So in lectures we derived the dispersion relation for the 1D monatomic chain as \omega ~ |sin(1/2 k a)|. That could be defined as having a Brillouin zone as either between -\pi / a and \pi / a or 0 and 2\pi / a, and it is exactly equivalent in either, as it should be.
But what about the phase velocity \omega / k? If we use the first definition of the Brillouin zone we'll have a large (negative) value of the velocity just below zero, as it's roughly a sinc function, but if we use the second definition in the equivalent place (just below 2\pi / a) we'll have a tiny phase velocity. The first one seems more realistic, but shouldn't these be reproducing exactly the same physics?