  For each $n \geq 3$ the authors provide an $n$-dimensional rigid compact
complex manifold of Kodaira dimension $1$. First they construct a series of
singular quotients of products of $(n-1)$ Fermat curves with the Klein quartic,
which are rigid. Then using toric geometry a suitable resolution of
singularities is constructed and the deformation theories of the singular model
and of the resolutions are compared, showing the rigidity of the resolutions.
