We produce a nonpositively curved square complex containing exactly four squares. Its universal cover is isomorphic to the product of two –valent trees. The group is a lattice in but is not virtually a nontrivial product of free groups. There is no such example with fewer than four squares. The main ingredient in our analysis is that contains an “anti-torus” which is a certain aperiodically tiled plane.
"A smallest irreducible lattice in the product of trees." Algebr. Geom. Topol. 9 (4) 2191 - 2201, 2009. https://doi.org/10.2140/agt.2009.9.2191