The fundamental group of 2-dimensional random cubical complexes.

We study the fundamental group of certain random 2-dimensional cubical complexes which contain the complete 1-skeleton of the d-dimensional cube, and where every 2-dimensional square face has probability p. These are cubical analogues of Linial--Meshulam random simplicial complexes, and also simultaneously are 2-dimensional versions of bond percolation on the hypercube. Our main result is that if p is less than or equal to 1/2, then with high probability the fundamental group contains a nontrivial free group, and if p is greater than 1/2 then with high probability it is trivial. As a corollary, we get the same result for homology with any coefficient ring. We also study the structure of the fundamental group below the transition point, especially its free factorization.