Geometry and topology of random 2-complexes

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 (Formula Presented.) the fundamental group π₁(Y) has elements of order two and is of infinite cohomological dimension. We also prove that for (Formula Presented.) 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 (Formula Presented.)) 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).