Babson, Hoffman, and Kahle have written a paper on fundamental groups of random 2-complexes

Babson, Hoffman, and Kahle have written a paper on fundamental groups of random 2-complexes. They worked with the Linial-Meshulam model whereby you begin with a complete graph on $n$ vertices and then add independently uniformly random 2-simplices.

Babson has just written a paper on the fundamental groups of clique complexes of Erdős–Rényi random graphs using similar techniques.