  We will show that if a proper complete CAT(0) space X has a visual boundary
homeomorphic to the join of two Cantor sets, and X admits a geometric group
action by a group containing a subgroup isomorphic to Z^2, then its Tits
boundary is the spherical join of two uncountable discrete sets. If X is
geodesically complete, then X is a product, and the group has a finite index
subgroup isomorphic to a lattice in the product of two isometry groups of
bounded valence bushy trees.
