The Fundamental Group of 2-Dimensional Random Cubical Complexes

Explore the fundamental group of 2-dimensional random cubical complexes in this 27-minute lecture by Érika Roldán Roa. Delve into the study of cubical complexes containing the complete 1-skeleton of the d-dimensional cube, where 2-dimensional square faces are added independently with probability p. Examine these structures as cubical analogues of Linial–Meshulam random simplicial complexes and 2-dimensional versions of bond percolation on the hypercube. Discover the main result showing that for p ≤ 1/2, the fundamental group of a random cubical complex is likely nontrivial, while for p < 1/2, it is likely trivial. Learn about the implications for homology with any coefficient ring and investigate the structure of the fundamental group below the transition point, focusing on its free factorization.

Érika Roldán Roa (7/13/20): The fundamental group of 2-dimensional random cubical complexes