  Given an affine isometry of $\R^3$ with hyperbolic linear part, its Margulis
invariant measures signed Lorentzian displacement along an invariant spacelike
line. In order for a group generated by hyperbolic isometries to act properly
on $\R^3$, the sign of the Margulis invariant must be constant over the group.
We show that, in the case when the linear part is the fundamental group of a
punctured torus, positivity of the Margulis invariant over any finite
generating set does not imply that the group acts properly. This contrasts with
the case of a pair of pants, where it suffices to check the sign of the
Margulis invariant for a certain triple of generators.
