Abstract. We classify Hantzsche-Wendt manifolds in dimension $7$, proving that there are exactly $62$ diffeomorphism classes. We consider three different notions of isospectrality, showing that all three are equivalent to each other. We also show that a finite part of the spectrum determines the full spectrum. We list the isospectral classes in dimension $7$ and give a formula to compute the multiplicities of the eigenvalues.
AMSclassification. Primary 58G25; Secondary 20H15
Keywords. Bieberbach groups, isospectral Hantzsche-Wendt manifolds