Geometry and Topology of Random 2-complexes

Understand

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.