#### Volume 9, issue 4 (2009)

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A smallest irreducible lattice in the product of trees

### David Janzen and Daniel T Wise

Algebraic & Geometric Topology 9 (2009) 2191–2201
##### Abstract

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

##### Keywords
irreducible lattice, CAT(0) cube complex
Primary: 20F67