Abstract
We study random 2-dimensional complexes in the Linial-Meshulam model and prove that the fundamental group of a random 2-complex Y has cohomological dimension ≤ 2 if the probability parameter satisfies p≪ n−3/5. Besides, for \({n^{ - 3/5}} \ll p \ll {n^{ - 1/2 - \epsilon }}\) the fundamental group π1(Y) has elements of order two and is of infinite cohomological dimension. We also prove that for \(p \ll {n^{ - 1/2 - \epsilon }}\) the fundamental group of a random 2-complex has no m-torsion, for any given odd prime m≥ 3. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be aspherical; this implies that (for \(p \ll {n^{ - 1/2 - \epsilon }}\)) any aspherical subcomplex of a random 2-complex satisfies the Whitehead conjecture. We use inequalities for Cheeger constants and systoles of simplicial surfaces to analyse spheres and projective planes lying in random 2-complexes. Our proofs exploit the uniform hyperbolicity property of random 2-complexes (Theorem 3.4).