  We study the Neumann Laplacian operator $-\Delta_\Omega^N$ restricted to a
twisted waveguide $\Omega$. The goal is to find the effective operator when the
diameter of $\Omega$ tends to zero. However, when $\Omega$ is "squeezed" there
are divergent eigenvalues due to the transverse oscillations. We show that each
one of these eigenvalues influences the action of the effective operator in a
different way. In the case where $\Omega$ is periodic and sufficiently thin, we
find information about the absolutely continuous spectrum of $-\Delta_\Omega^N$
and the existence and location of band gaps in its structure.
