  We study random 2-dimensional complexes in the Linial - Meshulam model and
find torsion in their fundamental groups at various regimes. We find a simple
algorithmically testable criterion for a subcomplex of a random 2-complex to be
aspherical; this implies that 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 strong hyperbolicity
property of random 2-complexes.
