Abstract: Abstract polytopes and maniplexes are combinatorial objects that generalize the face lattice of convex polytopes. Informally n-polytopes (resp. n-maniplexes) are constructed by glueing together (n-1)-polytopes (resp. (n-1)-maniplexes), which are called its facets. If all the facets of an n-polytope P are isomorphic to an (n-1)-polytope K we say that P is an extension of K. We say that it is a Cayley extension if in addition there is a group of automorphisms of P acting regularly on its facets. We will see how to construct Cayley extensions of a polytope (or maniplex) 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.
