  Using the cut-and-paste procedure, we construct static and dynamic plane
symmetric wormholes by surgically grafting together two spacetimes of plane
symmetric vacuum solutions with a negative cosmological constant. These plane
symmetric wormholes may be viewed as domain walls connecting different
universes, having planar topology, and upon compactification of one or two
coordinates, cylindrical topology or toroidal topology, respectively. A
stability analysis is carried out for the dynamic case by taking into account
specific equations of state, and a linearized stability analysis around static
solutions is also explored. It is found that thin shell wormholes made of a
dark energy fluid or of a cosmological constant fluid are stable, while thin
shell wormholes made of phantom energy are unstable.
