Abstract
An affine isometry of ℝ3, with hyperbolic linear part, admits a measure of signed Lorentzian displacement along an invariant line, called its Margulis invariant. In order for a group generated by hyperbolic isometries to act properly on ℝ3, the sign of the Margulis invariant must be constant over the group. One might ask whether positivity of the Margulis invariant over some finite generating set implies that the group acts properly on ℝ3. We show that in general no so such set exists, in the case when the hyperbolic structure determined by the linear holonomy representation in SO(2, 1)0 corresponds to a punctured torus. 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.
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