  We examine the question of which finitely generated groups act properly on a
finite product of simplicial trees, considering both arbitrary trees and where
all trees are locally finite. In the second case we present evidence in favour
of hyperbolic surface groups having such an action. However we also present
evidence that many RAAGs do not admit such an action and we give an example of
a virtually special group which does not act properly preserving factors on any
finite product of locally finite trees, even though it does so on a product of
three trees without the local finiteness condition.
