  A simple surface amalgam is the union of a finite collection of surfaces with
precisely one boundary component each and which have their boundary curves
identified. We prove if two fundamental groups of simple surface amalgams act
properly and cocompactly by isometries on the same proper geodesic metric
space, then the groups are commensurable. Consequently, there are infinitely
many fundamental groups of simple surface amalgams that are quasi-isometric,
but which do not act properly and cocompactly on the same proper geodesic
metric space.
