Consider the solvable group \Gamma_n := \Lambda_n\rtimes {\Bbb Z}, with {\Bbb Z} acting on \Lambda_n as φ.
Inoue surfaces of type S^- are defined in the same way as for S+, but two eigenvalues a, b of φ acting on {\Bbb Z}^2 have opposite sign and satisfy ab = &minus;1.