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In this talk I will discuss recent work by Hoffman, Paquette, and myself, where we find a sharp threshold for the fundamental group of random 2-complexes to have Kazhdan’s property (T). Our result de- pends on a spectral gap criterion due to Zuk, but we also require new concentration results for spectral gaps of Erdos-Renyi random graphs. I will also put this work in the context of other work on random fundamen- tal groups (hyperbolicity, cohomological dimension, etc.), and compare what is know for this model with what is known for Gromov’s density model and Zuk’s triangular model.