Graphs, triangulations of manifolds, and expanders

I will explain a precise way to associate a closed manifold to a graph and two a 2-dimensional complex. As a consequence I will explain, for $n\ge 3$, how to build sequences of triangulations of the $n$-sphere which form an expander. It is classical that such sequences do not exist in dimension 2. This is joint work with Karim Adiprasito, Marc Lackenby, and Geva Yashfe.