We study the fundamental group of certain random 2-dimensional cubical complexes. We show a 2-dimensional generalization of a theorem of Burtin and Erdos-Spencer on the connectivity threshold for bond percolation. In the 2-dimensional setting, the natural analogue is a transition for the simple-connectivity of the space. This is in contrast to the 2-dimensional analogue of simplicial complexes, in which the natural analogue of the Erdos-Renyi theorem is the threshold for homological connectivity of the space (due to Linial–Meshulam). We also show that below the connectivity threshold, the fundamental group factors as a product of a finitely generated pieces, and that as the density parameter goes to 0, every finitely generated group appears. Based on joint work with Matt Kahle (The Ohio State University) and Érika Roldán (TU Munich).