Zero mass as a Borel structure

The Lorentz group Lor$_{1,3}=$SO$_0(1,3)$ has two point fixgroups, namely SO$(3)$ for time-like translations and SO$_0(1,1)\times R^2$ for light-like translations. However, for light-like translations it is reasonable to consider a line fixgroup that leads to the Borel structure of the Lorentz group and gives appropriate helicities for massless particles. Therefore, whether a particle is massless or massive is not so much a physical question but rather a question of the underlying Lie group symmetry.