Non-proper Actions of the Fundamental Group of a Punctured Torus

Given an affine isometry of ${\mathbb{R}}^3$ 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 ${\mathbb{R}}^3$, 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.