of local finiteness.
In the affine case, you will get a fundamental domain in the hyperboloid (and therefore in its cone), but infinitely many domain walls meet at ideal points on the boundary.  I have heard that reflection groups in signature $(m,n)$ do not usually admit fundamental domains for $m,n > 1$, but I do not know of a reference or a counterexample.
