Elías Mochán: Cayley extensions of polytopes: symmetric puzzles in all dimensions.

Cayley extensions of polytopes: symmetric puzzles in all dimensions.

Elías Mochán

Abstract: Abstract polytopes are combinatorial objects that generalize the face lattice of convex polytopes. Informally, $n$-polytopes are constructed by glueing together $(n-1)$-polytopes, which are called its facets. If all the facets of an n-polytope $\mathcal{P}$ are isomorphic to an $(n-1)$-polytope $\mathcal{K}$ we say that $\mathcal{P}$ is an extension of $\mathcal{K}$ We say that it is a Cayley extension if in addition there is a group of automorphisms of $\mathcal{P}$ acting regularly on its facets. We will see how to construct Cayley extensions of a polytope using a group and a set of generators in a similar way to how we construct the Cayley graph of a group. If time allows it, we will also talk about universal extensions of polytopes and how to know their symmetry types and when two universal extensions are isomorphic.