Strong marked isospectrality of affine Lorentzian groups

The Margulis invariant is a function defined on a group of Lorentzian transformations $G$ acting on Minkowski space $\R^{2,1}$, that contains no elliptic elements. The spectrum of $G$ is the sequence of values of the Margulis invariant for all its elements. If the underlying linear group of $G$ is fixed, Drumm and Goldman proved that the spectrum defines the translational part completely. In this note, we strengthen this result by showing that isospectrality holds for any free product of cyclic groups of given rank, up to conjugation in the group of affine transformations of $R^{2,1}$, as long as it is non-radiant and that its linear part is discrete and non-elementary. In particular, isospectrality holds when the linear part is a Schottky group.