A smallest irreducible lattice in the product of trees

Abstract

We produce a nonpositively curved square complex $X$ containing exactly four squares. Its universal cover $\tilde{X}\cong T_4\times T_4$ is isomorphic to the product of two $4$–valent trees. The group ${\pi }_{1}X$ is a lattice in $Aut\left(\tilde{X}\right)$ but ${\pi }_{1}X$ 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 $\tilde{X}$ contains an “anti-torus” which is a certain aperiodically tiled plane.